CN102075664B - A method and system for constructing non-progressive dots - Google Patents

Abstract

The invention relates to a method and system for constructing non-progressive screen dots, belonging to the field of printing and publishing. Multiple layers of annularly embedded non-progressive screen dots are generated through a defined function cluster F, a direct network-hanging method or screen dot optimizing method by using collimation errors that deep color is easy to invade light color on the vision of the human body on the basis of network hanging of the non-progressive screen dots. By adopting the method and the system disclosed by the invention, the printing effect is bright, the progressive effect is smooth and thin, and the new screen dot greatly reduces the continuous area of ink, therefore, the ink consumption is greatly lowered, the printing cost is saved and the environment is protected.

Description

A method and system for constructing non-progressive dots

technical field

The invention belongs to the field of publishing and printing, and in particular relates to a method and system for constructing non-progressive dots.

Background technique

In color printing, CMYK4 color overprinting is often used, relying on the overprinting of cyan, magenta, yellow, and black pigments to produce color images. These 4 basic colors are combined to present up to 8 colors: red, green, blue, cyan, magenta, yellow, black, and white, and a color image contains far more than 8 colors. By color-separating and correcting the color image into a 4-color grayscale image, the image after color separation is 4 grayscale images with continuous tones. In 4-color printing, only one ink can be used for each printing, and the ink concentration remains unchanged. In order to obtain continuous tone when printing, the grayscale image needs to be screened. Screening, also called screening, is the process of decomposing continuous tone images into dots. For the screened image, the size and density of the dots reflect the depth of the actual color of the image.

Based on human vision, when viewing an image from a close distance, the dots and the space around them create continuous tone artifacts, with larger dots appearing darker and smaller dots appearing brighter; densely dotted areas appear Dark, sparsely dotted areas look bright.

With the continuous improvement of printing paper flatness, improvement of paper ink adsorption characteristics, printing equipment precision and other technologies, the accuracy of dot processing has been greatly improved at present. At present, the dots of the simple mesh shape are solid, and the larger the dots are, the higher the thickness of the accumulated ink is.

The improvement of printing requirements for image accuracy requires not only to provide printing accuracy, but also to reduce the amount of ink used and be more environmentally friendly.

Among the dot generation technologies mentioned in existing documents or systems, there is no technology or method for constructing non-progressive dots for progressive dots.

Contents of the invention

Aiming at the problems existing in the prior art, the object of the present invention is to provide a method and system for constructing non-progressive dots through progressive dots. The method and system can be based on the generation of existing progressive dots, by defining the function cluster F , within its dot range, construct nested ring-shaped dots, that is, on the basis of enhancing the visual color and making it brighter, reduce the ink accumulation surface, thereby greatly reducing ink consumption.

The object of the present invention is achieved like this, a kind of method for constructing non-progressive network, comprises the following steps:

Step 1, first determine the progressive dot hanging area, and then obtain the gradient ratio along the progressive direction in the progressive dot hanging area;

Step 2, judging whether the gradient ratio along the gradual direction is greater than the set threshold, if yes, then proceed to step 3 to start constructing non-progressive dots, otherwise the process ends;

Step 3: Construct non-progressive dots, and construct nested ring-shaped dots within the range of progressive dots by defining a function cluster F.

Further, in Step 3, construct non-progressive outlets by directly hanging the net, the specific method is:

Construct the non-progressive network function Ψ Φ _{, F} (P) to generate the non-progressive network through the function cluster F and the progressive network function Φ, and the non-progressive network function Ψ _{Φ, F} (P)={Ψ _{Φ , F} (P) _ij }, said Ψ _{Φ, F} (P) _ij is a direct network function, Ψ _{Φ, F} (P) _ij = ∑ _{k = 0} ^n-1 (-1) ^k Φ(F _k ( P)) _ij , where F _k (P) _ij =F _k (P _ij ).

Or, in step 3, construct non-progressive dots by way of dot optimization, the specific method is:

Firstly, progressive dots are directly generated according to the progressive dot hanging function Φ, and then new dots are generated according to the function cluster F, and XOR superposition is performed on the progressive dots to generate new non-progressive dots.

Furthermore, when performing XOR superposition, the area, size, and gradient direction of the two images are consistent with the angle of the hanging net.

