CN101833777B - Real-time marble vein glaze pattern generating method based on mathematical function - Google Patents

Abstract

The invention discloses a marble vein glaze pattern real-time generating method, which comprises the following steps of: appointing the size of a vein pattern and the color of a background pattern layer; building a base pattern by means of dropping, spraying, overturning and the like; building a marble vein glaze pattern by means of user operations such as comb, stylus, wave, rotation and the like; and multiplying the density field of each layer by the color of the corresponding pattern layer and adding to obtain a final display color. The method overcomes the defects which are commonly existed in a computer simulation system of the existing marble vein glaze pattern technology such as slower speed, fuzzy generating pattern, unfriendly user interface, over-abstract physical equation parameter and the like, adopts the real-time marble vein glaze pattern generating method based on mathematical function, and guarantees that the generated marble vein glaze pattern has the characteristics of the clear boundary and the elegant pattern of the real marble vein glaze pattern by adopting the density field expressing method by means of reverse mapping, supersampling and multi-pattern layer and the iterative density field updating arithmetic.

Description

Real-time marble grain glaze pattern generating method based on mathematical function

Technical field

The present invention relates to computing machine non-photorealistic rendering and image and generate the field, particularly relate to a kind of real-time marble grain glaze pattern generating method based on mathematical function.

Background technology

Marbleizing glaze technology is a traditional art of on paper or fabric, making colored flow-like pattern.As a time-honored handicraft technology; The initial application of marbleizing glaze technology is to play ornamental purpose; The background patterns that can be used for the official communication is simultaneously revised and is forged preventing, develops into a kind of ancient anti-piracy technique afterwards again, promptly is the side that the marbleizing glaze pattern is printed on book; All can make pattern look discontinuous owing to tear any one page of book, thereby effectively prevent the artificial destruction of books.The making flow process of marbleizing glaze technology mainly contains following three steps: a container is prepared in (1), pours the dense thick substrate liquid of one deck at container bottom, and substrate liquid normally is made up of the adhesion agent of water and interpolation.(2) spray the pigment of various colors on substrate liquid top layer with instruments such as eyedrops drop bottle, transfer pipet or bamboo brushes, just obtained the basic pattern of a width of cloth.Carrying must be immiscible between substrate liquid and the pigment of these pigment, and wants enough thickness, to guarantee that pigment can swim in the surface.(3) come pattern-making with instruments such as wooden stick, comb, stylus, the back and forth stirring of instrument in container makes pigment be stretched and is out of shape, thereby form various decorative patterns.A kind of complicated patterns often need be carried out repeatable operation to these pigment.(4) after design producing is accomplished, be positioned over liquid-covered carefully, absorb floating color, start then with material (normally paper or fabric) with characterization of adsorption, rinsing, and dry.

Traditional marbleizing glaze technological requirement user possesses rich experience, adds the restriction (like the viscosity of substrate liquid, the floatability of pigment etc.) of various physical conditions, to a great extent limit the quality of marbleizing glaze pattern.Therefore, flourish along with computer graphics and image processing techniques, the marbleizing glaze process simulation system of some computer simulations also arises at the historic moment.

Traditional marbleizing glaze process simulation system mostly is that the utilization numerical method is found the solution the physics flow equation.2004, Akgun designed the Core Generator that a computer based has the marbleizing glaze pattern of alla turca.Referring to AKGUN, digital aft of marbled paper.Leonardo B.2004.The, 37,1,49-52..For the pattern that makes generation keeps the sharp-pointed characteristic of true marbleizing glaze pattern, researchists such as Acar and Boulanger have proposed to come this technology of emulation with the fluid model of multi-layer image, referring to ACAR; R.; AND BOULANGER, marbling:amulti-scale fluid model.IEEE Transactions on Visualization and ComputerGraphics P.2006.Digital, 12; 4,600-614.Since this method can not the emulation comb etc. classical marbleizing glaze process tool, after 1 year, Acar has proposed the fluid simulation algorithm based on level set method again, and an interactively marbleizing glaze pattern generation system is provided.Referring to ACAR, driven flows.ACM Trans.Graph.26 R.2007.Levelset, 4, Article No.15.Yet because these methods all are based on the CPU realization, travelling speed is slower, can not reach real-time user interactions requirement.Permitted the good grand people of grade afterwards and proposed on GPU, to find the solution Navie-Stokes physics flow equation, though on speed, can reach the requirement of real-time basically, but can not fundamentally solve the marbleizing glaze pattern edge fuzzy problem of generation; Referring to XU, J., MAO; X., AND JIN, X.2008.Nondissipative marbling.IEEE Computer Graphics and Applications; 28,2,35-43.Except the fluid method of emulation, business software Corel Painter provides picture editting's method to generate the marbleizing glaze pattern, but the speed of this software is slower, and the marbleizing glaze pattern kind that provides is less.

