Dyeing Theory based Liquid Diffusion Model on Woven ClothYuki Morimoto, *Kyushu Universityreo@verygood.aid.design.kyushu-u.ac.jp

Masayuki Tanaka, The University of Tokyotanaka.research@gmail.com

Reiji Tsuruno *tsuruno@design.kyushu-u.ac.jp

Kiyoshi Tomimatsu *tomimatsu@design.kyushu-u.ac.jp

Outline of Dyeing = Overview of Our Method

Visual Characteristics of Dyeing on Woven Cloth

1. Prepare the cloth →Cloth model

2, Decide distribution of dye →Protecting & Dyeing Table

3, Add dye →Set the initial amount of dye

4, Dye diffusion →Diffusion model

5, Finish dye diffusion →Finish at a arbitral time

(a) thin spots, (b) bleeding ("nijimi"), (c) mottles, and (d) different shapes caused by the amount of dye.Left:Batik simulation[1],

Right:Chinise ink simulation[2].

Cloth Model

Diffusion ModelDiffusion Equation with Variable D

D by Tortuosity and Porosity

Capacity of Dye Amount for Diffusion and Adsorption

Examples of Tie-dyeing(left) and Yuzen-dyeing(right).

① ② ③

④

⑤

Some studies have been done in this area into watercolor painting and Chinese ink painting, both of which involve diffusion of

pigment in paper. While paper consists of small fibers, the cloth consists of weaving wefts and warps. Because of the warp and weft

threads the characteristics of dye diffusion in cloth are found, which is different to that of paper. In cloth, as there are thin colored

threads, the mottles that occur between threads are small because of the diffusion coefficient, Different shapes of stains are caused by

the amount of dye used. Our method can simulate these visual characteristics.

Vmax

VVu

Vd

Diffusion cell⑦

Figure 1. Computer generated dye stains with various parameters.

This poster describes a method for simulating and visualizing dyeing based on weave patterns. Methods for simulating painting implements and drawing strokes are being advanced in the field of NPR. There have been several studies into methods for simulating of painting techniques on cloth, including the batik and

Chinese ink painting techniques. However, characteristics of dyeing on woven cloth were not represented in these researches. We describe the characteristics of the diffusion of pigments in cloth, and propose a way to simulate these features. For this we apply Fick's second law (diffusion equation) with a variable

diffusion coefficient. We calculate the diffusion coefficient along the cloth structure and so on to model the liquid diffusion on a wide variety of woven cloths. We define dye diffusion, diffusion coefficient, adsorption based on the dyeing theories. We describe a system to simulate a successive dyeing process.

(d)

Figure 3. The differences between dye stains on different weave patterns. (a) Plain weave, (b) diagonal weave, (c) satin weave.

Figure 5. Comparison of simulations with some adsorption models. (a), (b), (c) are calcurated with adsorption model (a” ), (b” ), (c” ) in ⑨. (a'), (b'), (c') show only the absorbed dye density of (a)(b)(c).

We represent the cloth by cells. It is apparent that the cloth structure can be represented using a two-layer model

because the woven cloth consists of a combination of weft and warp threads. In our model, we first produce ①the

weft layer and then the warp layer. Each layer is then divided into widths of threads before being subdivided into

cells. We refer to these cells as ②“cloth cell” . The definition of a weave pattern is done by setting ④the smallest

and basic pattern which is then arranged iteratively. Furthermore, each cloth cell is divided into diffusion cells, and

the factor of the gap is added. These ⑤diffusion cells are used to calculate dye diffusion.

Our dye diffusion model is based on Fick’ s second law which can be represented as diffusion equation (eq 1)[3]. Usually, it is

discretized to make a simple form by constant D value. However, we discretized it to obtain (eq 2) which can consider the variable D.

⑨a” is the case of b=1 in eq 6, b” is the case of b>1 in eq 6,c” is the case in eq 7.