Further, the function cluster F={F ₀ , F ₁ , ..., F _n-1 }, n is an integer greater than 1, and F ₀ , F ₁ , ..., F _n-1 are all A continuous function whose definition domain is [0, 1] and value range is [0, 1], satisfying F ₀ (x)≥F ₁ (x)≥...≥F _n-1 (x), and ∑ _{k= 0} ^n-1 (-1) ^k F _k (x) = x, where 0≤x≤1;

The definition of the progressive dot-hanging function Φ is as follows: for a certain discrete space Ω={(i, j)|0≤i<Width, 0≤j<Height}, the grayscale image space on it is defined as I _G = {Ω→[0, 1]}, the binary image space is I ₂ ={Ω→{0, 1}}, if the grayscale image P, P'∈I _G , and for any i, j there exists P _ij ≤P' _ij , then it is defined as P≤P', and the progressive grid linking function Φ: I _G →I ₂ , satisfying Φ(P)≤Φ(P');

The grayscale image P=∑ _ij α(P _ij ), where α(P _ij ) is the brightness of the pixel with coordinates (i, j) in the grayscale image P.

Further, the threshold described in step 2 is [α(P _i'j' )-α(P _ij )]/α(P _i'j' ), which is greater than or equal to 25% and less than or equal to 1, where P _i'j 'is the maximum brightness pixel point in the grayscale image P, and P _ij is the minimum brightness pixel point in the grayscale image P.

Furthermore, the function cluster F supports recursive calls, that is, a new dot is generated on the basis of the progressive dot, and a second dot is generated based on the new dot once again, so that it stops until the ink ratio of the dot generated by the function cluster F exceeds the limit value, and then reverses the difference. Or superimposed to generate non-progressive dots, that is, the XOR of the second dot and the first dot, and the XOR of the last dot and the progressive dot to generate a non-progressive dot.

A system for constructing non-progressive dots, comprising the following modules:

Progressive dot construction module, used to determine the progressive dot hanging area, and then obtain the gradient ratio along the progressive direction in the progressive dot hanging area;

Threshold judgment module, used to judge whether the gradient ratio along the gradual direction is greater than the set threshold;

The non-progressive dot construction module is used for constructing non-progressive dots when the gradient ratio is greater than a set threshold, by defining a function cluster F to construct nested ring dots within the range of progressive dots.

The effect of the present invention is: the present invention hangs the network by defining the function cluster F, or optimizes progressive network point, generates the network point of new circular multi-layer nesting, and new network point has following remarkable effect:

1) By increasing the whiteness in the dots, the bright effect of the picture is enhanced, and the image is more delicate and smooth;

2) The new dot shape reduces the continuous area and reduces the amount of ink used in specific printing;

3) According to the characteristics of ink and paper, the new dots can flexibly adjust the annular interval area of dots, which can widely adapt to the needs of various papers and inks.

The reason why the present invention has the above-mentioned remarkable effects is that the present invention only adds the processing of the function cluster F to the original hanging net processing, the processing method is simple, and does not affect the mesh shape and mesh angle in the original hanging net processing. And other settings, it is easy to upgrade and improve the original hanging net processing, and the application is very simple. Moreover, the above-mentioned function cluster F is a processing function accumulated through a large number of practical tests, which can effectively enhance the bright effect of the picture, make the picture more delicate and smooth, and save ink significantly. With the development requirements of higher printing precision and more environmental protection, this technology transforms the original screen hanging technology and will become the mainstream dot structure technology in printing in the future.

Description of drawings

Fig. 1 is a schematic flow chart of a method for constructing non-progressive dots in an embodiment of the present invention;

Figure 2 is a schematic diagram of the comparison between the size of the circular dot and the thickness of the ink;

Figure 3 is a continuous tone image;

Fig. 4 is the image after hanging the net of pure circle dots;

Fig. 5 is the image after hanging the net of the circular dot image.

Detailed ways

The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments.

A system for constructing non-progressive dots, comprising the following modules:

Progressive dot construction module, used to determine the progressive dot hanging area, and then obtain the gradient ratio along the progressive direction in the progressive dot hanging area;

Threshold judgment module, used to judge whether the gradient ratio along the gradual direction is greater than the set threshold;

The non-progressive dot construction module is used for constructing non-progressive dots when the gradient ratio is greater than a set threshold.