Summary of the invention

The present invention provides a kind of real-time marble grain glaze pattern generating method based on mathematical function.This method is regarded marbleizing glaze technology as fluid motion that the Two-dimensional incompressible on the unlimited extending plane contracts, and uses the technological visual this fluid motion of mathematical function driven image distortion to form the marbleizing glaze pattern.The present invention provides real-time feedback and method for generating pattern that have lively fluid motion effect for design marbleizing glaze pattern.

A kind of real-time marble grain glaze pattern generating method based on mathematical function comprises following four steps:

1) user specifies the size of the marbleizing glaze pattern that needs generation and the color of Background From Layer, and wherein the color of Background From Layer is used C _bExpression;

2) user specifies the color and the relevant parameters of figure layer, figure layer, modes of operation such as selecting dropping liquid then, spray or topple over, and the mathematical function corresponding according to operation generates a width of cloth basic pattern;

3) user selects modes of operation such as comb, stylus, wave or rotation; Adjust corresponding parameters then; The mathematical function corresponding according to operation; Adopt the method for reverse mapping, super and iteration regeneration density field to review the position of the pixel of each pixel in basic pattern in the current pattern, the density value that will put then copies to current pixel point, thereby obtains the density field of current pattern;

A kind of complicated patterns often need be carried out operation repeatedly to liquid, and the user can be according to the mode of operation in the effect repeat number of wanting the to generate the marbleizing glaze pattern time step (3).

4) color of the density field of each figure layer (comprising Background From Layer) and corresponding appointment is multiplied each other and every layer result is added up to the end color value and shows that the density field of i layer is used D _iExpression, the paint color of every i layer is used C _iExpression, the color of Background From Layer is used C _bExpression.

Described pattern of the present invention, by Background From Layer and on several figure layers form, any color of certain in the pattern is to obtain after this color adds up on the All Layers.

User of the present invention specifies the size of the marbleizing glaze pattern of generation to refer to the height and the width of marbleizing glaze pattern, the paint color of the substrate liquid in the corresponding marbleizing glaze technology of the color of Background From Layer.

Dropping liquid mode of operation of the present invention is to generate the drop of specifying size and color.Described spraying operation mode is the drop that generates at random in some and the radius.It is described that to topple over mode of operation be to generate a succession of continuous drop.For the squeezing effect between the expression drop of image, use following method:

The color relation that is positioned at the inner pixel of drop is the color of user's appointment figure layer, and remaining pixel is done reverse mapping in order to minor function:

$P^{\′}=C+(P-C)\sqrt{1-\frac{r^2}{{|P-C|}^2}}$

P representes the pixel in the current pattern in the formula, and P ' expression is traced back to the corresponding pixel points in the last pattern by P, and C representes the home position of drop, and r representes the radii size of drop.

Comb mode of operation of the present invention is to be formed by the combination of function that some straight lines are represented, wherein each tooth of comb is represented straight line, and corresponding function is following:

$P^{\′}=P-\frac{\mathrm{\α\λ}}{d+\mathrm{\λ}}M$

d＝|(P-A)·N|

P representes the pixel in the current pattern in the following formula; P ' expression is traced back to the corresponding pixel points in the basic pattern by P; M representes the unit vector of this straight line, and d representes the distance of P point to straight line, and α and λ be the maximum displacement and the displacement gradient that move of remarked pixel respectively; A represent on the straight line more arbitrarily, the per unit system that N representes straight line to;

The function of stylus mode of operation correspondence of the present invention is following:

$P^{\′}=P-\frac{\mathrm{\α\λ}}{d+\mathrm{\λ}}M$

P representes the pixel in the current pattern in the formula; P ' expression is traced back to the corresponding pixel points in the basic pattern by P; M representes that stylus streaks the unit vector of line segment place straight line; D representes the bee-line that the P point streaks line segment to stylus, and α and λ be the maximum displacement and the displacement gradient that move of remarked pixel respectively.

