CN103440417B - Ink diffusion model modeling method based on Lattice Random Walk - Google Patents

Abstract

The invention discloses a kind of based on Lattice Random the ink diffusion model modeling method of Walk, for solving the poor technical problem of existing ink diffusion model modeling method real-time. Does is technical scheme by the Random for liquid or the research of gas molecule motion track does is Walk theory applied to the modeling of ink diffusion model, by classical Random is Walk diffusion equation set up the Lattice for ink diffusion modeling Random Walk diffusion equation; Does the real physical characteristics spreading on paper according to ink, determine Lattice Random two main constraints of Walk ink diffusion equation, range of scatter R and diffusion step number N, realize the modeling of different diffusion property paper ink diffusion model. Due to Lattice Random Walk algorithm is realized simply, can be walked abreast, and has improved the real-time of ink diffusion model modeling.

Description

Ink diffusion model modeling method based on Lattice Random Walk

Technical field

The present invention relates to a kind of ink diffusion model modeling method, particularly relate to a kind of based on LatticeRandomThe ink diffusion model modeling method of Walk.

Background technology

Document 1 " Q.Guo, ANDT.L.Kunii, Modelingthediffusepaintingofsumie, IFIPModelinginComputerGraphics, 1991,329-338. " one is disclosed based on east writing brush ink and wash inkThe modeling method of diffusion effect. Carbon dust in this model assumption ink is made up of the carbon powder particle of different sizes, and ink existsDiffusion on paper adopts the Method Modeling of one dimension filter screen. Calendar year 2001, the people such as kunii adopt partial differential equation (partialDifferentialequations) original method is improved. Improved method adopts Fick theorem to describe ink diffusionPhenomenon, its major defect is algorithm complexity, is difficult to real-time implementation.

Document 2 " N.S.H.Chu, andC.L.Tai, MoXi:Real-TimeInkDispersioninAbsorbentPaper,ACMTransactionsonGraphics (SIGGRAPH2005), August, 2005,24 (3): 504-511 " openA kind of based on LatticeBoltzmanneqution(LBE) ink model modelling approach. LBE method is seen to be situated between(mesoscopic is between microcosmic and macroscopic view) rank is described fluid molecule motion, but LBE method needs meterThe sub-velocity component data of point counting, cause this modeling method implementation efficiency not high.

Summary of the invention

In order to overcome the poor deficiency of existing ink diffusion model modeling method real-time, the invention provides a kind of based onThe ink diffusion model modeling method of LatticeRandomWalk. The method will be widely used in liquid or gas molecule fortuneThe RandomWalk theory of moving track research is applied to the modeling of ink diffusion model, is spread by classical RandomWalkEstablishing equation is for the LatticeRandomWalk diffusion equation of ink diffusion modeling; Spread on paper according to inkReal physical characteristics, determine two main constraints of LatticeRandomWalk ink diffusion equation, diffusionScope R and diffusion step number N, realize the modeling of different diffusion property paper ink diffusion model, can improve ink diffusionThe real-time of model modeling.

The technical solution adopted for the present invention to solve the technical problems is: a kind of based on LatticeRandomWalkInk diffusion model modeling method, is characterized in adopting following steps:

Step 1, d dimension space isotropism RandomWalk Limit Distribution after strolling step for a long time trend towards heightThis distributes, and its probability-distribution function is:

$P_N\left(R_d\right)~\frac{e^{{-\mathrm{dR}}^2/2\⟨r^2\⟩N}}{{(2\mathrm{\π}\⟨r^2\⟩N/d)}^{d/2}}---\left(1\right)$

In formula, < r²> be the limited two rank squares of random file amount, R is range of scatter, N is diffusion step number.

