JP4243633B2 - Discrete pattern generation method, program, recording medium, and discrete pattern generation system - Google Patents

Description

  The present invention relates to a discrete pattern, and more particularly, to a method for generating a discrete pattern including a pattern of dots having low discrepancy and arranged so as not to overlap each other, and to generate the discrete pattern And a computer-readable recording medium on which a computer-readable program for generating the discrete pattern is recorded, and a discrete pattern generation system.

  In addition, the present invention provides a method for generating a discrete pattern that enables automatic generation and deletion of dots so that good characteristics can be given even to a discrete pattern including a portion in which the dot filling rate changes rapidly. The present invention also relates to a program for generating the discrete pattern, a computer-readable recording medium on which a computer-readable program for generating the discrete pattern is recorded, and a discrete pattern generation system.

  Furthermore, the present invention provides a method for generating a discrete pattern capable of providing a dithering pattern for binarization of a grayscale image, a program for generating the discrete pattern, and a computer for generating the discrete pattern The present invention relates to a computer-readable recording medium on which a readable program is recorded and a discrete pattern generation system.

  Various techniques using discrete patterns have been known so far. As a technique using a discrete pattern, for example, a light guide plate of a transmissive liquid crystal display device, a scattering plate, a dithering pattern, a photomask pattern for lithography, a pattern for anti-slip, etc. can be mentioned in recent years. The application of discrete patterns to the arrangement of DNA in a DNA chip has also been studied.

  Conventionally, the discrete pattern described above is formed by randomly arranging dots using a so-called random number generator or by arranging dots on a regular orthogonal grid such as graph paper. However, according to the conventional method, the random pattern generated simply by using a normal random number generator has the following disadvantages. In other words, even if dots are randomly arranged, the dots have a finite size, resulting in unevenness in the overlap between dots and the dot density. This impairs the appearance of the dot pattern, causing unevenness in brightness and density. This would cause an optical inconvenience such as unevenness. In the case of regular arrangement, optical inconvenience such as moire occurs due to interference between dots or an external regular pattern.

  In view of this, Japanese Patent Laid-Open No. 10-153779 proposes a method for generating an irregular pattern without overlapping between dots or without excessive proximity between dots. First, as an “absolute random number arrangement method”, (1) the initial position (x, y) is given to all dots by a random number generator, and (2) random numbers are generated again for overlapping dots to correct the position. A procedure has been proposed. However, it is known that the calculation method disclosed in Japanese Patent Laid-Open No. 10-153779 has a disadvantage that the calculation for eliminating the overlap between dots does not converge in a region with a high filling rate exceeding 50%. In other words, the technique described in Japanese Patent Laid-Open No. 10-153779 has a disadvantage that it is substantially difficult to generate an irregular pattern that does not have an abnormal approach between dots while maintaining randomness. In addition, such a method based on multiple generations of pseudo-random numbers has a disadvantage that it is difficult to remove the unevenness of the dot pattern itself even if the overlap between dots can be removed in the low filling rate region.

  FIG. 1 shows an example of a dot pattern formed by the technique described in JP-A-10-153779. In the dot pattern shown in FIG. 1, (1) a dot is placed on a two-dimensional regular lattice point formed from a straight line or a curve, and this is used as an initial position. (3) The overlapping dots are generated by the process of generating random numbers again and correcting the positions. As shown in FIG. 1, according to this method, dots can be arranged without overlapping between dots as long as the displacement from the lattice points is kept small. However, in the method of generating random positions as perturbations from lattice points, a pattern having sufficient irregularity is generated while avoiding the generation of moire patterns in a region where the filling rate is relatively high, for example, exceeding 50%. Is difficult. In addition, with such a method based on multiple generations of pseudo-random numbers, even if the overlap between dots can be removed, agglomeration between the dots is often seen, and it is difficult to generate a highly uniform random pattern. .

  The reason for this will be discussed below. In the above-described known example using random dots, the case where the dot size is set to about 100 μm and the filling rate is set to 70% will be considered. Here, it is assumed that the dot shape is a perfect square as shown in FIG. In the case of the above-described dot dimensions, the above-described filling rate produces only a 20 μm spacing between the dots. FIG. 1 shows the dots 100 and the spacing 102 between the dots at an accurate scale. Starting from this regular lattice, when the perturbation is given randomly, the dot 104 shown by the broken line in FIG. 1 is obtained. It can be seen that these dots 104 can only generate patterns having extremely limited irregularities (hereinafter referred to as randomness in the present invention) as shown in FIG. This is because the dots adjacent to each other do not jump over each other, and the correction of the position can be performed only within a limited range because the filling rate is high.

  In the conventional example shown in FIG. 1, a square lattice is used. However, under the condition that the dots are not too close to each other, other types of regular lattices can be used more or less. Randomness will be limited. In other words, the method of giving a predetermined dot as an initial position and generating a random arrangement as a perturbation to this has the disadvantage that, in principle, a dot pattern that is nearly truly random cannot be given when the filling rate increases. This is not sufficient from the viewpoint of generating a random pattern regardless of the filling rate.

  Further, considering the optical characteristics of the dot pattern generated by the above-described method, there also arises a problem of generation of a moire pattern when light rays are transmitted or reflected through the dot pattern. Conventionally, in the field of printing technology, many studies and proposals have been made on methods for eliminating moire patterns. For example, in Japanese Patent Laid-Open No. 2000-94756, moire (so-called feed unevenness / streaks unevenness) generated by regular printing fluctuations due to rotation of a paper feed drum and a printing dot pattern in halftone processing of a printer. A technique for avoiding this problem has been proposed.

  For this reason, in Japanese Patent Laid-Open No. 2000-94756, randomness is introduced in the dot arrangement. In other words, by randomly perturbing the printing halftone dots arranged on the regular lattice, the above-described unevenness generated by moire or the like is improved satisfactorily. However, for the reasons described above, it is difficult to directly apply the above-described method to a uniform discrete pattern having randomness as it is.

  In addition to printing technologies such as printers, the above-described problems such as generation of random dot patterns and reduction of moire patterns have caused problems in various fields, for example, backlighting devices (hereinafter referred to as backlights in the present invention). It also occurs in the display device that contains it.

  For example, as a representative example, transmissive liquid crystal display devices can be made small and light, and have low power consumption. Therefore, these technologies are becoming increasingly important as a core technology in the so-called IT revolution hardware. Yes. However, unlike conventional displays, ie CRTs, the liquid crystal itself does not emit light, so it is necessary to illuminate the entire liquid crystal cell using a backlight unit. In particular, recently, there is an increasing demand for color display and high definition for liquid crystal display devices, and accordingly, the backlight unit is also strongly required to have a characteristic capable of uniformly illuminating a wide area uniformly.

  FIG. 2 shows a transmissive liquid crystal display as a typical display device including a backlight unit. Taking the transmissive liquid crystal display device shown in FIG. 2 as an example, countermeasures against the above-described random dot pattern and moire will be described below. As shown in FIG. 2, the conventional transmissive liquid crystal display device includes a backlight unit. The backlight unit includes a light guide plate 108 on which dot patterns 106 are randomly formed, A fluorescent tube CFL disposed adjacent to the light guide plate 108, a reflector 112 for covering the fluorescent tube CFL and allowing light emitted from the fluorescent tube CFL to efficiently enter the light guide plate 108, and the light guide plate A reflection sheet 114 for efficiently reflecting, for example, a light beam scattered by 108 to a liquid crystal panel (not shown) is configured.

  The dot pattern 106 formed on the light guide plate 108 is formed to be as random as possible, and the problem of moire pattern is improved. As shown in FIG. 2, the backlight unit is provided with a diffusion sheet 116, a prism sheet 118a, and 118b, and regulates the light rays irradiated to the liquid crystal panel.

  As the backlight unit described above, a side lighting system (side light system) is widely adopted as shown in FIG. In the backlight unit shown in FIG. 2, the light emitted from the cold cathode ray tube (CFL) such as a fluorescent tube is a dot pattern formed on the bottom surface of the light guide plate 108 formed of acrylic resin or the like, or the light guide plate. The light is scattered by the reflection sheet 114 disposed below the light 108, passes through the diffusion sheet 116 disposed on the upper surface of the light guide plate, and further passes through the prism sheets 118 a and 118 b to the liquid crystal panel (not shown) and is observed by the user. Will be. That is, it can be said that the backlight unit shown in FIG. 2 is a device for converting a linear light source into a planar light source.

  As long as the so-called sidelight system is adopted as shown in FIG. 2, a mechanism for scattering the light from the light source and reflecting it to the liquid crystal panel is indispensable. This mechanism improves the brightness of the backlight unit. It will be an important elemental technology to achieve For this reason, various investigations on the manufacturing method have been conventionally made on the bottom surface of the light guide plate 108 or the reflection sheet 114. For example, in Japanese Patent Laid-Open No. 8-085001, a study is performed to obtain a scattering effect by intentionally forming a scattering surface by machining a bottom surface of a light guide plate using a cutting tool having a negative rake angle. However, in the above-described method, it is difficult to quantitatively control the luminance uniformity of the backlight unit, and the scattered light is wasted due to the bias in the light scattering direction. There is a disadvantage that it is not suitable for obtaining a high quality backlight unit.

  Furthermore, in Japanese Patent Laid-Open No. 7-294745, a method of forming a groove having a concave cross-sectional shape on the bottom surface of the light guide plate to scatter light rays on the top surface of the light guide plate, and in Japanese Patent Laid-Open No. 6-242320, titanium dioxide is disclosed. For example, a method of forming a pattern coated with a particulate pigment on the bottom surface of a light guide plate has been proposed. The conventional method described above has a common feature in that a scatterer having a certain geometric periodicity, that is, a dot pattern is formed on the light guide plate. However, since liquid crystal display devices normally include elements including fine periodic patterns such as color filters and prism sheets, if the arrangement of dot structures is periodic, the dot structure and the light beam are optical. Resulting in moire patterns. Such a moire pattern significantly reduces the value as a light source, and therefore it is desirable to prevent it from occurring as completely or completely as possible.

  As described above, the moiré pattern generated by the dot pattern formed on the light guide plate 108 in the display device such as the transmissive liquid crystal display device using the backlight unit is also various element technologies as described above. Its reduction is being investigated using

  For example, Japanese Patent Application Laid-Open No. 9-269489 discloses a method in which a large number of minute scatterers called micro dots are randomly arranged on the bottom surface of a light guide plate to cause light scattering, and further, Japanese Patent Application Laid-Open No. 2000-171797. The publication discusses the improvement. In Japanese Patent Laid-Open No. 11-250713, an attempt is made to randomly arrange dots on the upper surface of the light guide plate for application to a reflective liquid crystal display device. FIG. 3 shows a conventional example in which a randomly generated dot pattern is applied to a reflective liquid crystal display device.

  As shown in FIG. 3, conventionally, a light guide plate including a dot pattern randomly formed by using a pseudo random number by a conventional method is used to constitute a backlight unit. In the backlight unit shown in FIG. 3, the fluorescent tube CFL and the reflector 124 are arranged adjacent to the light guide plate 122 on which a plurality of pseudo-random dots 120 are formed. The light guide plate 120, the fluorescent tube CFL, and the reflector 124 shown in FIG. 4 are held by a frame 126 and constitute a backlight unit that reflects light rays in the direction of the arrow A. As described with reference to FIG. 2, the backlight unit shown in FIG. 3 includes a reflection sheet, a diffusion sheet, and a prism sheet, which are omitted in FIG.

  The light guide plate 122 used in the conventional backlight unit shown in FIG. 3 has an optical requirement that the scattering intensity of the light guide plate 122 is as uniform as possible throughout the light guide plate 122. In some cases, it is required to continuously change the dot filling rate distribution in the central region and the four corner regions of the light guide plate 122. For this reason, the establishment of a simple method for generating the initial position so as to satisfy a given continuous filling rate distribution is also being studied. For example, conventionally, formation of a pattern in which the filling rate is continuously changed by connecting regions having different lattice intervals has been attempted. However, with this method, defects such as joints are often observed at the boundary where the filling rate changes.

  It is considered that such a joint can be reduced by generating a two-dimensional lattice whose shape continuously changes so as to match a given filling rate distribution over the entire surface on which dots are formed. However, except for the case where the distribution is simple and given by a simple analytic function, the above-described lattice generation itself requires a high-level and large-scale calculation. That is, the conventional method of considering the perturbation from the lattice point is not satisfactory in terms of not only the irregularity but also the correspondence to the filling rate distribution.

  Further, for the above-described backlight unit, it has been proposed to change the structure of the backlight unit itself in order to improve the brightness or to improve the angle dependency on the light beam. For example, Koji Okita and Minoru Katsuda, '99 latest liquid crystal process technology, press journal, published on September 10, 1998, page 441, proposed a backlight unit in which a prism is directly formed on the upper surface of a light guide plate. Yes. In addition to the above, it has also been proposed to omit optical sheets such as diffusion sheets and prism sheets.

  However, all of the above-described studies require that the scattering mechanism of the light guide plate be accurately controlled, which increases the possibility of occurrence of moire patterns and interference fringes. It becomes necessary to take all the measures more strictly. Furthermore, the above-described conventional pseudo-random dot pattern is insufficient in terms of the correspondence to the above-described high filling rate distribution, and the uniformity of the dot pattern itself is insufficient. Further, depending on the structure of the backlight unit, it is necessary to provide a discrete pattern including a higher quality dot pattern, such as that some kinds of interference fringes are unavoidable. For this purpose, in addition to the randomness of the discrete pattern, it has been desired to provide a method for generating a random discrete pattern that satisfies a stricter condition after newly introducing an index for uniformity.

  On the other hand, in recent years, the use of the LDS method for the problem of sampling sample points irregularly and uniformly from a predetermined region in a multidimensional space is particularly important in mathematical fields such as numerical integration. It has been studied. For example, in the information processing by Shoichi Ninomiya et al., Vol. 39, p. 794, 1998, when calculating the price of financial derivatives by approximation of multiple numerical integration similar to the Monte Carlo method, instead of pseudo-random numbers It has been suggested that deterministic LDS such as a FAURE sequence and a SOBOL sequence can perform highly accurate calculations by using irregularly and uniformly distributed samples in a multidimensional space.

  Also, as described in US Pat. No. 5,872,725 and JP-A-11-259542, these are defined as discrepancies that mean mathematically biased or non-uniformity. It is used because it is a number sequence that has an upper limit on the amount to be performed and that dramatically speeds up the convergence of multiple integral calculations such as the Monte Carlo method. It has also been reported that the numerical integration using the LDS method described above is used to improve the rendering speed in the ray tracing method.

  That is, as described above, in the pseudo-random dot pattern formed by the dot pattern that directly depends only on the randomness, while improving the above-described disadvantages, a good light guide plate, a backlight that uses the light guide plate, Insufficient optical members such as providing a unit, a transmissive liquid crystal display device including the backlight unit, and therefore, a new method for generating an initial distribution by controlling discrepancy is required It was said. In the present invention, the term “discrepancy” is described in, for example, Tezuka, “About Discrepancy of Point Sequences”, “Discrete Structure and Algorithm IV”, Kazuo Murota, Modern Sciences, Chapter 3. As used herein, it means an indicator of the uniformity of the distribution of discrete dots.

  In addition, it is a technique for expressing an image including continuous gradation, such as a silver halide photographic film, that is, a continuous gradation image on a medium capable of only binary expression as in printing or copying. In halftoning, it is necessary to handle a discrete dot pattern. In the fields of printing and plate making, halftoning is also called shading. The above-described technique can be said to be a binarization technique that more generally expresses the density of an image with the density of minute dots.

  In recent years, due to the progress of digital signal processing technology, printer and facsimile technology, industrial requirements for such a binarization method are very large, and various methods have been proposed. Although the expression varies depending on the literature, a method of expressing the density with the density of minute dots having the same size is hereinafter referred to as an FM (frequency modulated) screen method. The FM screen method is roughly classified into an error diffusion method and a mask method when roughly divided from the viewpoint of a calculation method.

  The error diffusion method converts an input image into a binary or multi-valued image by comparing it with a threshold value for each pixel, and the error generated between the output value and the input value at that time The image density is stored by weighting and diffusing the pixel group. In this case, the pixels of the input image and the output image have a one-to-one correspondence. The output image obtained by this method has a relatively high image quality and good resolution, and is widely used. However, a slightly more complicated calculation is required for each pixel as compared with the mask method in which binarization is advanced in comparison with a given threshold value. Further, since the positions where dots are shot differ depending on the input image, it is difficult to predict the degree of color mixing, and color reproducibility is considered to be poor. In addition, the appearance of a transient region in the boundary region to which the error diffusion method is applied, and the fact that a virtual image called a worm or serpentine raster is often observed are considered to be unsolved problems (Tatsumi Tanizaki, Multimedia and digital signal processing, Corona, 1997, 295 pages). In addition, the progress of electronic hardware has been remarkable in recent years, and the amount of calculation due to error diffusion is not necessarily a fatal problem, but the existence of problems in image quality is recognized as a problem that requires improvement. Yes.