Wave user mode of operation of the present invention is divided into the wave of horizontal direction and the wave of vertical direction, and corresponding mathematical function is distinguished as follows:

Horizontal direction:

Vertical direction:

X, y represent the horizontal ordinate of the pixel P in the current pattern and the value of ordinate respectively in the formula; P ' expression is traced back to the corresponding pixel points in the basic pattern by P; A representes sinusoidal wave amplitude; ω representes sinusoidal wavelength, and

representes sinusoidal phase deviation.

The function of user's mode of operation correspondence of rotation of the present invention is following:

$P^{\′}=C+(P-C)\left[\begin{array}{cc}\cos \mathrm{\θ}& \sin \mathrm{\θ}\\ -\sin \mathrm{\θ}& \cos \mathrm{\θ}\end{array}\right]$

P representes the pixel in the current pattern in the following formula, and P ' expression is traced back to the corresponding pixel points in the basic pattern by P, and C representes that the home position that rotates, θ represent anglec of rotation size, when θ turns clockwise for the negative interval scale, otherwise then for being rotated counterclockwise.

The mathematical function corresponding of the present invention according to operation; Adopt the method for reverse mapping, super and iteration regeneration density field to review the position of the pixel of each pixel in basic pattern in the current pattern, the density value that will put then copies to the method for current pixel point following:

In order to calculate the density value that the active user operates any pixel in the back pattern that generates, adopt the method for reverse mapping to review the pixel position of this pixel in basic pattern, and the density value that will put copy to current pixel point.Pattern for fear of generating has the phenomenon of losing shape; Adopt the anti-aliasing pattern of super; Be about to current pixel and be divided into the experimental process pixel; Try to achieve the pixel in the corresponding basic pattern of these sub-pixels according to function respectively then, then the density value weighted mean of these pixels is composed to current pixel point, formula is following:

$D(x,y)=\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{\mathrm{\ω}}_i{\mathrm{\ρ}}_i(x_i,y_i)$

D representes the density value of current pixel point in the formula, and n representes the number of sub-pixel point, (x _i, y _i) represent i subpixels point, ρ _iAnd ω _iDensity value and the weight of representing i subpixels point respectively, the weight of all sub-pixel points is 1/n here.

The color of density field and the corresponding appointment with each figure layer (comprising Background From Layer) of the present invention multiply each other and with every layer result add up to the end color value and the method that shows following:

Carry out the calculating of color value according to following formula:

$C=\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{C}_i{D}_i+C_b(1-\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{D}_i)$

C is the color of output in the formula, C _bThe color of expression Background From Layer, D _iThe density value of representing the i layer, C _iThe color value of representing the i layer, n=16 presentation graphs number of layers.

It is slower that the present invention has improved in the computer simulation system of existing marbleizing glaze technology ubiquitous speed; Generate dim pattern; User interface is unfriendly; Shortcomings such as the physical equation parameter is too abstract; Adopted the real-time marble grain glaze pattern generating method based on mathematical function, it is clear that this method adopts the density field update algorithm of density field method for expressing and the iteration of reverse mapping, super, multi-layer image to guarantee that the marbleizing glaze pattern that generates has true marbleizing glaze pattern boundaries, the characteristics that pattern is exquisite.Algorithm of the present invention is clear and definite, friendly interface, robust as a result.

Description of drawings

Fig. 1 is the process flow diagram of the inventive method.

Fig. 2 is the corresponding synoptic diagram of single straight line in the comb operation.

Fig. 3 is the corresponding synoptic diagram of rotary manipulation.

Embodiment

Below in conjunction with accompanying drawing the real-time marble grain glaze pattern generating method that the present invention is based on mathematical function is elaborated through embodiment.