Note random file amount is a, for have < r of two-dimensional space²>=a², d=2, obtains two-dimensional space isotropism RandomWalk probability-distribution function:

$P_N\left(R_2\right)~\frac{e^{{-R}^2/a^{2}N}}{{\mathrm{\&pi;a}}^{2}N}---\left(2\right)$

D dimension space isotropism RandomWalk, the probability density function from initial point to R utilizes following formula to obtain

P_N(R)=A_dR^d-1P_N(R_d)（3）

In formula, A_dD dimension space unit ball area, A₁=1,A₂=2π,A₃=4π...。

Obtain two-dimensional space isotropism RandomWalk ink diffusion equation:

$P_N\left(R\right)=\frac{2R}{a^{2}N}{e}^{{-R}^2/a^{2}N}---\left(4\right)$

In formula, a is for strolling step-length, and R is range of scatter, and N is diffusion step number.

Adopt the mode of Lattice grid to divide virtual paper, definition step-length is unit grids, i.e. a=1.

Obtain the ink diffusion equation of bidimensional LatticeRandomWalk:

$P_N\left(R\right)=\frac{2R}{N}{e}^{{-R}^2/N}---\left(5\right)$

Step 2, according to the water absorption character of paper, calculate the range of scatter R in ink diffusion equation. Based on LatticeThe real-time ink diffusion model of RandomWalk. The ink saturation capacity of definition unit grids isUnit grids can be protectedHolding quantity of ink isSuppose that the total China ink amount that is subject in this region is I, maximum expansion diffusion length is calculated as

$R=I/I_{x,y}^s---\left(6\right)$

Step 3, the difference that measured by China ink according to virtual paper, will be subject to territory, ink-covered area to be subject to ink-condition to be divided into the paper of dry brush stateUndersaturation is subject to the paper supersaturation of ink-condition and rich ink-condition to be subject to ink-condition, and modeling respectively; For different diffusion propertiesPaper, set up the corresponding paper attenuation coefficient β of UNICOM, the current grid connectivity of virtual paper is by following iterative formulaCalculate:

Continuity(p_n-1top_n)=βContinuity(p_n-2top_n-1)

Different diffusion property paper define different coefficients, determine the diffusion step number N in ink diffusion equation.

The invention has the beneficial effects as follows: the method will be widely used in the Random of liquid or the research of gas molecule motion trackWalk theory is applied to the modeling of ink diffusion model, is set up for ink and is spread by classical RandomWalk diffusion equationThe LatticeRandomWalk diffusion equation of modeling; The real physical characteristics spreading on paper according to ink, determinesTwo main constraints of LatticeRandomWalk ink diffusion equation, range of scatter R and diffusion step number N,Realize the modeling of different diffusion property paper ink diffusion model. Because LatticeRandomWalk algorithm has the letter of realizationSingle, advantage that can Parallel Implementation, the ink diffusion model modeling method based on LatticeRandomWalk significantly reducesThe algorithm complex of ink diffusion simulations, improved the efficiency of ink diffusion simulations; By to paper ink, diffusion declinesSubtract characteristic modeling, realize dissimilar paper ink diffusion is accurately controlled, making ink spread modeling can be for realityBorder paper type modeling; By be subject to territory, ink-covered area to be subject to ink-condition modeling to virtual paper, solve real-time writing brush painting system dryEffect realizes a difficult problem, has improved the real-time of ink diffusion model modeling.

Below in conjunction with the drawings and specific embodiments, the present invention is elaborated.

Brief description of the drawings

Fig. 1 is the inventive method virtual paper Lattice spatial division schematic diagram.

Detailed description of the invention

With reference to Fig. 1. The ink diffusion model modeling method that the present invention is based on LatticeRandomWalk specifically comprise withLower step:

1, set up LatticeRandomWalk ink diffusion equation.

Ink diffusion meets the condition of isotropism rule diffusion, with etc. step-length, random direction, expand at the uniform velocity to the peripheryLoose. The final position of its stochastic variable set of rule diffusion isotropism RandomWalk is determined by diffusion equation.

D dimension space isotropism RandomWalk Limit Distribution after strolling step for a long time trends towards Gaussian distribution,Its probability-distribution function is:

$P_N\left(R_d\right)~\frac{e^{{-\mathrm{dR}}^2/2\&lang;r^2\&rang;N}}{{(2\mathrm{\&pi;}\&lang;r^2\&rang;N/d)}^{d/2}}---\left(1\right)$

In formula, < r²> be the limited two rank squares of random file amount, R is range of scatter, N is diffusion step number.