In order to alleviate the image quality problem of the error diffusion method, the distribution of error allocation to the surroundings is changed (there are various distributions including the Floyd-Steinberg method), and an attempt to optimize the scanning direction of diffusion Etc. (for example, T. Asano, “Digital halftoning algorithm based on random space-filling curve”, Proc. International
Conference on Image Processing, 1996, 545). However, with the technique of allocating the error of one pixel to the surrounding finite area, it is unavoidable to some extent that a problem in image quality occurs more or less fatally. This is because the error diffusion method introduces a distinction between a side that generates an error and a side that is responsible for the error depending on the scanning direction. This is because such distinction is only artificial as long as the dots constituting the binarized image should be equal.

  The same can be said for image binarization. Originally, the binarized image should be determined so as to correctly reproduce the original image as a whole. A technique for introducing a relationship in which only one of the pixels is considered can be essentially an approximation. As described above, the improvement of the error diffusion method has a long history, but all of them are symptomatic treatments based on the existence of an error diffusion algorithm. The best binarization algorithm is probably based on the mutual relationship of all dots, but no practical algorithm that can reproduce gradation gradients has been known so far.

In the following, Ulrichney's standard regarding the image quality of the binarized image and the conventional technique of the mask method will be described. Ulichney lists the following two conditions for visually preferred FM screen patterns (RA Ulichney, Proceedings of the IEEE, 76 (1988) 56.).
1. The radial Fourier component of the dot distribution has a "blue noise"characteristic;
2. The dot distribution is isotropic.

These standards are now widely accepted as standard indicators for image quality evaluation.
Ulichney's discovery has been proposed in the direction of improving the error diffusion algorithm, but it was later applied to the mask method to binarize to compensate for some of the disadvantages of the error diffusion method. It is being considered to construct a method. In other words, it was considered to produce a mask that gives a threshold value for binarization so as to satisfy the above-mentioned quantitative criteria shown by Ulichney. This is a technique often called “blue noise mask method”. A typical paper is Mitsa-Parker (T. Mitsa and KJ Parker, J. Opt. Soc. Am. A, 9 (1992) 1920). Similarly, Patent Publication No. 2622429 and US Pat. No. 5,111,310 can be cited as a review of conventional methods.

  FIG. 4 shows a conceptual diagram of processing of the mask method (organized dither method). 4A shows a read image, FIG. 4B shows a dither matrix, and FIG. 4C shows a display image. In general, the mask method is a method of binarizing a read image by comparing it with a threshold value of a given matrix. FIG. 4 shows the simplest mask method in which the original pixel and the binary image pixel have a one-to-one correspondence. There, a method is described in which the value shown in a numerical table called “dither matrix” is compared with the luminance value of the original pixel, and the black and white of the binarized image is determined based on the magnitude relationship. The aim of the blue noise mask method is that the resulting binary image (as shown in FIG. 4C) has isotropic blue noise characteristics. Therefore, it can be said that the binarization problem of the continuous tone image is basically equivalent to the dot pattern optimization problem. The blue noise characteristic ideally means a discrete pattern in which dots are distributed around a dominant wavelength related to the ratio between the area per dot and the filling rate at a predetermined gradation level.

  As another mask method, there is a method in which a dot pattern including a large number of dots (referred to as a dithering bitmap) is associated with one original pixel. For convenience, this is called a one-to-many mask method, and the method described in FIG. 4 is called a one-to-one mask method.

  As described above and as described in detail in Japanese Patent Laid-Open No. 2000-59626, almost all current binarization methods are designed based on the above-mentioned Ulichney criteria. As presented by Ulichney himself based on the error diffusion method, it is known that a binarized image that meets Ulichney's criteria is actually a smooth image without roughness or geometric patterns. However, Ulichney's standard is a limit on the amount of macro, and it has the disadvantage that it is not a direct standard for actual production. More specifically, even if the conditions for the desired power spectrum (equal to the Fourier transform of the autocorrelation function for the dot position distribution by Wiener-Khinchin's theorem) are presented, It is not easy to make a threshold mask based on the above. Moreover, even if a limit on isotropic property is given, a method for realizing it is not obvious. This inconvenience also applies to a series of studies on the “green noise mask” method of Lau et al. That inherited Ulichney's standards (for example, DL Lau, GR Arce, and NC Gallaghe, Proc. IEEE, 86 ( 1998) 2424).

  Furthermore, Ulichney's condition is unsatisfactory in that there is no effective index that directly represents dot pattern unevenness. In fact, according to Ulichney (RA Ulichney, Proceedings of the IEEE, 76 (1988) 56.)), a dot pattern based on white noise (that is, a pattern in which each dot position is extracted with a pseudo random number) is given. However, the amount of Ulichney's anisotropy is judged to be “completely isotropic (−10 dB)”.

  Furthermore, the current state of the one-to-multiple blue mask method will be described. For the purpose of simplifying the complicated calculation process of the error diffusion method, a dot pattern having a blue noise characteristic is prepared for each different luminance, US Pat. No. 4,920,501 discloses a binarization method in which the original pixel is replaced with its dot pattern in accordance with the luminance value. First, the outline of the one-to-many mask method is shown in the figure. As shown in FIG. 5, the original image OP is divided into small sections (FIG. 5A). An open-type dot pattern corresponding to the representative luminance value (denoted as J) is selected (FIG. 5B), which becomes a binarized image of the small section (FIG. 5C). It is assumed that the gradation is 256 gradations, and the 256 dot patterns are stored in the memory in advance, and Table 1 summarizes the methods disclosed in US Pat.

  In this dot pattern generation method, the initial arrangement is generated by random numbers and it is relaxed by the simulated annealing method (see Hidemori Nishimori, Spin Glass Theory and Information Statistical Mechanics, Iwanami Shoten, 1999, Section 9.4). There is. This conventional method shows that the frequency characteristics can be improved by using an optimization method for the initial arrangement with white noise characteristics. Furthermore, it is said that this algorithm can produce a blue mask that well meets Ulichney conditions. However, the initial arrangement generated by the pseudo random numbers is obviously uneven, and there is a problem that high quality binarization cannot be performed as it is.

  In Japanese Patent Laid-Open No. 10-275228, improvement of the above-described method is studied. The method disclosed in Japanese Patent Application Laid-Open No. 10-275228 is to sequentially determine other patterns from the intermediate gradation pattern as an initial pattern using a simulated annealing method so as to satisfy Mitsa-Parker conditions described later. is there. The initial pattern is not determined by “dart throwing” by random numbers, but merely points out that the optimum pattern is adopted using conventional technology. In any case, this is a method of creating a pattern by the procedure of initial position generation → optimization disclosed in JP-A-10-275228.

  However, as disclosed in Japanese Patent Laid-Open No. 10-275228, even if an intermediate gradation pattern is optimally selected in the meaning of the Ulichney standard, the multi-gradation pattern generated from it is always somewhat Is not optimal. For example, it is assumed that the dot pattern is optimized so that the interval between the nearest dots is almost concentrated on a certain value in the intermediate gradation. When one dot is added there, a field with a non-optimal dot interval always appears in the vicinity of the dot. This is because dots that have already been made in the past are considered fixed.

  In other words, these methods cause inconveniences in principle from the viewpoint of dot pattern optimization. Japanese Patent Laid-Open No. 10-275228 does not specifically consider improvement of the starting pattern optimization method.

  Furthermore, optimal binarization using the one-to-one blue mask method is not yet sufficient, and various improvements are being studied. Mitsa and Parker explain that the following conditions must be met in the production of a one-to-one blue mask (T. Mitsa and KJParker, J. Opt. Soc. Am. A, 9 (1992 1920.)

In the above equation, p (i, j, g ₂ ) represents the output value (0 or 1) of the (i, j) section in the g ₂ gradation mask pattern. This condition states that "in g ₁ gradation (i, j) if partition black, the partition is black at high g ₂ tone than". This is called the Mitsa-Parker condition for convenience. As detailed in JP 2000-59626 A, if this is satisfied,

Therefore, a threshold mask can be created (assuming 256 gradations). That is, under the Mitsa-Parker condition, it can be said that a threshold mask for a one-to-one mask can be manufactured using a threshold mask for the one-to-many mask method.

  On the premise of this, Japanese Patent Laid-Open No. 2001-298617 discloses a method of manufacturing a threshold mask having blue noise characteristics. It starts with a low-tone dot pattern (where the dots are hit using a regular dither method based on the Bayer matrix) and gives a repulsive potential to the dots in the previous (lower) gray-scale pattern, High-gradation patterns are produced sequentially by the procedure of placing new dots at the lowest point. It is said that a dot pattern having a uniform blue noise characteristic can be manufactured using this principle. This known example is substantially equivalent to that described in W. Purgathofer, R. F. Tobler, and M. Greler, Proceedings of the International Conference on Image Processing, 1032 (1994). they again,

The repulsive potential is proposed. r is the distance from the dot. This equation shows that the potential decreases rapidly when the distance exceeds the size of s. This can be interpreted as an attempt to paraphrase the Ulichney criteria microscopically. This is because Ulichney himself states that a pattern with blue noise characteristics can be created if the nearest dot spacing is moderately concentrated around the “main wavelength”. However, a method for obtaining a desirable spectral characteristic of a pattern from such a viewpoint has not been proposed so far.

  Their method and the method disclosed in Japanese Patent Laid-Open No. 2001-298617 are not sufficient as a mechanical relaxation process. As described above, it is assumed that the dot pattern is optimized so that the interval between the nearest dots is almost concentrated on the value s in a certain gradation. When one dot is added there, a field with a non-optimal dot interval always appears in the vicinity of the dot. This is because dots that have already been made in the past are considered fixed.

More recently, strong mitigation algorithms have been proposed by Hiller et al. (S.Hiller, O.Deussen, and A.Keller, "Tiled Blue Noise Samples", Proceedings of the 6 th International Fall Workshop on Vision, Modeling, and Visualization 2001, in press). After generating dot patterns with their method, pseudo-random numbers, they applied a relaxation algorithm called Lloyd method, and reported that they generated dot patterns with blue noise characteristics at high speed. The Lloyd method is a method of leveling the nearest dot interval by repeating the operation of moving dots to the center of gravity of each Voronoi polygon based on so-called Voronoi division. This is shown in FIG. FIG. 6A shows randomly generated initial positions and their Voronoi division, FIG. 6B shows a state where Lloyd relaxation is performed, and FIG. 6C shows a dot pattern after relaxation. Hiller et al. Further argue that by properly generating Voronoi shapes at the boundaries, uniform binarized images can be generated without problems such as seams between the compartments.

  This approach improves the problems inherent in the mitigation process described so far. That is, a uniform pattern is generated by relaxing the whole without dividing the dots into those that relax and those that relax. Moreover, the principle of relaxation is based on a method of directly optimizing the closest interval between dots.

  However, Hiller et al.'S method has also been found to have two disadvantages. First, in their method based on Voronoi division, it is theoretically difficult to create a dot pattern in which the gradation is changed stepwise. Second, since pseudorandom numbers are used for initial position generation, unevenness is observed in the relaxed pattern.

  As described above, there is a method that allows a pattern having a continuous filling rate distribution up to a high filling rate region exceeding 50% and generating no visually inconvenient pattern. Not proposed. In addition, imposing a hard upper limit on the filling rate is a major obstacle to controlling the brightness, transmittance, and reflectance of optical members, so even if there is a distribution of filling rate, mutual filling By generating a discrete pattern without being limited by the rate, it has been desired that the filling rate distribution can be favorably imparted without giving a moire pattern while imparting higher flexibility. In addition, the random dot pattern generation method using the pseudo-random number generation method cannot cope with a high filling rate, and in addition, the distribution of the dot distribution is large, and it requires labor and time to correct it. There was an inconvenience.

  In particular, when an arbitrary filling rate distribution is given from a low filling rate to a high filling rate, it is particularly suitable when there is a steep filling rate gradient in which the filling rate change per dot diameter exceeds 1%. It was necessary to present a dot pattern generation method. In addition, it has been necessary to produce an optical member that does not cause obstacles such as unevenness, coloring, and moire by generating dots on a light-shielding film, a photomask, or the like using this method. In addition, there is a need for a dot pattern generation method that adaptively adds and deletes dots even when the filling rate gradient is large and maintains the filling rate distribution appropriately.

  In addition, contents equivalent to the Ulichney standard described in macro language can be described using only the micro quantity that defines the dot position, and an algorithm that specifically realizes it has been required. In addition, if a dot pattern generation method that specifically satisfies the requirements can be presented after presenting a quantitative standard for unevenness that cannot be expressed sufficiently by the Ulichney standard, the practical benefit is extremely large.

  Conventionally, the size of the tile-like division is almost determined by the resolution on the output side and the number of bits necessary for gradation expression, which restricts the method for removing perceptible artifacts. Such a binarization technique has been required, although the restriction can be removed if the gradation in the tile is allowed.

  Furthermore, it has been necessary to propose an algorithm that expresses the Ulichney criterion in a form directly linked to the micro quantity that defines the dot position.

  In addition, it has been necessary to realize a high-quality binarization method that does not cause a virtual image or image quality deterioration due to tiled division of an original image by considering a continuous gradient of gradation.

  That is, there has been a need for a method for generating discrete patterns that are sufficiently randomly arranged, have low discrepancy, and do not overlap each other.

  Further, there has been a need for a discrete pattern generation method that can arrange the above-described dot patterns at random sufficiently regardless of the filling rate.

  Further, a program for generating the above-described discrete pattern and a recording medium on which the program is recorded are required.

  Furthermore, a discrete pattern generation system for generating the above-described discrete pattern has been required.

  The present invention has been made in view of the above problems, and in the first configuration of the present invention, the discrepancy is reduced in a predetermined section in order to form a discrete pattern with improved randomness. The arrangement is adopted as the initial position. For this reason, the uniform randomness of a discrete pattern is ensured. Further, in the present invention, the above-described low discrepancy dot pattern is used as the initial position, and the position is displaced as a repulsive force acts between dots having a finite size, thereby allowing the dot to be displaced. Eliminate overlap.

  In the present invention, a process for eliminating dot overlap on the assumption that repulsive force acts between dots is defined as a repulsive force relaxation method. For this reason, even when the filling rate is high, from the initial position where the discrepancy is low at the given filling rate, the overlap between dots is eliminated by the repulsive force relaxation method, and the discrepancy is kept low. A discrete pattern can be generated. In addition, since the present invention makes it possible to reduce the filling rate and discrepancy for each predetermined section, even if the filling rate changes continuously, the randomness of the dot pattern can be reduced. It is possible to provide a discrete pattern having sufficient randomness and continuously changing the filling rate without causing the disadvantage due to the change in uniformity, that is, the generation of the boundary where the filling rate changes. To do.

  In the second configuration of the present invention, it is possible to provide a good discrete pattern based on a predetermined filling rate by automatically generating and deleting dots. In the second configuration of the present invention, automatic generation of dots can be performed from generation of initial dots, or initial dots are generated by the above-described low discrepancy sequence method, and dot positions are relaxed. It is also possible to automatically generate and delete dots in relation to the filling rate after the process.

  In the third configuration of the present invention, first, an initial position of the dot pattern is generated and relaxed to obtain a binarized pattern. In particular, in the third configuration of the present invention, an isotropic property is considered in the Ulichney criterion. The isotropic nature is that, in everyday terms, there is no uneven dot pattern. For this purpose, the above-described low discrepancy sequence (LDS) can be used. This is a numerical sequence in which an upper limit is set for discrepancy as an index of unevenness, and irregularities can be introduced by appropriate mathematical operations. By incorporating a new discrepancy amount as a means of quantification, it is possible to effectively binarize a grayscale image.

  The optical member including the above-described discrete pattern provided by the present invention makes it possible to provide an optical modulation element that can uniformly provide high luminance without causing a problem such as a moire pattern.

  That is, the above-described problem of the present invention is to generate the discrete pattern of the present invention, an optical member using the discrete pattern, a light guide plate, a sidelight device, a transmissive liquid crystal display device, a method for generating the discrete pattern, and the discrete pattern. The invention is solved by providing a program for reading, a computer-readable recording medium on which a computer-readable program for generating the discrete pattern is recorded, and a discrete pattern generation system.

That is, according to the present invention, a discrete pattern formed from discrete dots arranged two-dimensionally using a computer system, wherein the light beam is scattered, transmitted by light, or absorbed. A method for generating a discrete pattern that is passively controlled by a light guide plate, a scattering plate, a dithering pattern, and a photomask pattern for lithography, the method comprising:
Providing a predetermined compartment;
Determining the number of dots to be arranged in the predetermined section;
When the dots included in a rectangular section of horizontal Lx and vertical Ly are 4000 or less in position coordinates of the dots, the following formula (1)

(In the above formula (1), N is the number of dots included in the section, and D is an arbitrary coordinate point (x, y) from the reference coordinates (0, 0) among the N dots) ), The number of dots included in the rectangular area with the line segment up to the diagonal line as A (x, y) is given by the following equation (2)):

And the following formula (3)

( ₁ ) is ₁ or less (in (3), g ₁ is an average radial distribution function g (r) of each dot in the region in the range of r _{1 to} r _2. divided by the integrated value, g _av is .r ₁ and r ₂ is the average value at r ₁ or r ₂ the following range of g ₁ are square lattice dots to meet a given fill factor The values of the lattice constant Δr are 1 and 4 respectively, and the dot filling rate is obtained by multiplying the square of the maximum dot diameter by the number of dots. Divided by the area of the area.)
Setting the determined position coordinates as an initial position;
Changing the position coordinates of the dots so that the dots for which the position coordinates are determined do not overlap each other;
Generating and storing a first generator matrix for each coordinate axis;
Generating and storing a second generator matrix by a product of the first generator matrix and a scramble matrix having a lower triangular element having a random value;
Generating a binary expansion of one natural number n in a set of N natural numbers respectively corresponding to N dots as a first vector;
Generating a second vector from a product of the first vector and the second generator matrix;
Generating the position coordinates of the dots with the elements of the second vector for each coordinate as values of each digit of a binary decimal;
generating a position coordinate until n is increased by a predetermined increment and the total number of repeatedly generated point coordinates reaches the number N of dots in the section.