As shown in Figure 1; The inventive method; Comprise the size of specifying textured pattern and the color of Background From Layer, through dropping liquid, mode such as spray, topple over creates basic pattern, through comb; Users such as stylus, wave, rotation operate and create the marbleizing glaze pattern, four steps of Show Color that every layer density field and corresponding figure layer color are multiplied each other and add up to the end:

1) user specifies size and the color of Background From Layer of the marbleizing glaze pattern of generation, the wherein width and the height of size decision marbleizing glaze pattern, and the color of Background From Layer is corresponding to the paint color of substrate liquid in the marbleizing glaze technology;

2) user specifies the color and the relevant parameters of figure layer, figure layer, selects dropping liquid then, mode of operation such as sprays or topple over, and the mathematical function corresponding according to operation generates a width of cloth basic pattern;

The dropping liquid mode of operation is the drop that in design section, forms corresponding size and color through the center of circle of the definite drop in the position of the radius of interactive setup drop and click.The spraying operation mode is through the radius of specifying drop and the amount of droplets of sprinkling, in design section, generates the drop of specifying the respective amount in the radius at random.Toppling over mode of operation is through specifying the radius of drop, on the path that mouse streaks, form a succession of continuous drop in design section.The effect that can push the drop of front for the follow-up drop that splashes into of lively embodiment; Principle is following: according to radius and home position; First drop of liquid of dripping and the border circular areas that to form an area be a; If the area of second drop of liquid is b; And dripping with first dropping liquid has the identical center of circle; The total area that is then covered by drop should reach a+b; Owing to receive the influence of second drop of liquid; Originally the point at circle centre position can move to the distance apart from the center of circle

, and the point at drop edge place can move to

distance from the distance apart from the center of circle .Therefore, be positioned at C for the set point P and the center of circle, radius is the drop of r; If | P-C|＜r, then putting P is exactly in circle, and the color relation that P is ordered is the color of appointment figure layer; Otherwise, ask the corresponding point P ' point in the last pattern that P orders with reverse reflection method, and the density value of P ' in copy is to the P point.Wherein P ' computing formula is following:

$P^{\′}=C+(P-C)\sqrt{1-\frac{r^2}{{|P-C|}^2}}$

According to above formula, the drop that drips at last always keeps circular outward appearance, and the drop that before drips then deformation can take place.

3) user selects modes of operation such as comb, stylus, wave, rotation; Adjust corresponding parameters then; The mathematical function corresponding according to operation adopts reverse mapping, and the method for super and iteration regeneration density field is reviewed the position of the pixel of each pixel in basic pattern in the current pattern; The density value assignment that to put is then given current pixel point, thereby obtains the density field of current pattern;

The comb instrument is a kind of instrument very general in the marbleizing glaze technology; It is that some typical marbleizing glaze patterns of generation must be obligato; User's mode of operation of comb is to be formed by the combination of function that several straight lines are represented, wherein each tooth of comb is just represented the corresponding operation of straight line.The displacement that each pixel moves and this pixel are inversely proportional to apart from d to straight line, and the stretching displacement amount with those pixels of same distance equates.As shown in Figure 2, L is this straight line (can have slope arbitrarily), N be L per unit system to, A be on the L a bit, M is the vector of unit length of straight line L, and is following for the mapping function of the pixel P in the current pattern:

$P^{\′}=P-\frac{\mathrm{\α\λ}}{d+\mathrm{\λ}}M$

d＝|(P-A)·N|

α and the λ maximum displacement and the displacement gradient that move of remarked pixel respectively wherein just can obtain the mapping function of whole comb through the mapping function of synthetic these single straight line.

The stylus instrument is mainly used in creates some meticulous patterns (petal etc.), and the corresponding mapping function of stylus mode of operation is identical with the corresponding mapping function of straight line:

$P^{\′}=P-\frac{\mathrm{\α\λ}}{d+\mathrm{\λ}}M$

Unique difference is exactly different apart from the computing formula of d, streaks the starting point and the terminal point in path if A and B are respectively stylus, and L is the straight line at line segment AB place, shows with the P point and has replaced the distance of P to straight line L to the distance of line segment AB.

User's mode of operation of wave is divided into the wave of horizontal direction and the wave of vertical direction, and corresponding mathematical function is distinguished as follows with sinusoidal curve:

Horizontal direction:

Vertical direction:

X, y represent the horizontal ordinate of the pixel P in the current pattern and the value of ordinate respectively in following formula; Through mutual definite sinusoidal wave amplitude A; Sinusoidal wavelength ω, sinusoidal phase deviation

can be in the hope of corresponding pixel points P '.