Note random file amount is a, for have < r of two-dimensional space²>=a², d=2, obtains two-dimensional space isotropism RandomWalk probability-distribution function:

$P_N\left(R_2\right)~\frac{e^{{-R}^2/a^{2}N}}{{\mathrm{\&pi;a}}^{2}N}---\left(2\right)$

D dimension space isotropism RandomWalk, the probability density function from initial point to R utilizes following formula to obtain

P_N(R)=A_dR^d-1P_N(R_d)（3）

In formula, A_dD dimension space unit ball area, A₁=1,A₂=2π,A₃=4π...。

In conjunction with formula 2, formula 3, obtains two-dimensional space isotropism RandomWalk ink diffusion equation:

$P_N\left(R\right)=\frac{2R}{a^{2}N}{e}^{{-R}^2/a^{2}N}---\left(4\right)$

In formula, a is for strolling step-length, and R is range of scatter, and N is diffusion step number.

Adopt the mode of Lattice grid to divide virtual paper, definition step-length is unit grids, i.e. a=1.

Obtain the ink diffusion equation of bidimensional LatticeRandomWalk:

$P_N\left(R\right)=\frac{2R}{N}{e}^{{-R}^2/N}---\left(5\right)$

2, determine range of scatter R.

Ink range of scatter R depends primarily on the water absorption character of paper, for can its water imbibition of bibulous paper being embodied inTwo aspects: ink saturation capacity (watersaturation) and can keep quantity of ink (fluid-holdingcapacity).

The ink saturation capacity of definition unit grids isUnit grids can keep quantity of ink to be

IfThe ink at this some place will continue diffusion to the periphery.

Otherwise this some place ink is in poised state or undersaturated condition.

For being subject to its maximum of territory, ink-covered area that is subject to that China ink completes to depend on that this region is subject to China ink amount and single grid by diffusion length RInk saturation capacity. Suppose this region total be subject to China ink amount for I, maximum diffusion length is calculated as

$R=I/I_{x,y}^s---\left(6\right)$

3, determine diffusion step number N.

The difference that is subject to China ink amount according to virtual paper, is subject to China ink by the paper undersaturation that is subject to territory, ink-covered area to be subject to ink-condition to be divided into dry brush stateThe paper supersaturation of state and rich ink-condition is subject to ink-condition.

Under dry brush state, paper is subject to territory, ink-covered area in undersaturated condition, does not have ink diffusion phenomena, now spreads step numberN=0。

Under rich ink-condition, the ink diffusion of paper has larger difference according to the difference of paper ink diffusion property. For a surnameStationery diffusivity good sheet, its ink diffusion process is only subject to the impact of China ink amount, the absorption of ink on paperDiffusion meeting is along with paper is subject to the saturation balance of territory, ink-covered area quantity of ink and stops. And for low diffusible paper, its inkDiffusion is except being subject to the impact of China ink amount, and paper connectivity also affects its diffusion process. Therefore, for rich ink-condition,The present invention is directed to different diffusion property paper and determine respectively diffusion step number N.

First according to the attenuation coefficient β of different definition paper UNICOM of paper diffusion property, different diffusion property paper defineDifferent coefficients, the present invention defines rice paper β=1, writing paper made from bamboo A4 paper β=0.1, A4 paper β=0.01. Virtual paper is worked asFront grid connectivity is calculated by following iterative formula:

Continuity(p_n-1top_n)=βContinuity(p_n-2top_n-1)（7）

(1) rice paper class ink diffusivity good sheet.