In the present embodiment, described non-progressive dot construction module is also provided with following submodules:

Construct the submodule of the non-progressive network by directly hanging the network mode, and concrete method is: construct the non-progressive network function Ψ _{Φ, F} (P) to generate the non-progressive network by the function cluster F and the network function Φ of the progressive network, and generate the non-progressive network, described Non-progressive network point hanging network function Ψ _{Φ, F} (P) = {Ψ _{Φ, F} (P) _ij }, said Ψ _{Φ, F} (P) _ij is a direct hanging network function, Ψ _{Φ, F} (P) _ij = ∑ _k＝0 ^n-1 (-1) ^k Φ(F _k (P)) _ij , where F _k (P) _ij ＝F _k (P _ij );

The sub-modules of non-progressive dots are constructed by means of dot optimization. The specific method is: firstly, the progressive dots are directly generated according to the progressive dot hanging function Φ, and then new dots are generated according to the function cluster F, and XOR superposition is performed on the progressive dots to obtain Generates new non-progressive dots.

Embodiment one:

The present embodiment is an example to generate a circular dot with a gradient ratio of 100%. Its flow process is as shown in Figure 1. A method for constructing non-progressive dots includes the following steps:

First determine the progressive network dot hanging network area (step S11), obtain the gradual change ratio along the progressive direction in the progressive network dot hanging network zone to be 100%, wherein said gradual change ratio along the progressive direction refers to minimum and maximum ink ratio along the progressive direction The ratio of the ink mentioned above is the percentage of the ink area and the blank area in the area where the dot is located. The gradient ratio varies depending on the ink and paper. In practice, the empirical value is generally set according to the printing parameters such as ink and paper. Generally, is greater than or equal to 25% and less than or equal to 1.

Step S12, judging whether the gradation ratio is greater than the threshold value, if yes, it meets the construction conditions for non-gradation dots. Generally speaking, the threshold is [α(P _i'j' )-α(P _ij )]/α(P _i'j' ), and its value is greater than or equal to 25%, less than or equal to 1, wherein, P _{i 'j'} is the maximum brightness pixel in the grayscale image P, P _ij is the minimum brightness pixel in the grayscale image P.

In this embodiment, the gradation ratio in the progressive dot hanging area is greater than 25% of the set threshold. After meeting the conditions for constructing non-gradient dots from gradient dots, start to construct non-gradient dots.

The next step is to select the generation method of non-progressive dots (step S13).

If the non-progressive network point generation method is set to be directly connected to the network, then according to step S14, directly connected to the network, the method is as follows:

Generate non-progressive dots according to the method of directly hanging the net, that is, the function module generates non-progressive dots according to the following function:

Construct the non-progressive network function Ψ _{Φ, F} (P) to generate network points through the function cluster F and the progressive network function Φ, wherein, the progressive network function Φ refers to a discrete space Ω={(i, j )|0≤i<Width, 0≤j<Height}, the grayscale image space above it is defined as I _G ＝{Ω→[0,1]}, and the binary image space is I ₂ ＝{Ω→{0 , 1}}, if the grayscale image P, P'∈I _G , and for any i, j there exists P _ij ≤ P' _ij , then it is defined as P≤P', the progressive grid mapping function Φ: I _G → I ₂ , satisfy Φ(P)≤Φ(P');

Wherein, the function cluster F={F ₀ , F ₁ , ..., F _n-1 }, n is an integer greater than 1, and F ₀ , F ₁ , ..., F _n-1 are defined A continuous function whose domain is [0, 1] and value range is [0, 1], satisfying F ₀ (x)≥F ₁ (x)≥...≥F _n-1 (x), and ∑ _k=0 ^n-1 (-1) ^k F _k (x) = x, where 0≤x≤1;

The grayscale image P=∑ _ij α(P _ij ), where α(P _ij ) is the brightness of the pixel with coordinates (i, j) in the grayscale image P.

The function cluster F is defined as follows:

1) F={F ₀ , F ₁ , F ₂ }, where

${\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left\{\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; if</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \&le;</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ \mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; if</span>}& \frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right.$

F ₁ (x)=x

${\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left\{\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; if</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \&le;</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ \mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; if</span>}& \frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right.$

2) F={F ₀ , F ₁ , F ₂ }, where

${\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left\{\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; if</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \&le;</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}\\ \mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; if</span>}& \frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \&le;</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ \mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; if</span>}& \frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right.$

${\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left\{\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; if</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \&le;</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}\\ \mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; if</span>}& \frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \&le;</span>\frac{\mathrm{<span\; class="notranslate">\; 3</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}\\ \mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; if</span>}& \frac{\mathrm{<span\; class="notranslate">\; 3</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right.$