The pattern that user's mode of operation of rotation forms is one of pattern very common in the marbleizing glaze technology, and as shown in Figure 3, the central coordinate of circle of known rotation is positioned at C, and the mapping function that radius of turn r, P are ordered is following:

$P^{\′}=C+(P-C)\left[\begin{array}{cc}\cos \mathrm{\θ}& \sin \mathrm{\θ}\\ -\sin \mathrm{\θ}& \cos \mathrm{\θ}\end{array}\right]$

$\mathrm{\θ}=\frac{l}{|P-C|}$

$l=\frac{\mathrm{\α\λ}}{(d+\mathrm{\λ})}$

d＝|P-C|-r

α in the formula and λ be the maximum displacement and the displacement gradient that move of remarked pixel respectively.θ representes anglec of rotation size, when θ generates the pattern that turns clockwise during for negative, otherwise then for being rotated counterclockwise pattern.

Ask the mapping function of P ' to be reverse mapping in the above formula, in the image mapped algorithm, the effect of reverse mapping is better than the forward mapping, and reverse mapping algorithm is highly suitable for upward realization of GPU.The principle of reverse mapping is following: review the pixel position of pixel in last pattern in the current pattern, and the density value that will put copies to current pixel point.Pattern for fear of generating has the phenomenon of losing shape; Adopt the anti-aliasing pattern of super; Be about to current pixel and be divided into the experimental process pixel; Try to achieve the pixel in the pattern on these sub-pixel correspondences according to function respectively then, then the density value weighted mean of these pixels is composed to current pixel point, formula is following:

$D(x,y)=\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{\mathrm{\ω}}_i{\mathrm{\ρ}}_i(x_i,y_i)$

D representes the density value of current pixel point in the formula, and n representes the number of sub-pixel point, (x _i, y _i) represent i subpixels point, ρ _iAnd ω _iDensity value and the weight of representing i subpixels point respectively, the weight of all sub-pixel points is 1/n here.

In addition, the density field update method of iteration can guarantee that the marbleizing glaze pattern that generates has clear sharp-pointed characteristic.If all want the regeneration density field after each operation, method is as follows:

D(P _n)＝D(P _n-1)＝...＝D(P ₁)＝D(P ₀)

N representes the number of times that the user operates, P _i(i=0,1 ..., n-1) expression P _I+1The back tracking point of individual pixel.After passing through multi-pass operations so, the problem that dim pattern and contrast descend can appear, even adopt the high-order interpolation method can not eliminate this blooming completely.In order to address this problem, this method has proposed the method for iteration regeneration density field, and formula is following:

D(P _n)＝D(P _n→P _n-1→...→P ₁→P ₀)

The mathematical function that the mode of operation of all user's appointments is corresponding is placed in the pixel coloring device of GPU each pixel P in the current pattern of iterative computation _nCorresponding pixel position P in basic pattern ₀, then with P ₀The density value at place copies P to _nThe place.Owing to reduced the number of times of regeneration density field, the marbleizing glaze pattern of generation is sharply marginated.

4) color of the density field of each figure layer (comprising Background From Layer) and corresponding appointment is multiplied each other and every layer result is added up to the end color value and shows.

Pigment in the true marbleizing glaze technology is immiscible each other, so the pattern that generates can keep border clearly between different pigment.This method has been introduced the density field method for expressing (always having 16 layers) of multi-layer image in order to prevent the mixing between different pigment, and the color value through density field and corresponding diagram layer color multiply each other and add up to the end calculates with following formula:

$C=\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{C}_i{D}_i+C_b(1-\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{D}_i)$

In following formula, C is the color of output, C _bThe color of expression Background From Layer, D _iThe density value of representing the i layer, C _iThe color value of representing the i layer, the quantity of n presentation graphs layer.