  The present invention may further include a step of generating a discrete pattern having a predetermined size by adjoining the predetermined sections using a plurality of predetermined boundary conditions. The step of changing the position coordinate may include calculating a repulsive force between the adjacent dots related to the size and distance of the adjacent dots.

The step of changing the position coordinates of the present invention includes:
Calculating a repulsive force from other dots existing in the vicinity of the dot for a predetermined dot;
Displacing the predetermined dot according to the magnitude of the repulsive force;
Calculating a repulsive force with respect to the predetermined dot displaced;
Summing the repulsive forces for dots within a predetermined range.

  The step of changing the position coordinates of the present invention further includes that the difference between the total of the first repulsive force and the total of the second repulsive force calculated immediately before the total of the first repulsive force is a predetermined value. A step of executing until a convergence condition is satisfied may be included. The repulsive force is constant below a predetermined value with respect to the distance between the dots, and when the distance between the dots exceeds a predetermined value, the repulsive force may decrease as the distance between the dots increases. preferable. In the present invention, the initial position of the dots can be given using a low discrepancy sequence method.

  According to the present invention, there is provided a program for causing a computer to execute any one of the steps described above.

  According to the present invention, there is provided a computer-readable recording medium on which the program described above is recorded.

According to the present invention,
Discrete patterns for passively controlling light rays by light scattering, light transmission, or light absorption to produce discrete patterns that are light guide plates, scattering plates, dithering patterns, lithographic photomask patterns A generation system,
Means for providing a discrete pattern by each of the steps described above;
Storage means for recording position coordinates of dots forming the discrete pattern;
Printer means for outputting the position coordinates contained in the storage means;
A discrete pattern generation system including a pattern receiving element in which a discrete pattern is formed by the printer means can be provided.

  In the present invention, the discrete pattern may be a light intensity modulation element of an optical member or a binary pattern of a grayscale image.

  Hereinafter, the present invention will be described based on the embodiments shown in the drawings, but the present invention is not limited to the embodiments described below. Hereinafter, as part I, after giving the initial arrangement which is the first configuration of the present invention, repulsion is applied by applying a repulsion model between dots, and formation of a random discrete pattern by the LDS method is described. In Part II, the formation of a discrete pattern by automatically generating / deleting dots, which is the second configuration of the present invention, will be described. Furthermore, as part III, generation of a binarized pattern of a grayscale image that can provide good blue noise characteristics according to the third configuration of the present invention will be described. Further, as part IV, an optical member, a sidelight device, and a transmissive liquid crystal display device that use discrete patterns obtained based on the above-described LDS method and dot automatic generation method will be described.

Part I: Formation of Discrete Pattern by Relaxing Dots by Assume LDS Method and Repulsive Force Between Dots Hereinafter, in Part A, the determination of the initial position of the discrete pattern will be described as Part I. Describes a method of forming a discrete pattern by eliminating overlap between dots, and in section C, a description is given of how to change the filling rate continuously. In section D, the discrete pattern given by the present invention is described. Explain the range of discrepancy and criteria for determining randomness

<Section A>
Generation of initial position of discrete pattern having low discrepancy (A-1) Mathematical formulation of discrepancy N points in a rectangular area such as a rectangle having a length Lx, a length Ly having a length Ly, and a square When distributed, the discrepancy can be given as follows.

  Consider a rectangular area whose diagonal is the origin (0, 0) and position (x, y) with respect to position (x, y) in the entire rectangular area. V = x × y / (Lx × Ly), where V is the proportion of the area occupied by the area in the square. Further, if the number of points included in the rectangular area among the N points considered is A (x, y), the ratio of the points included in the above-described rectangular area out of the N points. Becomes A (x, y) / N. If the distribution of points is ideally uniform, the proportion of the points contained in any rectangular region should be exactly the same as the proportion of the area of the rectangular region in the entire square.

  Therefore, by examining how much the difference between the two deviates from 0, the deviation of the points can be understood. In this specification, the discrepancy is defined as the square root of (A (x, y) / N−V squared, where (x, y) is integrated for all the positions in the square described above. That is, if the square of the discrepancy is D (Lx, Ly; N), D (Lx, Ly; N) is given by the following equation.

If the above-described discrepancy is used, the discrepancy is 0 in an ideal distribution, and is 1 which is the maximum value in a very biased situation where all points are fixed at the origin, for example. / 9.

  On the other hand, let us consider the case of a point sequence generated truly at random. The discrepancy is not so small for such a truly randomly generated point sequence. The reason for this is that, for example, when a random number is created by rolling the dice six times, it is more intuitive that one of the numbers appears more frequently than when the number is one at a time from 1 to 6. Can understand. When random and low-discrepancy point sequences are required, randomness is not the same as non-uniformity, i.e., uniformity. It is necessary to use a sequence of points generated to make it low. The present inventors pay attention to the fact that LDS has the characteristic of low discrepancy from the definition, and as a result of intensive studies, the LDS method can be applied to the actual problem of arranging dots in a predetermined section. They found it to be effective.

(A-2) Flowchart for Applying the LDS Method FIG. 7 shows a flowchart of the LDS method that can be used in the present invention. The LDS method used in the present invention starts from step S1, and the generator matrix 1 is calculated for each coordinate axis in step S2. Details of the generator matrix are described in detail, for example, in “Discrete Structure and Algorithm IV”, edited by Kazuo Murota, Modern Science. Next, a scramble matrix having a random value in the lower triangular element is generated in step S3, and the product of the generator matrix 1 and the scramble matrix is stored as a generator matrix 2 in step S4. In the present invention, it is also possible to generate an initial arrangement without using a scramble matrix. However, when the number of dots is large, it is preferable to use a scramble matrix together in terms of generating a more uniform discrete pattern.

  In step S5, binary expansion of the natural number n assigned to a predetermined position is performed, and the result is stored as a vector x. In step S6, a vector y is generated by multiplying the vector x by the generator matrix 2 for each coordinate axis. In step S7, a new coordinate is generated with the element of vector y as a binary decimal value for each coordinate axis, and in step S8, n is increased by 1 to a predetermined number, for example, N. Steps S5 to S8 are repeated until the coordinates generated by the N LDS methods are determined.

(A-3) Specific Procedure of LDS Method A procedure for generating a discrete pattern in the LDS method shown in FIG. 7 will be described in detail below when applied to one dimension. The procedure described below is used for explanation, and the present invention can be applied as it is in any higher dimension of two or more dimensions.

  In the initial position determination by the LDS method used in the present invention, a high speed is realized by using the Niederreiter configuration method which extends the net theory. In the present invention, randomness as an initial position is also important. Therefore, Tezuka, “About Discrepancy of Point Sequences”, “Discrete Structure and Algorithm IV”, edited by Kazuo Murota, Modern Science, Chapter 3 By using a generalized Niederreiter sequence with a random scramble, we generated a sequence with both randomness and small discrepancy.

  A typical LDS generation method uses net theory. Regardless of the size of the area as in the definition of discrepancy, the sum of the sizes of the points that fall within it must be close to the size of the partition. Therefore, a partition is divided into small partitions by a p-adic box whose side length is hierarchically reduced by a power of a prime number p, and the partitions are filled with a P-adic box by boxes of various sizes. Considering the situation, a position coordinate is determined based on a natural number p-adic expansion coefficient from among the positions where a point should be entered, thereby reducing the discrepancy.

  Specifically, when natural numbers arranged in order of 1, 2, 3, 4 are used, the same number of overlaps does not occur, but they are arranged in order from the end, so that they are widely scattered throughout the whole section. You cannot make a line.

  For this reason, in the present invention, a strictly deterministic method for associating coordinate values with the original natural number sequence is used to eliminate the bias like a random point sequence, and in order from the end like natural numbers. A method of giving a sequence of points that are not arranged is adopted. The basic idea is to first divide the compartment into smaller units, place points in each of the smaller compartments, and if the number of points increases, divide it into smaller subdivisions and separate them. It occupies the position of.

  Here, considering the nature of the sequence of natural numbers, the first place changes rapidly from 1 to 9, and then the upper digit changes slowly in order. Therefore, this digit arrangement is reversely arranged so that the one's place is higher, and the small division divided by different sizes for each digit is changed greatly every time the original natural number slightly changes to cover the whole division. Thus, the position coordinates can be generated without deviation. That is, in the generation method employed in the present invention, a natural number is expressed in binary not in decimal, and then a numerical value appearing in each digit is replaced by a predetermined method. In other words, after displaying the original natural number sequence in binary, the lower digit that changes faster causes the position change over the entire large section corresponding to the upper digit of the coordinate value, and the upper digit that changes slowly is the coordinate The irregularity is further increased by performing conversion so as to bring about a fine position change corresponding to the lower digit of the value.

  More specifically, for example, when natural numbers from 1 to 5 are expressed as binary numbers, they are 1, 10, 11, 100, and 101 in order. Here, for example, abcd is arranged in reverse from a natural number n expanded as abcd, and 0. It corresponds to dcba. In this case, in the embodiment in the case of the natural number described above, the order is 0.1, 0.01, 0.11, 0.001, 0.101, respectively, and is 1/2, 1/4, 3/4, Since 1/8 and 5/8, as shown in FIG. 8, a point sequence is obtained in which small sections that gradually become smaller between 0 and 1 are randomly distributed with respect to the center Cr.

  When obtaining coordinate values in two dimensions, the value of each digit is converted by applying a different conversion matrix for each dimension using the method described above so that it changes independently for each dimension, such as x coordinate and y coordinate. Then, the corresponding coordinate point is calculated for each dot from the original natural number sequence. FIG. 9 shows a two-dimensional discrete pattern of low discrepancy obtained by generating position coordinates by sequentially assigning natural numbers to the positions of coordinates to be used for each coordinate axis. In FIG. 9, with respect to the vertical axis, by performing the above-described conversion, the conversion indicated by the arrow line is performed to generate coordinates, and two-dimensional position coordinates are obtained. In FIG. 9, dots 10 are arranged at position coordinates where the horizontal axis and the vertical axis cross each other, and an initial arrangement generated by the LDS method is used.

  In the present invention, it is preferable to provide a variation in the arrangement of the point coordinates by performing a certain exchange called a scramble method for rearranging the digits. When the scramble method is adopted, a generator matrix previously given for each dimension is stored in a storage device, and multiplication is performed with a vector x composed of numerical values of binary numbers of natural numbers to be converted. Thus, it can be easily expanded by generating the vector y.

  FIG. 10 shows a discrete pattern including the dot pattern generated by the LDS method described above, assuming that 50 dots 10 are arranged in a section. In FIG. 10, the vertical axis and the horizontal axis are represented by standardized unit lengths, and the sections are squares.

  FIG. 11 is a diagram illustrating a low discrepancy dot pattern used in the present invention described above. In FIG. 11, the overlap between dots is not particularly processed, and corresponds to a so-called initial position. FIG. 11A shows a dot pattern obtained by using the LDS method according to the present invention, and FIG. 11B shows a dot pattern generated by using a pseudo random number generation method for comparison. .

  As can be seen by comparing FIG. 11 (a) and FIG. 11 (b), the initial position generated using the LDS method is not shown in FIG. Compared to the dot pattern generated by the pseudo-random number generation method shown in FIG. 11 (b), the uniformity is sufficiently high, and the location where the density of dots is extremely low or high is not generated. Recognize. On the other hand, in the dot pattern generated using the pseudo random number generation method, as shown in FIG. 11B, a locally high-density portion and a low-density portion are generated. It is shown that the aspect is insufficient as compared with the case of the present invention shown in FIG.

  In the present invention, when generating a discrete pattern with low discrepancy, in addition to using net theory, as another embodiment, a method such as a Good Lattice Point (GLP) method can be used. FIG. 12 shows a low discrepancy dot pattern obtained using the GLP method. In the dot pattern shown in FIG. 12, the dots 10 are arranged in the sections indicated by squares by the GLP method, and are initially arranged. When the dot pattern shown in FIG. 12 is obtained, when the generation vector is selected as (h1, h2), the GLP is (((h1 × k) modn) / n, ((h2 Xk) modn / n) For example, if two adjacent Fibonacci sequences are defined by F (m−1) and F (m), n = F (m) and the generated vector is (1, F (m-1)).

  FIG. 12 illustrates a dot pattern created using F (m−1) = 987 and F (m) = 1487. As shown in FIG. 12, even in the GLP method described above, a uniform distribution with small discrepancy can be obtained. However, as can be understood from the dot pattern of FIG. 12 using the GLP method, each dot is characterized by being relatively regularly arranged as compared with the LDS method. It is not suitable for applications that do not want to generate patterns.

  However, when the moire pattern is not an important factor, specifically, when the discrete pattern of the present invention is used as an anti-slip, design application, or dithering pattern, a discrete pattern having sufficient randomness is used. Can be given. Similarly, in other embodiments of the present invention, a Richtmeyer sequence using a fractional square root of a prime number and a Halton sequence using p-adic expansion with different prime numbers for each coordinate axis are also used to generate a low discrepancy pattern. Can be used.

<Section B>
Elimination of Overlapping of Adjacent Dots by Repulsive Force Relaxation Method (B-1) Outline of Repulsive Force Relaxation Method In the conventional method considering perturbation from a grid point, inconveniences are inherently generated in a high filling rate region as described above. That is, the method derived from the sequential generation of random numbers is substantially impossible to create an irregular pattern with no overlap between dots in a high filling rate region. Without introducing some kind of interaction between dots, it is difficult to generate irregular patterns with a high filling rate exceeding 50%. Needless to say, the present invention can be easily applied to a section having a high filling rate of 50% or more as well as a section having a low filling ratio. This repulsive force relaxation method can also be suitably applied to the second and third configurations of the present invention described later by appropriately changing the function shape.

  In the present invention, as the filling rate, a ratio of a dot area when a dot is arranged in a predetermined partition and a partition area can be used. At this time, in order to correspond to dots having various geometric shapes, the dot area is defined as the square of the maximum dot diameter multiplied by the number of dots. Alternatively, the ratio between the number of dots and the area of the compartment can be used as the definition of the filling rate.

  In principle, even when the initial position is determined by the pseudo random number generation method, it is possible to achieve the elimination of overlap between dots by the repulsive force relaxation method. However, when the initial position is determined by the pseudo-random number generation method, in many cases, as will be described later, a mesh-like unevenness appears in the pattern, and a large amount of processing and processing are required to remove this unevenness. It will take time. For this reason, an initial position with low discrepancy is generated in advance by, for example, the LDS method, and a repulsive force relaxation method is applied to this initial position, thereby generating a discrete pattern with even higher uniformity. It is preferable.

  As described above, for example, when the initial arrangement is generated only using the LDS method, the uniformity and randomness are sufficient, but due to the finite size of the actual dots, overlap occurs between the dots, and the dots appear. It causes inconvenience such as. Therefore, reconstruction processing (overlapping elimination) of these distributions is indispensable. In the present invention, the dots arranged at the initial position generated using the LDS method are regarded as two-dimensional particle systems that interact with each other, and the overlap between adjacent dots is reduced by repulsive force from the initial position. Eliminate it as a process. That is, in the present invention, an interaction model that applies a strong repulsive force at least in the vicinity of the distance between particles is zero with respect to the distance between dots. For this reason, between dots that overlap each other at the initial position, a high potential energy state is obtained, and only the dots that overlap each other can be moved efficiently, and the overlap between dots can be eliminated.

  When such a model is adopted, it can be considered that the adjacent dots are settled in a state having an appropriate interval by repulsion as the relaxation time elapses. In the present invention, the above-described process is defined as a repulsive force relaxation method. In the present invention, various patterns suitable for a specific application can be generated according to the degree of relaxation and depending on the mechanical mechanism that provides relaxation.

  FIG. 13 is a schematic diagram showing the principle of the repulsive force relaxation method when a repulsive force acts between the dots 12, 14 and 16 when the dot shape is a square. In FIG. 13, a dot is regarded as a two-dimensional particle system that interacts repulsively. It is assumed that dots 14 and 16 are arranged around the dots 12 at a certain time. In the present invention, a repulsive force model is provided so as to receive a larger repulsive force as the distance between adjacent dots is shorter.