${\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left\{\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; if</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \&le;</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ \mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; 3</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; if</span>}& \frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \&le;</span>\frac{\mathrm{<span\; class="notranslate">\; 3</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}\\ \mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; if</span>}& \frac{\mathrm{<span\; class="notranslate">\; 3</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right.$

3) F={F ₀ , F ₁ }, where

${\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left\{\begin{array}{ccc}\frac{\mathrm{<span\; class="notranslate">\; 3</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; if</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \&le;</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ \frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; if</span>}& \frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right.$

${\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left\{\begin{array}{ccc}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; if</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \&le;</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ \frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; if</span>}& \frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right.$

4) F={F ₀ , F ₁ }, where

F ₀ (x)=2x-x ²

F ₁ (x)=xx ²

That is, new non-progressive dots are generated, as shown in FIG. 5 . The original progressive network point is shown in Figure 4. By comparison, it can be found that the function cluster F disturbs the progressive network point during the process of generating non-progressive network point, so that each network point forms a concentric ring structure, and its ring bandwidth It varies according to the ink ratio.

Embodiment two:

Others are identical with embodiment one, and difference is: the threshold value of setting in step 2 is 60%; In step 3, if setting non-progressive dot generation mode is the dot optimization mode, then according to step S15, at first generate progressive dot, as As shown in Fig. 4, then according to the same function cluster F in the above-mentioned embodiment one, new network points are generated (step S16), and XOR superposition is carried out on the progressive network points generated in Fig. 4, to generate new non-progressive network points (step S14 ).

In this embodiment, the function cluster F supports recursive calls, that is, a new dot is generated on the basis of the progressive dot, and a second dot is generated based on the new dot once again, so that the dot ink ratio generated by the function cluster F exceeds the limit value, and then stops. XOR superposition generates non-progressive dots, that is, secondary dots are XORed with primary dots, and the last dots are XORed with progressive dots to generate non-progressive dots.

In this embodiment, the limited value of dot-ink ratio setting is 20% to 80%, then a new dot is generated on the basis of progressive dots, and the dot-ink ratio is judged to exceed the above-mentioned interval based on the regenerated dots of the first dot, and no longer generates Function cluster F new network point. The primary dot binary image is XOR-superimposed with the progressive dot binary image to finally generate a non-progressive dot, as shown in Figure 5.

This method has no specific restrictions on the dot shape, but for non-circular, such as rhombus, triangle and other hanging network dots, it is easy to understand. According to the method of this embodiment, when binary images are XOR superimposed, it is necessary to ensure that the regions of the two images, The size and gradient direction are consistent with the hanging net angle.

In the above-mentioned embodiments, the dot-to-ink ratio limit value is set according to empirical values to construct the boundary of non-progressive dots. The specific value varies with different inks and papers, and is generally set between 5% and 95%.

As can be seen from the above-mentioned embodiments, the present invention realizes the improvement on the progressive netting algorithm to generate non-progressive netting by proposing a simplified function cluster F calculation method, and can also optimize the progressive netting by this method, By generating ring-shaped multi-layer nested dot shapes, as shown in Figure 2, this non-progressive dot makes full use of the visual error that dark colors easily invade light colors in human vision, and has the following significant effects:

1) The printing effect is bright and the gradient effect is delicate and smooth;

2) The circular shape of the new dot reduces the continuous area of ink, and reduces the amount of ink used in specific printing. Figure 2 shows the principle and effect of saving ink;

3) According to the characteristics of ink and paper, the new dots can flexibly adjust the annular interval area of dots to meet the requirements of various papers and inks.

The method described in the present invention is not limited to the embodiments described in the specific implementation, and those skilled in the art can obtain other implementations according to the technical solution of the present invention, such as, by setting calculation rules, constructing ink dot rings The method of reducing the amount of ink used and improving the brightness of the screen also belongs to the technical innovation scope of the present invention.