Diffusivity good sheet is when being subject to after China ink rapidly by spread to the periphery by territory, ink-covered area, due to its paper UNICOM coefficientβ=1, is measured by the China ink that is subject in territory, ink-covered area so its diffusion process is mainly limited by paper. Based on diffuse fluid characteristic, work as paperOpen while being subject to territory, ink-covered area to start to spread from center to periphery, the grid in diffusion process is except continuing to unsaturation regionOutside diffusion, the ink that it receives from its contiguous grid simultaneously, comes from the ink that is subject to black central area. These feelingsUnder condition, should, in unsaturation state or critical saturation state, remove in the time that ink diffusion stops in spreading peripheral gridBeyond peripheral meshes, the whole grid that is subject to territory, ink-covered area should be in saturation balance state. Now its diffusion step number N substantially andR is consistent, and its diffusion end condition is

And current step number R=N,

In formula, I_x,yFor current grid quantity of ink.

(2) writing paper made from bamboo, the low ink diffusion property of A4 stationery paper.

Low ink diffusivity paper arranges the diffusion step number N primary data the same with diffusivity good sheet, by quantity of inkVariation and the decay of connectivity determine to stroll end condition. Low ink diffusion property paper is from ink dispersal center regionStart, its connectedness is attenuation trend gradually, supposes, after the roaming of n step, to arrive grid p_n, this some place fiberConnectivity is calculated by iterative formula 7. Arthmetic statement is as follows:

Initialize n the connective Continuity of being of grid place fiber_n=1，

Work as Continuity_n≤ 0 o'clock, orThe process of strolling stops.

In low diffusivity paper, conventionally stop step number r < N.

Claims (1)

1. the ink diffusion model modeling method based on LatticeRandomWalk, it is characterized in that comprising withLower step:

Step 1, d dimension space isotropism RandomWalk Limit Distribution after strolling step for a long time trend towards heightThis distributes, and its probability-distribution function is:

$P_N\left(R_d\right)~\frac{e^{-{\mathrm{dR}}^2/2<r^2>N}}{{(2\mathrm{\&pi;}<r^2>N/d)}^{d/2}}---\left(1\right)$

In formula, < r²> be the limited two rank squares of random file amount, R is range of scatter, N is diffusion step number;

Note random file amount is a, for have < r of two-dimensional space²>＝a², d=2, obtains two-dimensional space isotropism RandomWalk probability-distribution function:

$P_N\left(R_2\right)~\frac{e^{-R^2/a^{2}N}}{{\mathrm{\&pi;a}}^{2}N}---\left(2\right)$

D dimension space isotropism RandomWalk, the probability density function from initial point to R utilizes following formula to obtain

P_N(R)＝A_dR^d-1P_N(R_d)(3)

In formula, A_dD dimension space unit ball area, A₁＝1,A₂＝2π,A₃＝4π...；

Obtain two-dimensional space isotropism RandomWalk ink diffusion equation:

$P_N\left(R\right)=\frac{2R}{a^{2}N}{e}^{-R^2/a^{2}N}---\left(4\right)$

In formula, R is range of scatter, and N is diffusion step number;

Adopt the mode of Lattice grid to divide virtual paper, definition random file amount is unit grids,a＝1；

Obtain the ink diffusion equation of bidimensional LatticeRandomWalk:

$P_N\left(R\right)=\frac{2R}{N}{e}^{-R^2/N}---\left(5\right)$

Step 2, according to the water absorption character of paper, calculate the range of scatter R in ink diffusion equation; Based on LatticeThe real-time ink diffusion model of RandomWalk; The ink saturation capacity of definition unit grids isUnit grids can be protectedHolding quantity of ink isSuppose this region total be subject to China ink amount for I, maximum range of scatter is calculated as

$R=I/I_{x,y}^s---\left(6\right)$

Step 3, the difference that measured by China ink according to virtual paper, will be subject to territory, ink-covered area to be subject to ink-condition to be divided into the paper of dry brush stateUndersaturation is subject to the paper supersaturation of ink-condition and rich ink-condition to be subject to ink-condition, and modeling respectively; For different diffusion propertiesPaper, set up the corresponding paper attenuation coefficient β of UNICOM, the current grid connectivity of virtual paper is by following iterative formulaCalculate:

Continuity(p_n-1top_n)＝βContinuity(p_n-2top_n-1)

Different diffusion property paper define different coefficients, determine the diffusion step number N in ink diffusion equation.