  The coordinates of the dots after the relaxation process are described, for example, by Shiro Kanamori et al., “Solid-Structure and Physical Properties”, Iwanami Shoten, page 255; E. It is calculated based on the so-called molecular dynamics method described in Qussac, “Physics of Structurally Irregular Systems”, Yoshioka Shoten, page 110, 1994. However, for the purpose of creating an irregular pattern, it is not necessary to explicitly solve the equation of motion of dots in the present invention. Therefore, in this specification, the simplest case is described as an embodiment, and the repulsive force is described in, for example, Ryogo Kubo et al. It is assumed that it is based on a molecular field model. However, a more complex model can be adopted in the present invention.

The molecular field model is a method of calculating the force acting on one focused dot from the distribution at a certain point in time and calculating the displacement of the next step based on the force. As shown in FIG. 13, the dot 12 receives a repulsive force from the adjacent dots 14 and 16. Figure 13 shows the respective forces acting from the dot 14 and dot 16 in the dot 12 as _{B 14} and _{B 16,} shows the resultant force of the force _{B 14, B 16} at arrow line B. Similarly, repulsive force is applied to the dots 12 and 16. By doing so, the force received from the surroundings for each dot is calculated at a certain point in time. Displacement is given to each dot in proportion to this force, and the distribution at the next time point is obtained. By repeating this operation, it is possible to gradually eliminate the overlap between dots that generates a strong repulsive force.

(B-2) Repulsive force model in the repulsive force relaxation method The purpose of the repulsive force relaxation method used in the present invention is to eliminate overlapping or abnormal approach between dots having a finite size, and to keep the distance between dots appropriate. For this reason, the interaction model used has a large repulsive force when the distance between two dots approaches a certain limit D, and its size decreases rapidly with distance outside the limit. desirable. That is, as a parameter characterizing the repulsive force model, it is preferable that a repulsive force attenuation distance L exists in addition to the repulsive force reachable range D.

  In the present invention, a coordinate system shown in FIG. 14 is set in order to formulate the above-described model. That is, paying attention to an arbitrary dot on a two-dimensional plane (or curved surface) for generating a pattern, two-dimensional polar coordinates (r, θ) are set with the center of the dot as the origin. The positional relationship of the start line OO ′ with respect to the dots 18 and 19 can be arbitrary. The force acting when the center P of the dot 19 comes to the position (r, θ) is expressed by the following equation:

Defined in In the above formula,
Represents a vector OP of magnitude r.

  In the present invention, the reach range D of the repulsive force can be made to depend on the angle θ, and can be defined as D (θ) = OQ, for example, as shown in FIG. The above formula means that the force acting between the dots is a so-called central force. It has been found that this condition is effective in that the dot group is swirled with the progress of the relaxation process so as not to cause inconveniences such that the overlap removal cannot be performed efficiently.

A function F (r, θ) serving as an index of the magnitude of force is expressed by the following equation for a predetermined finite function F ₁ (θ) that does not depend on r.

Those satisfying the conditions can be selected. As for F (r, θ), it is assumed that positive indicates repulsive force and negative indicates attractive force. However, the function D (θ) is generally an arbitrary function on the order of the maximum dot diameter s or the average dot interval Δr. Here, the average interval of dots can be defined as a lattice interval when dots are arranged on a square lattice so as to satisfy a predetermined filling rate.

  Below, the repulsion model which can be used in embodiment of this invention is illustrated. For convenience, the repulsive force attenuation distance L is expressed by the following equation using the base e of the natural logarithm.

Determined by As is apparent from the above formula, in the present invention, L can generally depend on θ.

  (B-3) Specific example of repulsion model

(1) Isotropic Repulsive Model In the present invention, an isotropic repulsive model is defined by ignoring the θ dependence of F, F ₁ and D used in the above equation. In this case, when the center of a predetermined dot enters a circle having a radius D around the center dot, a strong repulsive force is generated between the dots. Although there are various function forms that satisfy the above-mentioned conditions, the following repulsive model combining a power function and an exponential function can be adopted from the viewpoint of numerical calculation.
(A) Exponential function type

(B) Yukawa type

(C) Power type

(D) Leonard Jones (LJ) type

(However, m, n, and α are real numbers, and m> n and α> 1.)
(E) Composite type

(However, m, n, and α are real numbers, and m> n and α> 1.)

FIG. 15 shows a graph in which each repulsion model described above is calculated using parameters such that the curves in the figure are located at similar positions. In FIG. 15, D = 0.1 mm and F ₁ = 5 for all models. The other parameters used are summarized in Table 2.

  In the embodiment of the repulsive force model described in the present invention, as a variation of the so-called hard sphere model, only those in which the magnitude of the force is constant when r <D are listed. However, it can be made dependent on r, and in this case, it is desirable that the repulsive force decreases monotonously according to r.

  As can be seen from FIG. 15, these functional differences do not give dramatic differences to the discrete pattern generated from the dot pattern. For the purpose of providing the discrete pattern of the present invention for optical use and creating a random discrete pattern in order to eliminate the moire pattern, an exponential function of n = 1 is sufficient. However, when it is desired to limit the distance between dots to a certain narrow range, the LJ type or each repulsive force model can be used, and for example, a composite type model that is linearly coupled to these repulsive force models is adopted. You can also

  For example, the above-described repulsive force model having the minimum value is effective when it is desired to keep the displacement range of the dots below a certain value, because the force between the dots is attractively minimized at a predetermined position. However, for example, an LJ-type repulsive model having these minimum points tends to accumulate numerical errors as compared with other models, so that it is necessary to pay attention to errors in numerical calculations.

(2) Anisotropic Repulsion Model In the above-described repulsive force model, a model in which the repulsive force has θ dependence is referred to as an anisotropic repulsive force model in the present invention. The anisotropic repulsion models that can be used in the embodiments of the present invention are listed below as an example. In making an actual pattern, the above-described θ dependency is an important factor. The isotropic repulsion model has a considerable utility value as long as it has a highly symmetrical shape such as a circle or a regular polygon. However, in order to more uniformly distribute dots having a low symmetry shape, it is preferable in the present invention that the reach of the repulsive force is anisotropic according to the dot shape. Many examples of the repulsion model in this case can be exemplified, but the following two models can be given as particularly useful repulsion models for a specific embodiment in which the dot shape is rectangular. .

(A) Ellipse model There are various types such as exponential type, power type, LJ type, etc. As an example, the simplest exponential elliptical model with n = 1 is represented by the following formula:

Given in. However, in the above formula,

It is. FIG. 16 shows the relationship between the parameters. In FIG. 16, D is the major axis of the ellipse, kD is the minor axis of the ellipse, and r is the distance to the center of another adjacent dot. The repulsive force field given by the above equation is shown in FIG. The repulsive force field shown in FIG. 17 is given when the parameter k is 0.3.

(B) Rectangular model Also in this model, various types such as exponential type, power type, and LJ type can be exemplified, but the simplest exponential rectangular model with n = 1 is represented by the following formula:

Given by. In the above formula, R (r, θ) is the following formula:

It is represented by FIG. 18 shows the repulsive force field when the rectangular model is used. However, in FIG. 18, k = 1.

  The repulsion model described so far is a model belonging to the class of so-called hard spheres. However, in the present invention, the model is not limited to the above-described model, and as long as it has a property of generating a strong repulsive force when the distance between dots is short, for example, N.I. E. Any repulsive model known so far can be used, such as described in Cusack, “Structural Irregular Physics”, Yoshioka Shoten, 1994.

(B-4) Convergence Judgment Condition in Repulsive Force Relaxation Method The most appropriate variable for numerically representing the progress of the relaxation process is the potential energy E per dot. Strictly speaking, this potential energy E is expressed by

For a scalar function V that satisfies

Is defined as follows. In the above formula, grad represents a two-dimensional gradient operator,

Represents the potential energy E of dot j viewed from dot i. N is the total number of dots. For the isotropic exponential model with n = 1, the function form of V is specifically exemplified as follows:

It is represented by FIG. 19 shows the shape of V with respect to r when D = 0.1, L ₁ = 0.04, and F ₁ = 5 in V (r) described above.

As a convergence determination condition in the present invention, when an exponential model of k = 0.5 is adopted, the dot shape is a rectangular dot, and the above V (r) is used as an evaluation function, it is uniform and does not overlap. the rate of decrease in potential energy E _n per dot in the n steps of the relaxation process for producing a pattern,

It has been found that it is preferable.

  The potential energy E can be calculated in the same manner for other models or anisotropic cases. However, when the calculation formula becomes complicated, it is practically possible to use a similar function for the convergence determination. In the present invention, the convergence determination can also be made by approximating the graph shown in FIG. 19 with several straight lines. Furthermore, the convergence determination of the elliptic model can be substituted with the function V of the isotropic model. This is because the present invention does not deal with dot dynamics itself, but V is merely a measure of how dots approach. Furthermore, in another embodiment of the present invention, an evaluation function that is relatively similar to the potential energy E can be used among functions that rapidly decrease in size with respect to the distance between dots. Note that the above-described convergence determination condition can be appropriately set as necessary depending on the calculation time of the repulsive force relaxation method, the evaluation function to be used, and the like.

(B-5) Actual Processing Process of Repulsive Force Relaxation Method FIG. 20 shows a flowchart of the repulsive force relaxation method used in the present invention. The process shown in FIG. 20 starts from step S11, and in step S12, the initial dot position is calculated by the above-described LDS method. Thereafter, in step S13, the repulsive force from the surroundings is calculated for each dot. In this case, any of the repulsive models described above can be selected as necessary. In step S14, for each dot, based on the resultant repulsive force, the dot position coordinate is displaced from the initial position, and in step S15, the position coordinate generated in step S14 is stored as a new dot distribution.

  In step S16, the potential energy is calculated based on the new dot arrangement, and it is determined in step S17 whether the potential energy has converged using the convergence determination condition described above. If the potential energy E is not sufficiently small, the repulsive force is calculated again for the dots, the displacement is performed, a new distribution is generated, the energy is calculated, and the potential energy E is lowered until convergence. If the change in energy becomes smaller than the convergence determination condition in the determination in step S17 (yes), the potential energy E has been converged and the final dot position is registered as a discrete pattern, and step S18. End with.

  Table 3 summarizes the results of generating discrete patterns using the repulsive force relaxation method for the initial positions obtained when the above-described LDS method is used and when the conventional pseudorandom numbers are used.

In Table 3, a total of N dots are formed in a rectangular region whose diagonal is a line segment drawn from the point (0, 0) to the point (x, L) on the two-dimensional orthogonal linear coordinate (x, y). The result of calculating D (Lx, Ly; N) for a certain pattern is shown as a value obtained by multiplying by 10 ⁶ . In the calculation of Table 3, the periodic boundary condition was applied, the small sections were calculated as square dots of 0.1 mm square, the filling rate was calculated as 50%, and the relaxation time was common to LDS and pseudorandom numbers. Note that the behavior of the discrepancy calculated by the pseudo random number generation method was determined by extending the relaxation time up to 5 times in order to determine the effect of the relaxation time. The result is indicated by the value in parentheses in the 1225-dot data of the pseudo random number generation method. As can be seen from Table 3, since the difference between LDS and pseudorandom numbers increases with N, this is a comparison with LDS at the best conditions for pseudorandom numbers. As shown in Table 2, when pseudorandom numbers are used, it can be seen that the discrepancy is greater both before and after relaxation than when the LDS method is used.

  In addition, it is understood that the discrete pattern formed by the repulsive force relaxation method using the LDS method has a sufficiently low discrepancy although the discrepancy tends to increase slightly. Is done. On the other hand, when the conventional pseudorandom number generation method is used, the discrepancy is increased or decreased by the repulsive force relaxation method. This is presumed to be due to inferiority and, in some cases, the distance traveled by the dots is long due to repulsion. Correspondingly, the dependence on relaxation time is large, and the repulsive force relaxation method is effective even in the conventional pseudorandom number generation method, but it can be said that it is preferable to use it together with the LDS method in terms of computer resources and computational effort.

  FIG. 21 is a diagram showing a comparison between a discrete pattern generated according to the present invention and a discrete pattern generated using a method using pseudo-random numbers. FIG. 21A is a discrete pattern generated using the LDS method and the repulsive force relaxation method according to the present invention, and FIG. 21B is a discrete pattern generated using the pseudorandom number generation method and the repulsive force relaxation method. It is. The same repulsive force model was used. As shown in FIG. 21A, the discrete pattern generated according to the present invention has a uniform randomness, and almost no dot density is observed.

  However, in the discrete pattern generated using the pseudo random number generation method shown in FIG. 21B, the density of the dots is visually observed, indicating that the uniformity is inferior. The results shown in FIG. 21 indicate that the LDS method used in the present invention gives a good initial position, and the repulsive force relaxation method employed in the present invention reduces the discrepancy while reducing the distance between adjacent dots. It shows that it is possible to eliminate overlap.

  FIG. 22 shows the discrete pattern of the present invention obtained using the process of FIG. 20 and using the isotropic exponential repulsion model with the initial position given by the LDS method. The parameters used in the repulsion model were n = 1, D = 0.8 mm, L1 = 0.4D, and the dot shape was a 0.1 mm square. FIG. 22 is an enlarged view of 4 mm × 5 mm among the discrete patterns generated in the section. Furthermore, in FIG. 22, the filling rate is set to 70%. As shown in FIG. 22, according to the present invention, it is shown that a random discrete pattern with good uniformity can be formed even at a high filling rate.

  In the calculation of FIG. 22, even if the relaxation time is very long, it does not result in a regular arrangement of square lattices. In other words, this means that the appropriate irregular pattern shown in FIG. 22 cannot be generated from the square lattice by perturbation based on pseudo-random numbers, and the LDS method and repulsive force relaxation employed by the present invention. It can be concluded that a good discrete pattern has been generated by the synergistic effect of the law.

  FIG. 23 shows a discrete pattern of the present invention obtained when a rectangular exponential model is employed at the same filling rate (70%). The parameters in the repulsion model used were n = 1, D = 0.1 mm, L = 0.15D, and the dot shape was a 0.1 mm square. The region shown in FIG. 23 is the same region as described in FIG. As shown in FIG. 23, it has been found that only by adjusting the parameters of the repulsion model, the aspect of the discrete pattern can be changed considerably freely depending on the application.

  The results shown in FIGS. 22 and 23 indicate that there is an optimum repulsive force model according to the dot shape. For example, for a rectangular dot, it can be said that a repulsion model having an elliptical repulsion range is suitable. The high flexibility of the present invention described above cannot be obtained by the conventional method at all, and gives much higher flexibility than the method disclosed in, for example, Japanese Patent Laid-Open No. 10-153379. It is possible to provide a discrete pattern with good uniformity even at a high filling rate.

  On the other hand, when the repulsive force relaxation method described above is applied, the handling of boundary conditions is important. This is because if no condition is imposed, the entire area of the dot spreads. In some cases, it may be possible to calculate an irregular pattern by dividing a region where dots are arranged into a plurality of sections and changing the conditions. For this reason, the selection of boundary conditions is particularly important from the viewpoint of eliminating joints and the like.

  If the distribution is uniform or close to that, a sufficiently good result can be obtained by calculating the repulsive force using a so-called periodic boundary condition. However, when the filling rate is maximized at the end, an error may occur in the filling rate distribution as a result of the rearrangement of dots by the above-described dynamic repulsion model. This is because, due to the use of the periodic boundary condition, a discontinuity of the filling rate occurs virtually at the boundary. In such a case, the initial position generated by the LDS method can be inverted at the boundary, stored as an external field, and used as a boundary condition thereafter. In the present specification, the setting of the boundary condition described above is defined as a self-similar boundary condition. The explanation is schematically shown in FIG. It has been found that using this boundary condition eliminates the discontinuity of the filling rate and gives good results. The reason why the initial position can be used as a boundary condition is that the initial position itself given by the LDS method has high uniformity and is random.

  In addition, once discrete patterns are calculated independently for each section and these discrete patterns are to be joined, first, a partial discrete pattern is calculated using self-similar boundary conditions, and then the valley portion is calculated. Calculations can be performed using repulsion relaxation methods. At that time, as the boundary condition, an external field as shown in FIG. 24 obtained from the surrounding pattern can be adopted. In FIG. 24, a dot in a predetermined section is indicated by 10a, and a dot inverted with respect to the boundary Bnd is indicated by 10b. As will be described later, the reason why a seamless pattern is obtained while maintaining a predetermined filling rate distribution is considered to be because an interaction by a repulsive force relaxation method is introduced between dots.

<Section C>
Continuous change of filling rate: probability extraction method When applying a discrete pattern to a practical application, it may be necessary to change the filling rate while maintaining uniform randomness. In the present invention, the probability extraction method is used in order to continuously change the filling rate of dots and not to give visible defects to the discrete pattern. FIG. 25 shows a filling rate distribution required when the present invention is applied to a light guide plate of a backlight unit of a transmissive liquid crystal display device. The filling rate in the lower left corner of FIG. 25 is about 60%, and the diagonal side is about 30%, and the contours of the filling rate are shown between them.