Claims (9)

1. A method for constructing non-progressive sites, comprising the following steps: Step 1, first determine the progressive dot hanging area, and then obtain the gradient ratio along the progressive direction in the progressive dot hanging area; Step 2, judging whether the gradient ratio along the gradual direction is greater than the set threshold, if yes, then proceed to step 3 to start constructing non-progressive dots, otherwise the process ends; Step 3: Construct non-progressive dots, and construct nested ring-shaped dots within the range of progressive dots by defining a function cluster F. 2. a kind of method for constructing non-progressive network as claimed in claim 1, is characterized in that, in step 3, constructs non-progressive network by directly hanging net mode, concrete method is: Construct the non-progressive network function Ψ Φ _{, F} (P) to generate the non-progressive network through the function cluster F and the progressive network function Φ, and the non-progressive network function Ψ _{Φ, F} (P)={Ψ _{Φ , F} (P) _ij }, said Ψ _{Φ, F} (P) _ij is a direct network function, Ψ _{Φ, F} (P) _ij = ∑ _{k = 0} ^n-1 (-1) ^k Φ(F _k ( P)) _ij , where F _k (P) _ij = F _k (P _ij ); The function cluster F={F ₀ , F ₁ , ..., F _n-1 }, n is an integer greater than 1, and the domains of F ₀ , F ₁ , ..., F _n-1 are [0, 1], a continuous function with a value range of [0, 1], satisfying F ₀ (x)≥F ₁ (x)≥...≥F _n-1 (x), and ∑ _k=0 ^{n- 1} (-1) ^k F _k (x)=x, wherein 0 ≤ x ≤ 1; the definition of the progressive netting function Φ is as follows: for a certain discrete space Ω={(i, j)|0≤i＜ Width, 0≤j<Height}, define the grayscale image space on it as I _G ={Ω→[0,1]}, and the binary image space as I ₂ ={Ω→{0,1}}, if Gray-scale image P, P'∈I _G , and for any i, j there is P _ij ≤ P' _ij , then it is defined as P≤P', and the progressive grid function Φ: I _G →I ₂ satisfies Φ( P)≤Φ(P'), the grayscale image P=∑ _ij α(P _ij ), where α(P _ij ) is the brightness of the pixel with coordinates (i, j) in the grayscale image P. 3. a kind of method for constructing non-progressive network as claimed in claim 1, is characterized in that, in step 3, constructs non-progressive network by network optimization mode, concrete method is: Firstly, the progressive network point is directly generated according to the progressive network point hanging function Φ, and then a new network point is generated according to the function cluster F, and XOR is superimposed on the progressive network point to generate a new non-progressive network point; The function cluster F={F ₀ , F ₁ , ..., F _n-1 }, n is an integer greater than 1, and the domains of F ₀ , F ₁ , ..., F _n-1 are [0, 1], a continuous function with a value range of [0, 1], satisfying F ₀ (x)≥F ₁ (x)≥...≥F _n-1 (x), and ∑ _k=0 ^{n- 1} (-1) ^k F _k (x)=x, wherein 0 ≤ x ≤ 1; the definition of the progressive netting function Φ is as follows: for a certain discrete space Ω={(i, j)|0≤i＜ Width, 0≤j<Height}, the grayscale image space above it is defined as I _G ＝{Ω→[0,1]}, the binary image space is I ₂ ＝{Ω→{0,1}}, if Gray-scale image P, P'∈I _G , and for any i, j there is P _ij ≤ P' _ij , then it is defined as P≤P', and the progressive grid function Φ: I _G →I ₂ satisfies Φ( P)≤Φ(P'), the grayscale image P=∑ _ij α(P _ij ), where α(P _ij ) is the brightness of the pixel with coordinates (i, j) in the grayscale image P. 4. A kind of method for constructing non-progressive dots as claimed in claim 2 or 3, characterized in that: the threshold described in step 2 is [α(P _i'j' )-α(P _ij )]/α (P _i'j' ), whose value is greater than or equal to 25% and less than or equal to 1, where P _i'j' is the maximum brightness pixel in the grayscale image P, and P _ij is the minimum brightness pixel in the grayscale image P. 5. a kind of method for constructing non-progressive mesh points as claimed in claim 2 or 3, is characterized in that: described function cluster F is defined as follows: 1) F={F ₀ , F ₁ , F ₂ }, where ${\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left\{\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; if</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \&le;</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ \mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; if</span>}& \frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right.$ F ₁ (x)=x ${\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left\{\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; if</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \&le;</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ \mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; if</span>}& \frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right.$ 2) F={F ₀ , F ₁ , F ₂ }, where ${\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left\{\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; if</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \&le;</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}\\ \mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; if</span>}& \frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \&le;</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ \mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; if</span>}& \frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right.