Figure 2. Computer generated different shapes caused by the amount of dye.

[3] A. Fick. On liquid diffusion, 1855.

[4] B. S.-H. Diffusion/adsorption behaviour of reactive dyes in cellulose. Dyes and Pigments, 1997.

[1] B. Wyvill, K. van Overveld, and S. Carpendale. Rendering cracks in batik, 2004.

[2] Jintae Lee, Diffusion rendering of black ink paintings using new paper and ink models, April 2001.

[5] B. R. van den. Human exposure to soil contamination: a qualitative and quantitative analysis towards proposals for human toxicological intervention values (partly revised edition), 1994.

[6] N. Adabala and N. Magnenat-Thalmann. A procedural thread texture model, 2003. (This is a reference for generating cloth textures.)

Direction of the weft

Direction of the warp

⑥τ3I :diffirent layerτ3II :two fibers are in the same layer, and are connected to each other 　 perpendicularlyτ3III : fiber and gapτ3IV : gap and gapτ3V : fiber and fiber in the same layer

Result

Figure 4. The simulation of seikaiha along the time step.

333

1111111555

4

⑨⑨⑨⑨⑨⑨⑨⑨⑨⑨⑨⑨⑨⑨

Parameters from Figs. 1,3,4,5. r is a random value in (0.5, 1] for each yarn. ※In Fig. 4,5, the initial amount of dye is kept constant during the simulation.

The diffusion coefficient is calculated between diffusion cells by eq 4 that is based on the Weisz-Zollinger model [4] in

accordance with our definition for tortuosity T (eq 3); where P is the porosity that can have any value in the range (0, 1], φo

denotes the dye concentration in the external solution when equilibrium is achieved and is given arbitrarily. Tortuosity T denotes

the degree of twist such that the smaller the value of the tortuosity is, the bigger the twist is. We define three kinds of tortuosity in

our method to calculate T (eq 3) : one is the twist of the thread (τ1), another is the position of a thread, including its orientation in

the weave pattern (τ2), the other is from the different orientations of fibers in neighboring diffusion cells (τ3). Each of these

tortuosities has values in the range (0, 1]. τ3 has five different conditions (see ⑥) which depend on properties such as the layer

neighboring diffusion cells are in and their porosity. It is important to represent various patterns of diffusion in cloth. Do is the

diffusion coefficient in free water and is calculated according to eq 5 defined by Berg in 1994, where M denotes the molecular

mass of dye [5]. We set P and τ1 in each thread within the cloth cell and set τ2 in only the part around diffusion cells in a different

position and set τ3 in all diffusion cells.

The total amount of dye in a diffusion cell V is defined by Vu and Vd (⑦, eq 9), where Vu is the dye capacity minus the

dye absorbed in the diffusion cell and enable to represent characteristics such as thin colored threads and some dyeing

techniques which use protection against dyeing such as with the resist paste using methods, tie-dyeing, batik and so on.

The parameter B that denotes the volume ratio in the range (0, 1]. We calculate the distribution of B based on the

normalized average RGB values in an input image. We call this image “Protecting Table.” We set the distribution of the

initial amount of dye by an input image. This is called the “Dyeing Table.” ⑧

　In chemical dyeing, the amount of adsorbed dye is significant in the visible effects of dyeing. So, the maximum amount

of dye adsorption (fixing dye) into fibers (Ad) is calculated each time step by adsorption isotherm based on dyeing theories

(⑨, eq 6,7) in our model. Also the volume capacity of the dye absorption in the diffusion cell Vd is determined by the

ratio of fiber (1-P) in the diffusion cell according to eq 6. These two things control the amount of dye adsorption（Vd >

Ad）.

Our Method

⑧Red region in (a) is distribution of the dye (Dyeing Table), while the black region is the pressure distribution (Protecting Table). (b) is the result with (a), (c) is a real dyed cloth showing the seikaiha pattern.