  The reason for providing such a filling factor distribution is to eliminate low luminance regions that tend to appear at the four corners of the screen of the transmissive liquid crystal display device by increasing the scattering properties at the four corners. The continuous filling rate distribution as shown in FIG. 22 is substantially difficult to realize without generating a moire pattern or a visible boundary in the conventional method in which random perturbation from a lattice point is considered. is there. In the present invention, the probability extraction method is adopted, and the entire region for generating the discrete pattern is divided into sections of a predetermined size, for example, about several mm, and each filling rate is first defined therein. At this time, the filling rate of the section i is defined as di. From the filling rate, the following formula

Define the variables given by. Here, the sum is performed over all sections. This amount is regarded as a probability defined for each section, and a section to be set to a predetermined filling rate is selected according to the probability. That is, let U be the discrepancy defined in interval (0, 1), and the equation

To select partition k. The discrete function F _k is an amount corresponding to the cumulative probability distribution. Then, the position is selected again using the two sets of discrepancy values in the section k. By selecting in this way, an arbitrary filling rate distribution can be realized at the initial position. Although the following reconstruction process causes dot displacement, as long as this filling rate is given as a continuous function, and as long as the reach and attenuation distance of the repulsion model are correctly scaled according to the filling rate, filling Deviations in the rate distribution are almost negligible. In the present invention, the above-described method is defined as a probability extraction method.

  When using a combination of probability extraction and relaxation, it is important to scale the repulsive parameters by the filling factor. For applications such as light guide plates where the filling rate varies from about 70% to about 10%,

It is desirable to take in the range. Here, Δr is an average interval of dots, and is defined as a lattice interval when dots are arranged on a square lattice so as to satisfy a given filling rate. What is the filling factor α

It is tied in a relationship like Here, s is the maximum dot diameter. For example, if the dot is 0.1 mm and the filling rate is 50%, it is about 0.14 mm. If an appropriate parameter is selected in accordance with the repulsion model within the above range, the average distance between the dots is distributed about Δr. It has been found that when D which is several times larger than Δr is taken, dot overlap and the like are not eliminated well. By performing this scaling, not only when the filling rate changes but also when the dot diameter changes continuously, a uniform irregular pattern without overlapping dots is used by using the common D and L. Obtainable.

FIG. 26 is a diagram showing a discrete pattern with a continuously changing filling rate generated by using the probability extraction method and a partially enlarged view thereof in the present invention. The discrete pattern shown in FIG. 26 has a filling rate with a filling rate distribution proportional to FIG. 25 for rectangular dots. In the calculation of FIG. 26, the repulsive force model is an elliptical exponent type with n = 1, k = 0.5, D = Δr [mm], and L ₁ = 0.4D. The filling rate was calculated with a dot diameter of 0.1 mm. As shown in FIG. 26, the discrete pattern with the continuously changing filling rate generated according to the present invention has no deterioration in uniformity seen in the boundary region where the filling rate changes, and there is a problem such as a moire pattern. It can be seen that a good discrete pattern can be obtained without occurrence.

<Section D>
Discrepancy range in the LDS method of discrete patterns (D-1) Discrepancy range

  FIG. 27 shows the results of calculation for discrepancy for various patterns. Here, the horizontal axis indicates N, that is, the number of dots in the calculation area. As shown in FIG. 27, each discrepancy changes in accordance with the number of dots arranged in a predetermined section. In FIG. 27, + indicates the value of D when dots are distributed by pseudo-random numbers for a uniform pattern with a filling rate of 50%. As shown by the one-dot chain line in FIG. 27, this value is distributed around a straight line of D = 0.15 / N within the range of N shown in FIG. Here D takes a square area for a random pattern and is calculated for the center coordinates of each dot. N is the number of dots that fall within the square area. In FIG. 27, ◇ represents the initial arrangement determined by the pseudo random number at the uniform filling rate (50%) and the overlap removal is performed by the repulsive force relaxation method. In this case, since the unevenness is reduced by the repulsive force relaxation method, the value of D tends to be smaller than the pseudorandom number itself within the range of the fluctuation of the distribution. However, as described with reference to FIG. 21, the distribution is more or less biased compared to the case where the initial position is determined by LDS (◯ and black triangle), and the value of D is within the range of fluctuation of the distribution. Can be larger than them. From various calculation results including FIG. 27, it was found that the value of D needs to satisfy a predetermined condition in order to prevent the unevenness as shown in FIG. 21 from appearing. Here D takes a square area for a random pattern and is calculated for the center coordinates of each dot. N is the number of dots that fall within the square area. Here D takes a square area for a random pattern and is calculated for the center coordinates of each dot. N is the number of dots that fall within the square area. However, in the case of a random dot pattern with a filling rate distribution, the value of [(maximum filling rate)-(minimum filling rate)] / (maximum filling rate) should not exceed 0.05 in the calculation area. To. The reason for this is to prevent the distribution of the filling rate from being reflected in D as unevenness. The filling rate of the calculation area is obtained from the number of dots entering the section when the calculation area is divided into sections having a size of 10 to 100 dots.

  As a result of the above-described examination, in order to provide the uniformity to be provided in the present invention, when the number of dots arranged in a predetermined section is 4000 or less, D is the square of the discrepancy. Is the following formula

It has been found necessary to satisfy Here, the range of N is decided to be 4000 or less. If it exceeds this, the area for calculating the discrepancy increases, and if there is a filling rate distribution, it causes an error of D. Because there are things. Further, in the present invention, the discrete pattern generated by the repulsive force relaxation method has no unevenness that is clearly visible when N is in the range of 50 to 4000 and D ≦ 0.13 / N ^1.15 is satisfied. It turned out to be a uniform discrete pattern.

Further, in the present invention, in addition to the above formula, when N is in the range of 50 to 4000 and D ≦ 0.30 / N ^1.15 is satisfied, a very uniform random with no visible unevenness is present.・ It was found that a pattern was obtained. As shown in the figure, the discrete pattern (◯) when LDS is used as the initial value for uniform filling rate distribution (50%) and overlap elimination is performed by the repulsive force relaxation method satisfies this condition. . The discrepancy in this case tends to increase slightly compared to that due to LDS, but the change is much smaller than the degree of decrease in the case of pseudorandom numbers. When the filling rate distribution is given by the probability extraction method and the initial position is given by the LDS (black triangle), the value of D tends to vary greatly, but the above condition is satisfied within the variation range.

  It should be noted that the low discrepancy does not directly give a random discrete pattern. A low discrepancy is good in terms of uniformity, but there can be a discrete pattern with a discrepancy as low as a regular lattice. In such a case, a moire pattern is generated when the discrepancy is suppressed to a low level, which is inappropriate for application to optical applications. That is, a low discrepancy is a necessary condition for providing uniformity in the present invention, but is not a sufficient condition for providing a uniform and random discrete pattern.

  Therefore, the inventors of the present invention have intensively studied by generating a plurality of types of discrete patterns that are uniform in visual observation and give randomness with no practical problem by the method of the present invention.

  FIG. 28 is a diagram showing a discrete pattern generated using the pseudo random number generation method, and FIG. 29 is a diagram showing a discrete pattern generated using the LDS and the repulsive force relaxation method. As shown in FIGS. 28 and 29, it can be seen that there is a great difference between the two in terms of uniformity. The present inventors calculated D actually measured from the discrete patterns shown in FIGS. In the calculation, as shown in FIGS. 28 and 29, the point P was fixed, and D was calculated by changing the lengths x1 and y1 of one side of the rectangular area.

  In order to show the outline of the procedure, the area is a square (x1 = y1), and the relationship between one side and D is shown in FIG. FIG. 30A shows the discrepancy obtained for the discrete pattern shown in FIG. 29, and FIG. 30B shows the discrepancy actually measured in FIG. It can be clearly seen that D of the pattern of FIG. 29 is smaller than the pattern of FIG. This is a result reflecting the existence of unevenness in FIG. That is, D proves that it is actually an index of the uniformity of the irregular pattern.

(D-2) Randomness Index The present inventors defined the total variation S ₁ as a randomness index by the following equation.

In the above formula, g is a radial distribution function centered on a predetermined dot,

Defined by Here, δ represents the Dirac delta function, R represents the distance to other dots when the predetermined dot is the origin, and the sum is the sum of all dots (except the origin dot) in the target region. Calculate over the center position. In the above formula, N is the total number of dots entering the area. The area to be summed is a square area centered on the origin dot, and for example, it can be a guideline that 50 or more dots are included in the square.

  The above function can be interpreted as the existence probability density of dots per unit length on the circumference of the radius r. This is a standardized condition

Can be understood from the fact that For example, a case where dots are arranged at each lattice point of a 5 × 5 square lattice shown in FIG. 31 will be described. In this case, the radial distribution function around the center position is

Given in. In the above formula, a is a lattice constant. It can be easily confirmed that this satisfies the above-mentioned standardization condition.
Actually, the delta function is

Replace with a Gaussian function like This is partly due to finite measurement errors. Where s is the following formula

Select to be. What is related to N is that it takes into account that the number of surrounding dots is finite when calculating g. Δr is a lattice constant when a square lattice satisfying a given filling factor is considered.

FIG. 32A shows g of the square lattice described above. FIG. 32A shows g when s = a / 24. As shown in FIG. 32A, the fact that the peak changes to a sharp shape corresponds to the regular arrangement of dots. This is because the positions of surrounding dots are regularly dispersed with respect to the center dot. Conversely, if the dot arrangement is irregular, for a sufficiently large N, a sharp peak as shown in FIG. 32A does not appear in g averaged with respect to the origin dot, and smooth. It is expected to be a simple function. Therefore, in the range of r ₁ to r ₂ ,

For the function defined in

This is defined as an index of regularity, that is, the total variation. However, g or g ₁ here is regarded as an average distribution function when the dot position of the origin is changed variously.

Where g _av is the following formula

It was. This value is indicated by a dotted line in FIG. S ₁ is the area in FIG. 32 (a) A, B, C, D. . . It corresponds to the sum of

FIG. 32B shows the result of simulation performed by changing s with respect to the above-described square lattice. In a regular grid, the distribution function averaged over the origin dot is exactly equal to the distribution function around any dot. Here, the values of s were set to a / 24, a / 10, and a / 2 in order from the top, and r ₁ = 1 and r ₂ = 3. Although the calculation example shown in FIG. 32B is an example using a regular lattice, the Gaussian function can be interpreted as a function representing the fluctuation of the dot center position, so it is not possible to increase s in a pseudo manner. The regularity will be reproduced. The broken line drawn horizontally in FIG. 32B is g _av , that is, the average value of g ₁ . Here, since a 5 × 5 lattice is considered as an example, g _av is calculated in the range of 1 to 3 in FIG. As shown in FIG. 32 (b), it can be seen that the value of S ₁ as a function is gradually becomes smaller. That is, the value of S ₁ the lower regularity is concluded that small.

Here, when the discrete pattern of the present invention is provided for optical applications, the regularity of the dots must be sufficiently low in order not to cause inconvenience such as a moire pattern. As a result of intensive studies on the actually generated random dot pattern, the present inventors satisfy the above-described condition for D, and when r ₁ = 1.0Δr and r ₂ = 4.0Δr, It is found that when S ₁ is 0.7 or less, it has randomness with no practical problem, and S ₁ is represented by the following formula:

It has been found that satisfying the above conditions is preferable in that a moiré pattern is not reliably generated. In addition, the inventors of the present invention have made extensive studies, and the pattern that does not satisfy this condition has the disadvantage that the moire pattern is generated to the extent that the moire pattern can be visually confirmed because the dots remain more or less regular. confirmed.

(D-3) Calculation of Discrepancy and Total Variation from Discrete Pattern In order to calculate the above-described discrepancy and total variation from an actual discrete pattern, various methods can be employed. . For example, a discrete pattern is converted into digital data by a digital camera, a scanner, or the like. Thereafter, the discrepancy is calculated from the obtained digital data including the discrete pattern using the method described with reference to FIGS. Moreover, to calculate the parameters described in the above-described from the digital data of discrete patterns (D-2), by obtaining a total change amount S _1, discrete patterns from an object containing a discrete pattern, as defined in the present invention Can be determined.

  The discrete pattern can be given as a two-dimensional shape. When a discrete pattern is attached to an object by means such as photolithography, the discrete pattern can be formed as a three-dimensional shape. When the discrete pattern is formed as a three-dimensional shape, the uniformity and randomness of the present invention can be determined using any part of the structure forming the discrete pattern as a reference. Further, it is possible to perform image processing such as appropriate discrimination and trimming, specify a region dot shape that gives necessary characteristics, and determine that the dot has the uniformity and randomness of the present invention.

  In addition, in the present invention, it is necessary that the moire pattern does not occur particularly when used for optical applications. In this case, the radial distribution function of the discrete pattern obtained from the digital data of the discrete pattern measured as described above can be examined. In the present invention, in order not to generate a moire pattern in particular, it is preferable that the average radial distribution function of the discrete pattern described above does not have a sharp peak. However, it is not necessary to pay particular attention to the restriction on the radial distribution function in the case where the moire pattern is not a problem, for example, a slip prevention pattern or a dithering pattern.

Part II: Generation of Discrete Pattern by Automatic Dot Generation Method In the first configuration of the present invention, as described above, the entire region is divided into small sections, and the filling rate is assumed to be uniform within each small section. After generating dots using a deterministic point sequence generation method for a small section, the dots can be reduced by using the relaxation method to mitigate defects in the filling rate that occurred inside and between the small sections. A pattern can also be generated. However, the number of dots to be generated has already been determined at the stage of generating dots in units of small blocks, and the number of dots does not increase or decrease during the relaxation method. Therefore, it may be desirable to generate dots so that the filling rate is changed as necessary instead of the predetermined filling rate in the portion where the gradient of the filling rate distribution function changes rapidly.

  FIG. 33 shows an embodiment in which the filling rate is determined in advance, and the dots are relaxed by the repulsive force relaxation method so as to be randomized. In the dot pattern shown in FIG. 33, since the gradient of the filling rate is large, the degree of freedom of movement of the dots due to relaxation is limited in the valley portion of the filling rate, and as a result, a desirable temporal change also in the relaxation process. This is because it does not show. Such a defect has a disadvantage that it is more or less seen in the method of fixing the initial number of dots for each small section as described above. In this case, the inconvenience shown in FIG. 33 can be prevented by generating or deleting dots for each small section.

  Hereinafter, generation of a discrete pattern by automatic dot generation in the present invention will be described in detail. In the present invention, as described in Part I, dots are arranged in a predetermined area. However, in the automatic dot generation method of the present invention, 1) a rectangular area for generating dots, 2) an area of one dot, and 3) a filling rate distribution function defined on the rectangular area are given as input data. FIG. 34 shows a rectangular area and a filling rate distribution function used in the dot automatic generation / deletion method of the present invention. 34A shows a rectangular area, and FIG. 34B shows a filling rate distribution function defined on the rectangular area shown in FIG. 34A.

  In FIG. 35, a bubble which is a concept related to dots used in the present invention will be described. Here, in FIG. 35, the dots are Pi (i = 1,..., N). In a particular embodiment of the invention, all dots are assumed to have the same area A. When the center (xi, yi) of the dot Pi is specified, the filling rate Gi (xi, yi)% at that position can be calculated using the filling rate distribution function. The filling rate is a percentage of the total area of dots per unit area. When one dot position is considered, since the filling rate specified at that position is satisfied, it is possible to determine a circular area in which other dots should not be included. This circular area is defined as bubble B. In the embodiment shown in FIG. 35, a dot is arranged at the center (xi, yi), and a bubble B having a diameter di has an area S.

  The bubble B has a center of [xi, yi] and a diameter of di (xi, yi). In the present invention, specifically, for example, it can be given by the following equation.

In the above formula, G _i (x _i , y _i ) is a filling rate expressed as a percentage (for example, 80%), and A is an area of a dot. As described above, in the present invention, since the dot area is given in advance, if one point [xi, yi] in the rectangular region is designated, the bubble diameter d at that position is determined using the filling factor distribution function. Can be requested.

The present invention uses the bubbles described above, and
1) Initial dot generation step (optional)
2) Dot coordinate value update step using attractive repulsion model
3) By using the adaptive dot addition / deletion step, automatic dot creation / deletion is performed, and dynamic relaxation including the generated dots is performed. A good discrete pattern is generated in a region that changes to.

  Hereinafter, the initial dot generation step will be described. The initial dot generation step can be used or omitted in the present invention. If the initial dot generation step is omitted, the dot starts from the initial state of zero. Thereafter, the dots can be sequentially increased, and the dots can be sequentially added to the rectangular area in the dot automatic generation / deletion step. However, in the present invention, by generating a dot pattern close to the desired dot pattern first, it is possible to save hardware resources such as calculation time and storage capacity until the desired dot pattern is generated. In the embodiment in which the initial dot generation step is omitted in the present invention, the position coordinate update step and the dot addition / deletion step by dot attraction / repulsive force relaxation, which will be described later, are repeatedly applied, so that the hardware resource capacity is increased. Hereinafter, in the embodiment of the present invention, a procedure including an initial dot generation step will be described.