$ ${\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left\{\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; if</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \&le;</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}\\ \mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; if</span>}& \frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \&le;</span>\frac{\mathrm{<span\; class="notranslate">\; 3</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}\\ \mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; if</span>}& \frac{\mathrm{<span\; class="notranslate">\; 3</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right.$ ${\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left\{\begin{array}{ccc}\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; if</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \&le;</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ \mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; 3</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; if</span>}& \frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \&le;</span>\frac{\mathrm{<span\; class="notranslate">\; 3</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}\\ \mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; if</span>}& \frac{\mathrm{<span\; class="notranslate">\; 3</span>}}{\mathrm{<span\; class="notranslate">\; 4</span>}}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right.$ 3) F={F ₀ , F ₁ }, where ${\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left\{\begin{array}{ccc}\frac{\mathrm{<span\; class="notranslate">\; 3</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; if</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \&le;</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ \frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; if</span>}& \frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right.$ ${\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left\{\begin{array}{ccc}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; if</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \&le;</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ \frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; if</span>}& \frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right.$ 4) F={F ₀ , F ₁ }, where F ₀ (x)=2x-x ² F ₁ (x)=xx ² . 6. A method for constructing non-progressive dots as claimed in claim 3, characterized in that: when XOR superimposition is performed, the area, size, and gradient direction of the two images are kept consistent with the hanging net angle. 7. a kind of method for constructing non-progressive network points as claimed in claim 3, is characterized in that: described function cluster F supports recursive call, promptly generates a new network point on the basis of progressive network point, generates two again according to a new network point Secondary dots, so stop until the ink ratio of the dots generated by the function cluster F exceeds the limit value, and then reverse XOR superposition to generate non-progressive dots, that is, the second dot and the first dot are XORed, and the last dot and the progressive dot are XORed to generate non-progressive dots outlets. 8. A system for constructing non-progressive dots, comprising the following modules: Progressive dot construction module, used to determine the progressive dot hanging area, and then obtain the gradient ratio along the progressive direction in the progressive dot hanging area; Threshold judgment module, used to judge whether the gradient ratio along the gradual direction is greater than the set threshold; The non-progressive dot construction module is used for constructing non-progressive dots when the gradient ratio is greater than a set threshold, by defining a function cluster F to construct nested ring dots within the range of progressive dots. 9. a kind of system of constructing non-progressive network dot as claimed in claim 8, is characterized in that, also be provided with following submodule in described non-progressive network dot construction module: Construct the submodule of the non-progressive network by directly hanging the network mode, and concrete method is: construct the non-progressive network function Ψ _{Φ, F} (P) to generate the non-progressive network by the function cluster F and the network function Φ of the progressive network, and generate the non-progressive network, described Non-progressive network point hanging network function Ψ _{Φ, F} (P) = {Ψ _{Φ, F} (P) _ij }, said Ψ _{Φ, F} (P) _ij is a direct hanging network function, Ψ _{Φ, F} (P) _ij = Σ _k＝0 ^n-1 (-1) ^k Φ(F _k (P)) _ij where, F _k (P) _ij ＝F _k (P _ij ); The sub-modules of non-progressive dots are constructed by means of dot optimization. The specific method is: firstly, the progressive dots are directly generated according to the progressive dot hanging function Φ, and then new dots are generated according to the function cluster F, and XOR superposition is performed on the progressive dots to obtain Generate new non-progressive dots; The function cluster F={F ₀ , F ₁ , ..., F _n-1 }, n is an integer greater than 1, and the domains of F ₀ , F ₁ , ..., F _n-1 are [0, 1], a continuous function with a value range of [0, 1], satisfying F ₀ (x)≥F ₁ (x)≥...≥F _n-1 (x), and ∑ _k=0 ^{n- 1} (-1) ^k F _k (x)=x, wherein 0 ≤ x ≤ 1; the definition of the progressive netting function Φ is as follows: for a certain discrete space Ω={(i, j)|0≤i＜ Width, 0≤j<Height}, define the grayscale image space on it as I _G ={Ω→[0,1]}, and the binary image space as I ₂ ={Ω→{0,1}}, if Gray-scale image P, P'∈I _G , and for any i, j there is P _ij ≤ P' _ij , then it is defined as P≤P', and the progressive grid function Φ: I _G →I ₂ satisfies Φ( P)≤Φ(P'), the grayscale image P=∑ _ij α(P _ij ), where α(P _ij ) is the brightness of the pixel with coordinates (i, j) in the grayscale image P.

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