Claims (3)

1. the real-time marble grain glaze pattern generating method based on mathematical function is characterized in that, may further comprise the steps:

(1). the user specifies the size of the marbleizing glaze pattern that needs generation and the paint color of Background From Layer;

(2). the user specifies the color and the relevant parameters of figure layer, figure layer, selects dropping liquid then, sprays or topple over mode of operation, generates a width of cloth basic pattern according to the mathematical function of correspondence;

Described dropping liquid mode of operation is to generate the drop of specifying size and color; Described spraying operation mode is the drop that generates at random in some and the radius; It is described that to topple over mode of operation be to generate a succession of continuous drop;

(3). the user selects comb, stylus, wave or rotary manipulation mode; Mathematical function according to correspondence; The method of employing super is calculated the density value of each pixel in the current pattern; Thereby obtain the density field of current pattern, be about to current pixel and be divided into the experimental process pixel, try to achieve the pixel in the pattern on these sub-pixel correspondences according to function respectively then; Then the density value weighted mean of these sub-pixel points is composed to current pixel point, formula is following:

$D(x,y)=\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{\mathrm{\ω}}_i{\mathrm{\ρ}}_i(x_i,y_i)$

The density value of D remarked pixel point in the formula, n representes the number of sub-pixel point, (x _i, y _i) represent i subpixels point, ρ _iAnd ω _iDensity value and the weight of representing i subpixels point respectively, the weight of all sub-pixel points is 1/n here;

The density value of described each sub-pixel point is to review this position in basic pattern through the method for reverse mapping and iteration regeneration density field, and the density value that will put then copies to current sub-pixel point;

Described comb mode of operation is to be formed by the combination of function that some straight lines are represented, wherein each tooth of comb is just represented straight line, and corresponding function is following:

$P^{\′}=P-\frac{\mathrm{\α\λ}}{d+\mathrm{\λ}}M$

P representes the pixel in the current pattern in the formula, and P ' expression is traced back to the corresponding pixel points in the last pattern by P, and M representes the unit vector of this straight line, and d representes the distance of P point to straight line, and α and λ be the maximum displacement and the displacement gradient that move of remarked pixel respectively;

The function of described stylus mode of operation correspondence is following:

$P^{\′}=P-\frac{\mathrm{\α\λ}}{d+\mathrm{\λ}}M$

P representes the pixel in the current pattern in the formula; P ' expression is traced back to the corresponding pixel points in the last pattern by P; M representes that stylus streaks the unit vector of line segment place straight line, and d representes the distance that the P point streaks line segment to stylus, and α and λ be the maximum displacement and the displacement gradient that move of remarked pixel respectively;

Described wave mode of operation is divided into the wave of horizontal direction and the wave of vertical direction, and corresponding mathematical function is distinguished as follows:

Horizontal direction:

Vertical direction:

X, y represent the horizontal ordinate of the pixel P in the current pattern and the value of ordinate respectively in the formula; P ' expression is traced back to the corresponding pixel points in the last pattern by P; A representes sinusoidal wave amplitude; ω representes sinusoidal wavelength, and

representes sinusoidal phase deviation;

The function of described rotary manipulation mode correspondence is following:

$P^{\′}=C+(P-C)\left[\begin{array}{cc}\cos \mathrm{\θ}& \sin \mathrm{\θ}\\ -\sin \mathrm{\θ}& \cos \mathrm{\θ}\end{array}\right]$

P representes the pixel in the current pattern in the formula, and P ' expression is traced back to the corresponding pixel points in the last pattern by P, and C representes that the home position that rotates, θ represent anglec of rotation size, when θ turns clockwise for the negative interval scale, otherwise then for being rotated counterclockwise;

(4). the color of the density field of each figure layer and corresponding appointment is multiplied each other and every layer result is added up to the end color value and shows;

Described pattern is made up of several figure layers on Background From Layer and the said Background From Layer.

2. the real-time marble grain glaze pattern generating method based on mathematical function according to claim 1 is characterized in that: when the method for employing iteration regeneration density field is reviewed the position of the pixel of each pixel in basic pattern in the current pattern:

The mathematical function that the mode of operation of all user's appointments is corresponding is placed in the pixel coloring device of GPU the pixel position of each pixel correspondence in basic pattern in the current pattern of iterative computation.

3. the real-time marble grain glaze pattern generating method based on mathematical function according to claim 1, it is characterized in that: step is carried out the calculating of color value according to following formula in (4):

$C=\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{C}_i{D}_i+C_b(1-\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{D}_i)$

In the formula, C is the color of output, C _bThe color of expression Background From Layer, D _iThe density value of representing the i layer, C _iThe color of representing the i layer, the quantity of n=16 presentation graphs layer.

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