  As described above, the initial dot generation step employed in the present invention is a step of generating a dot pattern close to the desired dot pattern for the purpose of shortening the calculation time. In the present invention, several methods can be used in the initial dot generation step described above. FIG. 36 shows a method of using a quadtree that can be used in the present invention. In the method using the quadtree shown in FIG. 36, a bubble is first placed at the center of a rectangular area. The bubble diameter is calculated from the above formula using the filling rate at that point. In the section indicated by s where the bubble covers a sufficient area of the rectangular area, the bubble is left and no further division is performed. In the section indicated by is not covering enough area, the quadrangular tree is used to divide the rectangular area into four subsections, and the same processing is performed for each subsection. Place a bubble. By repeating this process recursively, a bubble is given as shown in FIG. 36C, and an initial dot is generated by determining the center position of the given bubble as the dot position. In still another embodiment of the present invention, the initial dots can be generated by the LDS method using the filling rate distribution as described in Part I above.

  In the present invention, the dot position coordinates are then adjusted using the attractive force repulsion model to correct the initial dot arrangement within the optimum attractive force / repulsive potential. In the dot coordinate value update step, if two bubbles are close to each other, a repulsive force is applied to them, and if they are separated from each other, an attractive force is applied to move the bubbles to a position where these forces are balanced. Then, the bubble center position is set to the position coordinates where the potential energy after movement becomes the lowest.

  To explain more specifically, consider the potential between two bubbles B1 and B2. The centers of the two bubbles are [x1, y1] and [x2, y2], respectively, and the diameters are d1 (x1, y1) and d2 (x2, y2), respectively. When the distance between the centers of the two bubbles is represented by a variable r, when r is a value r0 calculated by the following equation, the two bubbles are in contact with each other, and both the attractive force and the repulsive force are balanced. This state is shown in FIG. 37 together with the repulsive function. As shown in FIG. 37, when bubbles overlap each other, repulsive force acts when the bubbles overlap, and when the bubble boundary is in contact, neither repulsive force nor attractive force acts, and the bubbles are separated from each other. In some cases, an attractive force is applied.

This state is considered as a stable state, and a state in which no force is applied. Also, as shown in FIG. 37, the inter-bubble force is determined so that the attractive force works when r is larger than r0 and the repulsive force works when r is smaller than r0. As one example, the inter-bubble force f (r) can be determined as in the following expression f (r).

  The function f (r) represented by the above equation shown in FIG. 37 has a repulsive force reach distance D, and includes a repulsive force term and an attractive force term. In the present invention, when dots are automatically generated, it is preferable to consider not only repulsive force but also attractive force shown in FIG. For this reason, among the various potentials shown in Part I, a function including an attractive term can be used in the dot automatic generation step. This is because when using the automatic dot generation method, it is necessary to consider not only the repulsive force between the generated dot and other dots, but also approaching to a distant dot. It is.

  In the automatic dot generation step employed in the present invention, a newly generated bubble Bi [xi, yi] receives a force given by the above formula from a plurality of bubbles located around and acquires new coordinates. Will do. At this time, for example, the following equation can be used as the equation of motion by potential energy.

Here, t is time, m is dot mass, and c is a coefficient representing resistance at the time of dot movement. The position of the movement destination is calculated by increasing the time Δt by a certain amount for each iteration using an expression obtained by rewriting the above expression using the difference. The center position of the bubble thus obtained is determined as the dot position.

  Further, in the dot generation method of the present invention, an adaptive dot addition / deletion step, which is the third step, is executed. This automatic dot generation step is a process that is repeatedly performed in the iterative process along with the movement of the generated dots. In the dot addition / deletion step, in order to generate an appropriate number of bubbles in the rectangular area, a bubble having a large overlap is deleted during the iterative calculation, and a new bubble is added to the gap. In the present invention, various methods that can be used in the above-described dot addition / deletion step are conceivable, but one method is a method of performing bubble addition / deletion using Delaunay triangulation.

  In the present invention, in the method using Delaunay triangulation, a Delaunay triangle mesh connecting existing bubble centers is first generated. Compare the side length of the mesh with the r0 value (sum of bubble radii) calculated from the bubbles at both ends. If the side length is greater than a certain percentage compared to r0, a new bubble is generated at the midpoint of that side. On the other hand, if the side length is smaller than a certain ratio compared to r0, either one of the bubbles at both ends is deleted. It is not always necessary to perform this bubble addition / deletion every time it is repeated.

  In the present invention, bubble generation / deletion processing can be performed by any method other than the Delaunay triangulation described above. For example, a certain threshold is set for the distance between adjacent bubbles, and a new bubble is generated at the center of gravity between bubbles that are more than this threshold, and the distance between bubbles is minimized. Any method can be used in the present invention as long as a bubble can be added or deleted appropriately, such as a method of extinguishing one of bubbles that are separated by a distance smaller than the minimum threshold by setting a threshold. can do.

  Hereinafter, specific embodiments of the automatic dot generation method of the present invention will be described in detail. In the present invention, the automatic dot generation method is performed for the purpose of adaptively performing automatic dot generation and deletion and keeping the filling rate distribution appropriate. A processing process that can be used in the present invention for this purpose is shown in FIG. FIG. 38A shows a schematic procedure for generating dots in a predetermined rectangular area, and FIG. 38B shows a schematic procedure of the dot addition / deletion process. There are three types of input data in the automatic dot generation process of the present invention, which are a rectangular area where dots are generated, a dot area, and a filling rate distribution function. In the specific embodiment of the present invention, this rectangular region is given as a region with a margin near the end so that the filling factor distribution can be accurately obtained in the region where dots are to be generated. The dot area is used when calculating the bubble diameter from the filling rate using the above formula. The filling rate distribution function which is the third input data is a filling rate distribution function defined on a rectangular area where dots are generated. A rectangular region that can be used in this dot generation is shown in FIG. 39 (a), and a filling rate distribution function is shown in FIG. 39 (b). Given a point within the rectangular area, the fill factor at that point is calculated from this fill factor distribution function.

  The automatic dot generation / deletion process will be described using the flowchart of FIG. 38. The automatic dot generation / deletion process starts from step S20, and in step S21, a rectangular area, a dot area, a filling rate distribution function, Enter. Next, in the procedure of FIG. 38, a declaration for generating dots on the boundary of the rectangular area is made in step S22. After this declaration, the dot automatic generation / deletion procedure shown in FIG. 38B is executed for the boundary of the rectangular area. After the procedure of FIG. 38 (b) has converged, the result is passed to step S23 shown in FIG. 38 (a) to declare that a dot is generated inside the rectangular area. Thereafter, using the procedure shown in FIG. 38 (b) again, dots are generated inside the rectangular area, the result is passed to step S24, and the result is output. In step S26, the automatic dot generation of the present invention is performed. -Terminate the automatic deletion process.

  The dot coordinate value on the rectangular boundary generated in step S22 described above is fixed, and step S23 is executed by generating a bubble in the boundary area of the rectangular area. This is to prevent movement outside the rectangle in the iterative calculation process. That is, by executing the procedure of step S22 in advance, the dots located near the boundary are repelled from the dots on the boundary, so that they do not protrude outside the rectangle.

  Further, as shown in FIG. 39A, the rectangular area used at this time is given with a margin to the area actually used. This is because the execution of step S22 aligns dots along the boundary line of the area to be used, and if there is no margin, it causes moiré or the like at the boundary between adjacent rectangular areas. Therefore, a rectangular area is generated with a margin in order to prevent dots from being aligned with the area boundary to be used. In FIG. 39 (a), an area for generating dots is illustrated as a rectangular area, but in the present invention, an area having an arbitrary shape can be used according to the purpose. Further, in FIG. 39A, the region for generating dots is shown as a planar region, but in the present invention, a three-dimensional region such as a curved surface shape can be used according to the purpose.

Furthermore, in the present invention, the process performed in step S22 in FIG. 38A is performed on a line segment (one-dimensional figure), while the process performed in step S23 is performed on a rectangular area (two-dimensional figure). Similarly, the process shown in FIG. 38B can be used for the processing used in each step. The procedure in FIG. 38B will be described. First, in step S28, initial dots are generated. In this case, in the present invention, either the LDS method or the random number generation method can be used. However, in particular embodiments of the present invention, initial dots can also be generated using the LDS method in order to obtain a good discrete pattern in a relatively short time. Next, in step S29, the coordinate value of the dot is updated in the process of iterative calculation. In this iterative calculation, when the update amount of the dot coordinate value in step S30 becomes smaller than a predetermined threshold value, it is determined that the movement of the dot is almost completed (yes), and the result is returned to step S23. hand over.

  In other cases (no), automatic dot generation / deletion is performed in step S31, and the dot coordinate value is updated again in step S29. The determination in step S30 is performed again, and when the update amount of the coordinate value of the dot is sufficiently small (yes), the result is passed to step S24 as output data, and the dot pattern filled in the rectangular area is output. In step S26, the process ends. Hereinafter, each step will be described in detail below.

  Hereinafter, the step of generating the initial dot in step S28 in FIG. 38B will be described in detail. As described above, this step is processing to reduce the calculation time by first generating a dot pattern close to the desired dot pattern. As shown in FIG. 36, the initial dot can be generated using the method using a quadtree as described above. Note that when a quadtree is used in the present invention, the target is a line segment in the process in step S22, and thus a binary tree is used. In the process in step S23, the target is a rectangular area, and thus a quadtree is used. To generate initial dots.

  In the present invention, as described above, a method in which initial dots are given in advance by the LDS method can be used. That is, the rectangular area is divided into a plurality of small sections, and the initial dot can be generated using LDS, assuming that the filling rate is constant for each small section. Further, in the present invention, when the initial dots are generated in advance, the initial dot is given fluctuations in the filling rate distribution for each small section, thereby improving the discreteness of the discrete pattern. You can also.

  Hereinafter, in FIG. 38, the step of updating the coordinate value of the dot in step S29 will be described. In this step, one bubble (circular region) is associated with each dot, and an attractive force / repulsive force is applied between the bubbles to calculate a new position of the dot. The force acting between bubbles is calculated using the above formula. The dot motion is analyzed using the above equation of motion, and the new dot coordinate is calculated so that the sum of the attractive and repulsive forces decreases.

  In the process in step S22, since the target for generating the bubble is a line segment, one bubble receives a force from the bubbles located on both sides thereof and moves to a new position on the line segment. On the other hand, in the process in step S24, since the target is a rectangular area, one bubble receives a force from the surrounding bubbles and moves two-dimensionally to a new position in the rectangular area. At this time, as shown in the above formula, the repulsive force and the attractive force between the bubbles should be determined so as not to receive a force from a bubble separated by a certain distance (1.5r0 in the specific embodiment of the present invention) or more. Since it is only necessary to consider the force from a bubble near a certain distance, stable convergence can be expected and calculation time can be saved.

  Further, step S30, which is a process for determining execution of iterative calculation in the automatic dot generation / automatic deletion process of the present invention, will be described. The moving distance of all dots is calculated by the above-described processing. If the maximum value of the distance is less than or equal to a predetermined threshold value, it is determined that the dot moving distance is very small (yes), and the iterative calculation is terminated. In other cases (no), the process proceeds to a dot automatic generation / automatic deletion step. In the present invention, the iterative calculation can be terminated when the amount of change in the sum of the attractive force and the repulsive force becomes a predetermined threshold value or less in addition to the moving distance.

  In step S31, automatic dot generation / deletion is executed. First, in step S31, the distance between two adjacent bubble centers is compared with the r0 value (sum of two bubble radii). When the distance r between the bubble centers is larger than the r0 value, for example, by a predetermined threshold value or more, a new bubble is added between the two bubbles. As a threshold value at that time, for example, a value such as a distance between bubbles of 1.8r0 can be used. On the other hand, if r is smaller than a certain ratio or minimum threshold value compared with r0, one of the bubbles is deleted. A value such as 0.5r0 can be used as the ratio or the minimum threshold.

  In the present invention, it is necessary to specify adjacent bubbles. As a technique for specifying this adjacent bubble, since the target is a line segment in the processing in step S22, the bubble columns on the line segment can be held in order. On the other hand, in the case of the processing in step S23, since the object is a rectangular area, it is not as easy as when a line segment is the object. For this reason, a triangular mesh having a bubble center as a node is first generated using Delaunay triangulation. As a result, the two bubbles connected by the sides of the triangle can be determined as adjacent bubbles.

  In addition, if the above-described addition / deletion determination is performed before the newly added bubble has moved to a stable location in the previous iteration, the newly added bubble may meet the deletion determination condition and be deleted. There is. Therefore, it is preferable that the dot automatic generation / deletion procedure in step S31 is not performed every time it is repeated, but only once every predetermined number of repetitions (about 5-10 times).

  FIG. 40 shows a dot pattern before being generated by the above-described automatic dot generation processing of the present invention, and FIG. 41 shows a dot pattern immediately after a dot is generated according to the present invention. For the new dots shown by hatching shown in FIG. 41, the position coordinates that become the most stable state are obtained in accordance with the attractive force / repulsive force from the surrounding dots, and for example, moire in a region where the distribution function changes abruptly. It is possible to suppress optical inconvenience.

Part III: Binarization of grayscale image based on mechanical scaling law In the present invention, it is possible to binarize a grayscale image satisfying a predetermined blue noise by using the above-described LDS and repulsive force relaxation method. It is. In the binarization of the grayscale image in the present invention, after the initial pattern is generated using the LDS method, the repulsive force relaxation method is applied to the initial dot pattern, and the binary of the grayscale image is obtained as a dynamic many-body problem. It is possible to provide a discrete pattern that provides a dithering pattern for optimization. Hereinafter, binarization of a grayscale image according to the present invention will be described in detail.

  The application of the grayscale image of the present invention to binarization is performed by applying a repulsive force between an arbitrary i dot and a dot adjacent to the i dot as described above, and following its mechanical time evolution. . Here, for i dots, in the present invention, m and c are constants as equations of motion,

Is used. The summation symbol on the rightmost side means a vector sum of repulsive forces over all dots adjacent to i dots. The above equation includes an attenuation term proportional to speed. Here, t ₀ is an initial time. The general solution of the above equation is for t> t ₀

Can be obtained as Here, the value of the constant ratio c / m can be arbitrarily set. When this ratio is increased, for example, when m is reduced with respect to c, specifically, it corresponds to a case where light particles are moved in a viscous fluid. In this case, since c / m is large, the exponential term is much smaller than 1 and can be ignored. For this reason, the position of the particles is preferably

It can be described by the difference equation shown by In the above formula, the unknown amount included in the term of the interaction force on the right side was replaced with the value of the past time. Speaking of the many-body problem, it is equivalent to the mean field approximation of the many-body correlation.

  In the present invention, it has been found that a good effect can be obtained by using the mechanical relaxation method in combination with LDS under the above-described conditions. Hereinafter, the binarization method of the grayscale image of the present invention is referred to as a DLDS (dynamical LDS) method. According to Ulichney (RA Ulichney, Proceedings of the IEEE, 76 (1988) 56.), the blue noise characteristic listed above as Ulichney's standard is that the closest dot spacing was originally near the principal wavelength. It was thought from the assumption that it should concentrate. For a gradation α normalized between 0 and 1 (ie, α = 1 is black and α = 0 is white), the dominant wavelength is:

Defined by Here, a is the area per dot. It is known that an isotropic dot pattern in which the closest dot interval is concentrated in the vicinity of the dominant wavelength has blue noise characteristics. In the present invention, the reach length D of the repulsive force between dots with respect to the above-described main wavelength is set to about the main wavelength. That is, let D be a repulsive force reachable range,

Set as. This is a direct rule for micro quantities, and ensures that the resulting dot pattern has blue noise characteristics while maintaining low discrepancy without complex processing such as Fourier transformation. . In the present invention, the repulsive force is an isotropic central force.

  The gradation dependence of the reach of repulsion has some degree of freedom of optimization. However, by using the above-described dominant wavelength for dynamic scaling and using the DLDS method, a binarized pattern having excellent characteristics can be obtained. This method is hereinafter referred to as a scaling DLDS method.

  The binarization method to which the above-described scaling DLDS method can be applied will be described in more detail. FIG. 42 shows a schematic diagram of binarization processing to which the present invention can be preferably applied. 42A shows the original image OP which is a grayscale image, FIG. 42B shows the density gradient for each section, and FIG. 42C shows the dot pattern generated in relation to the density gradient. Indicates. As shown in FIG. 42A, for a section covering a plurality of pixels of the original image OP, an initial dot is generated faithfully to the gradation gradient in the section, and is optimized by the scaling DLDS method. . In the original image shown in the figure, it is schematically shown that a density gradient exists as shown by the contour lines in FIG. In FIG. 42 (b), the higher the contour line, the lower the density, and the lower side, the higher the density. The dot pattern generated corresponding to the density gradient is shown in FIG. FIG. 42B and FIG. 42C have a relationship in which Pr and Pl correspond to each other.

  The scaling DLDS method of the present invention stores dot patterns in the surrounding sections, considers the repulsive field created by those dots as a fixed external field, and obtains the position coordinates of each dot by the repulsive force relaxation method. . According to the scaling DLDS method of the present invention, the joints between the sections can be substantially invisible, so that obstacles that tend to appear between the small sections can be substantially eliminated.

  In the present invention, an area including a continuous gradation change in the original image is considered as a basic section. The density is equally assigned to the gradations of 0 to 255, and the filling rate is determined for each gradation. The section includes a plurality of original pixels. In generating the initial arrangement in the section, the above-described probability extraction method is used. As a result, an initial dot pattern in harmony with the gradation change depending on the position is obtained.

  Next, the repulsion relaxation method is applied to the initial dot pattern generated using the LDS method to improve the frequency characteristics of the dot pattern. In this case, the reach range D of the repulsive force between dots is basically scaled as a function of the gradation value in relation to the scaling law with respect to the dominant wavelength. However, in the present invention, depending on the gradient of the gradation,

In this way, the reach of the repulsive force can be changed using a function related to the dominant wavelength.

  As a result of intensive studies by the inventors, when there is a gradation gradient of 0.2 / 100 or more per pixel of the dithering bitmap,

It was confirmed that the blue noise characteristic can be obtained effectively by using the function given by. FIG. 43 shows a dot pattern obtained using the repulsive force relaxation method using the above equation. As shown in FIG. 43, the interval between dots is scaled in accordance with the gradation gradient by the DLDS method of the present invention, and a good binarized image free from defects such as moire and unevenness can be obtained. Is shown.

  In the above-mentioned scaling DLDS method, the entire original image can be viewed as one small section to generate initial dots and binarization work can be performed. It is desirable to divide into two. FIG. 44 shows an embodiment in which division into small sections is used in binarization of a grayscale image in the present invention. When all the sections shown in FIG. 44 are scanned and there are no generated dots outside the boundary, the above-described self-similar boundary condition can be used to generate a dot pattern. That is, the repulsive force relaxation method can be applied by turning the initial arrangement around the boundary and adopting the repulsive field formed at the boundary as the external field. For a section having a section that has already generated a dot pattern as a boundary (for example, the left boundary of the section specified by “P3” in FIG. 44), a part of the dot pattern of the external section (the area that touches the boundary) is stored. It can be treated as a source of external fields. In this way, a seam or the like does not occur at the boundary between the sections. In the error diffusion method, there are various methods for scanning small sections in order to improve the image quality. However, in the method of the present invention, such a device is not necessary and simple software and hardware can be used. it can.

  FIG. 45 shows an enlarged view of the light and dark binary dot pattern obtained by using the self-similar boundary condition after being divided into the small sections shown in FIG. FIG. 45 includes a boundary, but is shown to be difficult to perceive. In FIG. 45, for example, a considerably steep gradation gradient is given in the lower part of FIG. 45, but the dot interval is correctly scaled by the degree of the main wavelength depending on the gradation, and there is a great optical disadvantage. It turns out that it has not occurred. As shown in FIG. 45, according to the present invention, it is possible to generate a dot pattern in harmony with surrounding dot patterns in each section, and no virtual images, moire patterns, etc. associated with image division appear at all. It is shown that a binarized image is obtained.

  The present inventors further examined the dot pattern generation in the one-to-many mask method using the above-described scaling DLDS method. A case where the present invention is applied to the one-to-many mask method shown in FIG. 5 will be described. That is, the scaling DLDS method was applied to create a dot pattern of each gradation in the one-to-many mask method. In this case, although the dot pattern of each gradation does not strictly satisfy the Mitsa-Parker condition, the present inventors have found that it can be sufficiently applied as a one-to-multiple blue mask method. It was done. Due to the characteristics of LDS as a definitive sequence used in the present invention, dot patterns between different gradations are not independent, and there is a correlation between adjacent gradation patterns. This correlation is a necessary condition for preferable binarization of the original image when the gradation changes continuously, and the scaling DLDS method of the present invention is sufficient for the one-to-multiple blue mask method. Can be applied.

  FIG. 46 is a diagram showing the correlation of LDS between different gradations. In FIG. 46, the gradations 0.32 and 0.30 are compared. In FIG. 46, when the gradation is 0.32, the dots added to the gradation 0.30 dots are indicated by ■ and the gradation 0.30 dots are indicated by □. In the figure, since the dot ■ is overwritten on the dot □, the portion that appears black is a dot added at the gradation 0.32. Such a correlation between different gradations is desirable in order to remove a pseudo image accompanying a gradation change.

  FIG. 47 shows a dot pattern relaxed by applying the repulsive force relaxation method to the dot pattern shown in FIG. By performing the rearrangement by the repulsive force relaxation method, a pattern having a well-aligned nearest dot characteristic, which is characteristic of a dot pattern having a blue noise characteristic, is generated. In FIG. 47, the overlap between two different gradations is not as clear as in the initial arrangement, but a relatively large gradation difference of 0.02 (in a specific embodiment described in the present invention, the gradation is entirely However, even though the gray level difference of 0.02 corresponds to the difference of about 5 gray levels), most of the dots are overlapped with □ and ■. . Therefore, it is shown that the pattern generated by the scaling DLDS method has very good characteristics as a one-to-multiple blue threshold mask. When this is used as a threshold mask, it is desirable to impose periodic boundary conditions in the dot relaxation process so as not to create seams at the boundaries.

  The present invention can also be applied to the generation of a blue noise dot pattern as a starting point in the conventional one-to-one blue mask generation method. In the one-to-one threshold mask generation method disclosed in the above-mentioned Mitsa-Parker et al. Paper or US Pat. No. 5,543,941 specification, a dot pattern having blue noise characteristics is used as a starting point in an intermediate gradation. . Since the dot pattern generated by the scaling DLDS method of the present invention has extremely excellent blue noise characteristics, it was confirmed that it can be used for its use.

  Furthermore, the present inventors applied the scaling DLDS method of the present invention to the generation of a dot pattern having blue noise characteristics based on Voronoi division. When the dot pattern is created by applying the probability extraction method and the scaling DLDS method in the present invention, it is necessary to obtain the interaction force on the right side in the differential expression of the equation of motion. In principle, this requires calculation for all dot pairs in the calculation area. For this reason, in the present invention, the calculation speed can be considerably increased by devising the data structure such as using a so-called bucket search method. Here, in specifying a dot pair, it is one of the preferable methods to use Voronoi division. That is, at a certain time step, Voronoi division as shown in FIG. 6 (a) is performed, and the repulsive force acting on a certain dot is calculated only for those existing inside the Voronoi polygon adjacent to the Voronoi polygon to which the dot belongs. To do. It has been confirmed that this approximation is a sufficiently realistic approximation in reproducing a continuous tone image.

  The present inventors further examined the application of the scaling DLDS method of the present invention to a method of creating a dot pattern without unevenness based on Hiller's relaxation method. It has been found that by performing the initial position generation in the method of Hiller et al. Using the scaling DLDS method of the present invention, it is possible to avoid the unevenness that has been considered to be unavoidable until now.

Part IV: Manufacture of optical members, sidelight devices, transmissive liquid crystal display devices and photomasks for CCDs using discrete patterns according to the present invention We have optical optics including discrete patterns using LDS and repulsive force relaxation methods Further studies were made on optical members such as members, light guide plates, sidelight units, CCD photomasks, and liquid crystal display devices and discrete pattern generation systems. The discrete pattern generated using the method for generating a discrete pattern described above in the present invention can be patterned on a plane or a solid by various methods, and can be an object including the discrete pattern of the present invention. Was found. FIG. 48 illustrates an embodiment of a computer system used to generate the discrete pattern of the present invention.

  The computer system shown in FIG. 48 stores a central processing unit (CPU) for executing the above-described discrete pattern generation method and data necessary for generating the discrete pattern such as various parameters and dot position coordinates. A computer 22 including at least a storage unit, a display unit 24 for performing necessary display, an input unit 26 for inputting various parameters, and a program for executing the discrete pattern generation method of the present invention. Read / write means 28 for reading a program from a recorded storage medium and writing digital data corresponding to the generated discrete pattern in a predetermined format.

  In order to manufacture an object including a discrete pattern from the discrete pattern generated by the discrete pattern generation method of the present invention, first, the discrete pattern obtained by the computer 22 is output as a hard copy by the printer means 29. Or read as digital data: output to writing means 28. When a discrete pattern is output as a hard copy, the recording paper itself on which the discrete pattern of the present invention is recorded is used as an object including the discrete pattern as it is. For example, a discrete pattern obtained by the computer 22 by a printer means 29 such as an ink jet printer or a color printer using an electrophotographic method is used as a dithering pattern, and is directly used as a pattern receiving element such as paper or a plastic sheet. It can also be printed. Further, as the printer unit, a printer unit that uses a laser in the visible light region can be exemplified.

  As a printer means using such a visible light laser, digital data is directly used to directly output to a photosensitive film used as a discrete pattern receiving element by a method such as a laser, and development and fixing. The thing which can be set as the object which can be used as optical members later, such as various photomasks, can be mentioned. Such a photomask can be provided for micro-photolithography using a photoresist, and makes it possible to apply the discrete pattern of the present invention to a photosensitive material. Furthermore, in the present invention, the pattern receiving element may include a photosensitive resin. The discrete pattern generation system of the present invention cures the photosensitive resin in layers with a visible light laser, and directly applies the discrete pattern of the present invention. It is also possible to manufacture a three-dimensional object including

  In FIG. 49, the top view of the photomask which is embodiment of the optical member containing the discrete pattern of this invention is shown. Photomask modulates light intensity by adding a photosensitive material such as silver halide to an optical sheet such as gelatin sheet, polyethylene terephthalate or composite sheet configured to contain optically appropriate properties. It is an optical member capable of This photomask can give a three-dimensional structure using a microlithography method suitable for a photosensitive material such as a positive photoresist or a negative photoresist.

  FIG. 50 shows a light guide plate 30 having a discrete pattern formed by micro-photolithography using the discrete pattern of the present invention. In the light guide plate shown in FIG. 50, light rays emitted from a fluorescent tube CFL (not shown) are scattered in the direction of arrow C. In the light guide plate 30 shown in FIG. 50, a discrete pattern 31 that scatters light rays for this purpose is formed on the surface of the light guide plate 30 opposite to the side from which the light rays indicated by arrow C are emitted. The discrete pattern 31 is generated by the discrete pattern generation method of the present invention, and is formed by microlithography using a photomask.

  In the embodiment shown in FIG. 50, the discrete pattern 31 has a quadrangular pyramid shape and is formed from dots 32 formed as concave portions in the light guide plate 30. The shape and size of the dot that can be used in the present invention is not particularly limited, and can be selected according to a specific application, for example, polygon, circle, square, rectangle, oval, truncated cone, It can be used by appropriately selecting from at least one two-dimensional or three-dimensional shape selected from a group including a polygon base. Further, in the present invention, a plurality of these shapes can be mixed and used, and the scattering property can be adjusted by changing the arrangement with respect to the fluorescent tube CFL.

  FIG. 51 is an enlarged top view showing in detail the dots 32 shown in FIG. 51, the long side 32a faces the fluorescent tube CFL, and the short side 32b is arranged perpendicular to the fluorescent tube CFL. These dots 32 are formed over the entire surface of the light guide plate 30 so that light rays can be scattered. However, in order to improve the scattering property, particularly at the four corners of the light guide plate 30, the probability extraction method used in the present invention is used for the dots 32 of the light guide plate shown in FIG. A filling rate distribution is given to the dots 32. For example, the filling rate distribution is increased so that the filling rate is low in the central portion, the filling rate is 50% or more toward the four corners, and about 67% in the embodiment shown in FIG.

  FIG. 52 shows a more detailed top view (a) of the dot 32 shown in FIG. 51 and a cross-sectional view (b) along the line DD. In the embodiment shown in FIG. 51, the dots 32 have a quadrangular pyramid shape. It is shown that the dot 32 has a three-dimensional shape formed by forming a recess with respect to the light guide plate 30, and the recess depth d is 8 μm. Further, the long side lg of the dot 32 is 100 μm in the illustrated embodiment, and the short side sht is 30 μm. In addition, the gradient β of the dots 32 formed as a quadrangular pyramid can be appropriately set in relation to the depth d so that necessary scattering properties can be given.

  FIG. 53 is a diagram showing the filling rate distribution of the dots 32 described above in the entire light guide plate 30 with contour lines. In the embodiment of the light guide plate 30 shown in FIG. 53, the filling rate is about 10% at the center of the light guide plate 30 and the filling rate is increased to increase the scattering effect toward the four corners of the light guide plate 30. In particular, in the corner portion 34 away from the fluorescent tube CFL indicated by a broken line, it is shown that the filling rate is increased to exceed 70%. In the discrete pattern generation method of the present invention, the LDS method and the repulsive force relaxation method are used, and the filling rate distribution is generated by the probability extraction method. Therefore, even if the filling rate distribution as described above is generated, the visible boundary is confirmed. A good light guide plate that cannot be provided can be provided.

  FIG. 54 is a diagram showing another embodiment of the light guide plate 30 used as the optical member of the present invention. In the light guide plate 30 shown in FIG. 54, the discrete pattern 31 of the present invention is formed by a plurality of dots 36 on the surface opposite to the light emitting side. In the embodiment shown in FIG. 54, the dots 36 are in the shape of a truncated cone. In the particular embodiment shown in FIG. 54, the truncated cone has a diameter of 100 μm and a height of 15 μm. Moreover, since it is a truncated cone, the dot 36 itself has no particular directionality with respect to the fluorescent tube CFL.

  FIG. 55 is a view showing still another embodiment of the light guide plate 30 of the present invention. The light guide plate 30 shown in FIG. 55 is provided with the discrete pattern of the present invention including dots 38 and dots 40 that are different from each other in arrangement with respect to the fluorescent tube CFL, and these are regions (a), (b), and regions. It is shown that it is classified as (c). The dot 38 is a quadrangular pyramid formed as a recess with respect to the light guide plate, the long side is 100 μm, the short side perpendicular to the fluorescent tube CFL is 70 μm, and the dot 40 has the same shape as the dot 38. However, the arrangement with respect to the fluorescent tube CFL is formed by rotating around the dot center. In the embodiment of the light guide plate 30 shown in FIG. 55, the center coordinates of the plurality of dots described above can be provided by using the discrete pattern generation method of the present invention, and then the dots can be rotated. . In the embodiment shown in FIG. 55, a configuration is adopted in which the scattering property at the corner is adjusted by forming a region in which dots 38 and dots 40 are mixed as region (b). In the embodiment shown in FIG. 55, it is possible to partially change the distribution of light emitted from the light guide plate 30 by adjusting only the relative arrangement of dots to the fluorescent tube CFL.

  FIG. 56 shows a partial sectional view of a sidelight device including the light guide plate 30 of the present invention. The sidelight device of the present invention can be used, for example, as a backlight unit of a transmissive liquid crystal display device, and the backlight unit shown in FIG. 56 further includes optical members such as a diffusion sheet and a prism sheet. However, for convenience of explanation, components that are not essential in the present invention such as a diffusion sheet and a prism sheet are omitted. As shown in FIG. 56, the side light device of the present invention covers the light source 42, the lamp socket 44 for holding the light source, the light guide plate 46 on which the discrete pattern of the present invention is formed, and the light source 42. , And a reflector 48 for efficiently allowing the light beam from the light source 42 to enter the light guide plate 46. As the light source described above, a fluorescent tube CFL can be used.

  A reflection sheet 50 made of a highly reflective material such as silver is disposed adjacent to the reflector 48 and below the light guide plate 46, and the light from the light source 42 is more efficiently indicated by the arrow E. It is set as the structure discharge | released to the direction shown. In the reflection sheet 50 used in the sidelight device shown in FIG. 55, a Newton ring is formed by the reflection sheet 50 being in close contact with the light guide plate 46, and in addition to the moire pattern when configured as a backlight unit. In some cases, it is necessary to prevent the deterioration of the quality due to the Newton ring pattern.

  Particularly in recent years, in order to increase the reflection efficiency of a backlight unit in a transmissive liquid crystal display device, a metal such as silver deposited on a sheet of polyethylene terephthalate by sputtering has been used. When the reflection sheet 50 having such a high reflectance is brought into close contact with the light guide plate 46, Newton ring-like stripe patterns are observed as described above due to the deflection of the reflection sheet 50, and the display quality of the transmissive liquid crystal display device is significantly deteriorated. I will let you. In order to suppress this interference fringe, it is effective to form a gap of about 20 μm between the reflection sheet 50 and the light guide plate.

  FIG. 57 shows a top view of the reflection sheet 50 described in FIG. As shown in FIG. 57, it is shown that the reflection sheet 50 is formed with unevenness including a discrete pattern generated by including the dots 52 according to the present invention. The dots 52 forming the discrete pattern formed on the reflective sheet 50 are formed by a method such as screen printing using UV curable ink, and the optical density is lowered so as not to deteriorate the optical characteristics more than necessary. ing. In the embodiment shown in FIG. 57, the discrete pattern is a dot 52 having a diameter of 100 μm and a height of about 15 μm.

  FIG. 58 shows a schematic cross-sectional view of the reflection sheet 50 including a discrete pattern generated according to the present invention. As shown in FIG. 58, the reflective sheet 50 formed to include the discrete pattern of the present invention includes a reflective layer 56 such as silver deposited by sputtering on a base material 54 such as polyethylene terephthalate, and the reflective layer 56. And the discrete pattern of the present invention comprising the dots 52 formed thereon. A light guide plate (not shown) is brought into contact with the reflective sheet 50 via the dots 52 to prevent inappropriate contact or proximity between the reflective sheet 50 and the light guide plate, and a Newton ring-like pattern is formed. Occurrence is reduced.

  If the filling rate of the dots 52 forming the discrete pattern described above is increased too much, the reflection efficiency is lowered. Therefore, the filling rate is preferably about 10% or less. In the embodiment shown in FIGS. 57 and 58, the filling rate is uniform over the entire surface of the reflection sheet and is 2.5%.

  The discrete pattern generated according to the present invention sufficiently satisfies both uniformity and randomness, so that overlapping dots are not visually observed or uneven. In addition, the light guide plate made of acrylic resin or the like is a dot made of resin formed by screen printing, so that it is possible to reduce inconvenience due to damage to the lower portion of the light guide plate.

  FIG. 59 is an exploded perspective view of a transmissive liquid crystal display device using a light guide plate including a discrete pattern according to the present invention. The transmissive liquid crystal display device 58 of the embodiment of the present invention shown in FIG. 59 includes an upper frame 62 that defines a display window 60 for defining an effective screen of the transmissive liquid crystal display device 58, and the present invention. A backlight unit 64 using a sidelight device, a liquid crystal display panel 66 disposed between the upper frame 62 and the backlight unit 64, a spacer 68, a diffusion sheet 70, and a prism sheet 72 are provided. It is configured to include.

  The backlight unit 64 is placed on the lower case 74 and is held integrally with the upper frame 62 to constitute a transmissive liquid crystal display device 58. The backlight unit in the transmissive liquid crystal display device shown in FIG. 59 is configured to include the light guide plate 76 of the present invention, and efficiently emits light emitted from the fluorescent tube CFL, changes in luminance at four corners, The liquid crystal display panel 66 is released without causing inconvenience such as a moiré pattern, and a good display can be performed.

  FIG. 60 shows a light-shielding sheet which is still another embodiment of the present invention. Such a light shielding sheet is used in a liquid crystal display device, and is used when the transmittance of the scattering plate is to be changed depending on the location for the purpose of improving the luminance uniformity of the liquid crystal display device. For example, in a so-called edge light type configuration, a bright and dark pattern called a bright line often occurs at the time of a lamp. This can be largely eliminated by locally reducing the transmittance at the bright line portion. To give this effect to the light-shielding sheet by printing a scattering plate or a transparent film that makes up a liquid crystal display device with white ink dissolved in a UV curable resin and forming a dot pattern. Can do. In particular, this type of dot pattern is characterized by a large filling rate gradient. In the stamper pattern of the light guide plate as described above, the filling rate gradient in the preferred embodiment is at most about 0.4% per dot diameter, but in light-shielding printing, it is necessary to add gradation with dots to a width of several millimeters. A filling rate gradient of several to 10 times is required. For this reason, the dot automatic generation / deletion method of the present invention can be applied particularly effectively.

  The light shielding sheet 80 formed according to the present invention shown in FIG. 60 is provided with a low transmittance region 80a along the end portion in order to erase the bright line near the lamp arranged in parallel with the surface on the front side. ing. Also, on the opposite side of the lamp, the luminance often increases in that region due to light leakage from the edge of the light guide plate and its reflected light. To correct this, a region with low transmittance in the vicinity 80b is provided along the end. Since the gradient of the filling rate distribution is large between the region 80c between these and the regions 80a and 80b, the dots tend to be localized. Here, by using the dot automatic generation / deletion method according to the present invention, it is possible to eliminate dot localization and provide a good discrete pattern.

  FIG. 61 shows an embodiment of a photomask 82 for a CCD lens, which is still another embodiment to which the present invention can be applied. The photomask 82 of the CCD lens is also used as a photomask by controlling the transmittance using a minute dot pattern as an optical modulation element. When the discrete pattern of the present invention is applied as a photomask, a method of creating a photomask for a micro CCD lens by arranging dots randomly and having a centrally symmetric density distribution is described. ing. The reason why random dot arrangement is avoided is to prevent the photomask from acting as a diffraction grating and causing problems such as coloring of transmitted light.

  The photomask 82 shown in FIG. 61 is formed with the discrete pattern 88 of the present invention formed by including the dots 86 of the present invention on the light transmissive support 84. The discrete pattern shown in FIG. 61 transfers the density gradient of dots as lens irregularities, so that the filling rate gradient is relatively large, and the filling rate gradient per dot diameter is about the same as that of the light-shielding printing pattern. . By using the discrete pattern of the present invention as a photomask, the problem of unevenness in transmitted light, which is inevitable in the prior art, does not occur, and a CCD lens can be manufactured with high accuracy.

  Moreover, the filter for equalizing the light for photolithography was created using the discrete pattern of this invention. The discrete pattern given by the scaling DLDS method of the present invention can provide a dot pattern that satisfies the restrictions on the dominant wavelength at any gradation while having a gradation gradient, and can provide a filter with good characteristics. Thus, it was possible to provide a filter with good characteristics in which inconveniences such as unevenness in transmitted light (including virtual images associated with partitioning) and coloring occurred.

  As described above, the discrete pattern of the present invention can be used for a member that can give various optical characteristics by any of light scattering, light transmission, and light absorption, and such optical characteristics. Specific examples of the optical member capable of providing a light guide plate include a light guide plate, a scattering plate, a dithering pattern, a photomask pattern for lithography, and the like.

  In addition, the discrete pattern of the present invention can be used for an anti-slip pattern such as a tire or a shoe sole that does not require optical characteristics.

  Further, the discrete pattern generation method of the present invention can be described in various programming languages, for example, C language, and the code describing the program of the present invention is a floppy (registered trademark) disk, hard disk, compact disk, etc. It can be stored in a storage medium such as a disk, a magneto-optical disk, a digital versatile disk (DVD), or a magnetic tape.

As mentioned above, according to the present invention,
(1) Arbitrary filling rate distribution from low filling rate to high filling rate of 50% or more, without causing optical quality degradation,
(2) No overlap between dots or abnormal approach,
(3) It is possible to provide a method for generating a discrete pattern that maintains sufficient randomness while maintaining high uniformity. The optical properties of the light guide plate, the scattering plate, the sidelight device, etc. including the discrete pattern according to the present invention can significantly improve the uniformity of light irradiation, the luminance distribution, etc., and the discrete pattern An optical member, a light guide plate, a sidelight device, a transmissive liquid crystal display device, a method for generating the discrete pattern, a program for generating the discrete pattern, and a computer-readable program for generating the discrete pattern are recorded A computer-readable recording medium and a discrete pattern generation system are provided.

  In addition, according to the present invention, when a steep filling rate gradient is required such that a change in filling rate per dot diameter exceeds 1%, this method is used to follow an arbitrary filling rate distribution. It is possible to obtain a discrete pattern with good characteristics. The discrete pattern generated in this manner can be suitably applied to a light shielding sheet for a liquid crystal display device, and a light shielding sheet for a liquid crystal display device having uniform luminance can be created. Similarly, by applying the present invention to a photomask for producing a minute CCD lens, a highly accurate CCD lens can be produced.

  Further, in the present invention, halftones are used in devices such as an ink jet printer that outputs a continuous tone black-and-white or color image after being converted into binary or multi-valued density, and laser beam printers, facsimiles, and printing machines. It can be expressed preferably. In other words, a uniform and smooth halftone can be expressed without any obstacles such as rough feeling, unevenness, and regular patterns. Generally speaking, the present invention makes it possible to provide a well-distributed and uniformly non-uniform discrete pattern, the discrete pattern of the present invention comprising an optical modulation element having good characteristics and the optical modulation element It can be said that it is possible to provide an optical member including a binary pattern of a grayscale image.

The figure which showed the dot pattern produced | generated by the conventional pseudorandom number generation method. The disassembled perspective view which showed the structure of the conventional transmissive liquid crystal display device. The disassembled perspective view which showed the reflection type liquid crystal display device to which the conventional random discrete pattern was applied. The figure which showed the binarization process of the conventional grayscale image. The figure which showed the binarization process of the conventional one-to-many mask method. The figure which showed the binarization process of the grayscale image using the conventional regular dither method. The flowchart of the LDS method used in this invention. The figure which showed the production | generation method of the point sequence in this invention. The figure explaining the production | generation of the initial position by LDS method in this invention. The figure which showed the initial position produced | generated in the case of 50 dots using the LDS method by this invention. The figure which showed the initial position produced | generated by LDS method, and the initial position produced | generated by pseudorandom numbers. The figure which showed the initial position which has the low discrepancy produced | generated using GLP method in this invention. Schematic of repulsive force relaxation method used in the present invention. Schematic which showed the coordinate system used for the repulsive force relaxation method in this invention. The figure which showed the dependence with respect to r of a repulsive force model. The figure which showed the relationship of each parameter of a repulsion model. The figure which showed the repulsive force field obtained using the ellipse model. The figure which showed the repulsion field obtained using the rectangular model. The figure which showed the convergence of V with respect to r. The flowchart of the repulsive force relaxation method used in this invention. The figure which showed the discrete pattern produced | generated by the LDS method and the repulsive force relaxation method, and the discrete pattern produced | generated by the pseudorandom numbers. The figure which showed the discrete pattern in this invention obtained by the process shown in FIG. The figure which showed the discrete pattern of this invention produced | generated using the rectangular model as a repulsion by the process shown in FIG. The figure which showed the self-boundary condition used by this invention. The figure which showed the filling rate distribution in a discrete pattern. The figure which showed the discrete pattern obtained using the probability extraction method. The figure which plotted the discrepancy of the discrete pattern produced | generated by each method with respect to the number of dots. The figure which showed the uniformity of the discrete pattern by the method using pseudorandom numbers. The figure which showed the uniformity of the discrete pattern by the LDS method and the repulsive force relaxation method. The figure which showed the discrepancy measured from the discrete pattern. The figure which showed the square lattice at the time of arrange | positioning a dot at each lattice point of a square lattice as a 5x5 structure. The figure which showed the value of g obtained about a square lattice. The figure which showed the discrete pattern in the abrupt part of the filling rate change produced | generated by the repulsion relaxation method. The figure which showed the rectangular area | region used in this invention, and a filling rate distribution function. The figure which defined the bubble used in this invention. The figure which showed the area division | segmentation method for the dot automatic generation in this invention The figure which showed the force made to act between the bubbles used in this invention. The flowchart of the dot automatic production | generation / automatic deletion method of this invention. The figure which showed embodiment of the rectangular area | region used when performing dot automatic generation and automatic deletion in this invention, and a filling rate distribution function. The figure which showed the previous dot pattern produced | generated by the dot automatic production | generation process of this invention. The figure which showed the dot pattern immediately after a dot was added according to this invention. Schematic of the binarization process which can apply this invention suitably. The figure which showed the dot pattern obtained using the repulsion relaxation method by this invention. The figure which showed embodiment in the case of using the division | segmentation into the subsection in the binarization of the gray image in this invention. The figure which showed the binarization pattern of the grayscale image obtained in this invention. The figure which showed the correlation of LDS between the different gradations of the gradations 0.32 and 0.30. The figure which showed the dot pattern relaxed by applying the repulsive force relaxation method with respect to the dot pattern shown in FIG. The figure which showed the computer system used in this invention. The figure which showed embodiment of the optical member of this invention. The figure which showed embodiment of the optical member of this invention. FIG. 33 is an enlarged detailed view of dots of the discrete pattern of FIG. 32. FIG. 34 is a more detailed enlarged detail view of the dots shown in FIG. 33. The figure which showed the filling rate distribution in embodiment of this invention. The figure which showed the dot shape in another embodiment of this invention. The figure which showed another embodiment of this invention. Sectional drawing which showed embodiment of the sidelight apparatus of this invention. The figure which showed the reflective sheet in which the discrete pattern of this invention was formed. FIG. 58 is an enlarged cross-sectional view of the reflection sheet shown in FIG. 57. 1 is an exploded perspective view of a transmissive liquid crystal display device of the present invention. The perspective view of the light shielding sheet in which the discrete pattern was formed by this invention. The figure which showed the photomask for CCD in which the discrete pattern was formed by this invention.

Explanation of symbols

10 ... dot 12 ... dot 14 ... dot 16 ... dot 18 ... dot 19 ... dot 22 ... computer 24 ... display means 26 ... input means 28 ... read / write means 29 ... printer means 30 ... light guide plate 31 ... discrete pattern 32 ... Dot 34 ... corner 36 of light guide plate ... dot 38 ... dot 40 ... dot 42 ... light source 44 ... lamp socket 46 ... light guide plate 48 ... reflector 50 ... reflection sheet 52 ... dot 54 ... substrate 56 ... reflection layer 58 ... transmission type Liquid crystal display 60 ... Display window 62 ... Upper frame 64 ... Backlight unit 66 ... Liquid crystal display panel 68 ... Spacer 70 ... Diffusion sheet 72 ... Prism sheet 74 ... Lower case 76 ... Light guide plate 80 ... Light shielding sheet 82 ... Photomask 84 ... light transmissive support 86 ... dot 88 ... discrete pattern

Claims (11)

A discrete pattern formed from discrete dots arranged two-dimensionally using a computer system, in which the light is passively controlled and guided by light scattering, light transmission, or light absorption. A method for generating a discrete pattern which is an optical plate, a scattering plate, a dithering pattern, a photomask pattern for lithography,
Providing a predetermined compartment;
Determining the number of dots to be arranged in the predetermined section;
The position coordinates of the dot, horizontal Lx, the number of dots the dots included in a rectangular form the compartments of the vertical Ly is the case of 4000 or less, the following equation (1)
(In the above formula (1), N is the number of dots included in the section, and D is an arbitrary coordinate point (x, y) from the reference coordinates (0, 0) among the N dots) ), The number of dots included in the rectangular area with the line segment up to the diagonal line as A (x, y) is given by the following equation (2)):
And the following formula (3)
( ₁ ) is ₁ or less (in (3), g ₁ is an average radial distribution function g (r) of each dot in the region in the range of r _{1 to} r _2. divided by the integrated value, g _av is .r ₁ and r ₂ is the average value at r ₁ or r ₂ the following range of g ₁ are square lattice dots to meet a given fill factor The values of the lattice constant Δr are 1 and 4 respectively, and the dot filling rate is obtained by multiplying the square of the maximum dot diameter by the number of dots. Divided by the area of the area.)
Setting the determined position coordinates as an initial position;
Changing the position coordinates of the dots so that the dots for which the position coordinates are determined do not overlap each other;
Generating and storing a first generator matrix for each coordinate axis;
Generating and storing a second generator matrix by a product of the first generator matrix and a scramble matrix having a lower triangular element having a random value;
Generating a binary expansion of one natural number n in a set of N natural numbers respectively corresponding to N dots as a first vector;
Generating a second vector from a product of the first vector and the second generator matrix;
Generating the position coordinates of the dots with the elements of the second vector for each coordinate as values of each digit of a binary decimal;
a step of generating position coordinates until n is increased by a predetermined increment and the total number of point coordinates repeatedly generated is equal to the number N of dots in the section.   The method of claim 1, further comprising: adjoining the predetermined partition using a plurality of predetermined boundary conditions to generate a discrete pattern of a predetermined size.   The discrete pattern generation method according to claim 1, wherein the step of changing the position coordinates includes a step of calculating a repulsive force between the adjacent dots related to a size and a distance of the adjacent dots. The step of changing the position coordinates includes:
Calculating a repulsive force from other dots present in the vicinity of the dot for a predetermined dot;
Displacing the predetermined dot according to the magnitude of the repulsive force;
Calculating a repulsive force with respect to the predetermined dot displaced;
The method of generating a discrete pattern according to claim 1, further comprising: summing up the repulsive force for dots within a predetermined range. The step of changing the position coordinate further includes the step of calculating a difference between a sum of the latest calculated repulsive forces and a sum of the repulsive forces calculated immediately before the sum of the most recently calculated repulsive forces. The discrete pattern generation method according to claim 4, comprising a step of executing until a condition is satisfied. The repulsive force is constant below a predetermined value with respect to the distance between the dots, and decreases when the distance between the dots exceeds a predetermined value when the distance between the dots exceeds a predetermined value. 4. The discrete pattern generation method according to 4. The discrete pattern generation method according to claim 1 , wherein the low discrepancy sequence method gives an initial position of the dot using the equation (2) .   The program for a computer to perform the step of any one of Claims 1-7.   A computer-readable recording medium on which the program according to claim 8 is recorded. Discrete patterns for passively controlling light rays by light scattering, light transmission, or light absorption to produce discrete patterns that are light guide plates, scattering plates, dithering patterns, photomask patterns for lithography A generation system,
Means for providing a discrete pattern by each step according to claims 1-7;
Storage means for recording position coordinates of dots forming the discrete pattern;
Printer means for outputting the position coordinates contained in the storage means;
A discrete pattern generation system including a pattern receiving element on which a discrete pattern is formed by the printer means. The discrete pattern generation system according to claim 10, wherein the discrete pattern is a light intensity modulation element of an optical member or a binarized pattern of a grayscale image.