## Abstract

This paper proposes an inverse approach for modeling the appearance of interwoven cloth. Creating the desired appearance in cloth is difficult because many factors, such as the type of thread and the weaving pattern, have to be considered. Design tools that enable the desired visual appearance of the cloth to be replicated are therefore beneficial for many computer graphics applications. In this paper, we focus on the design of the appearance of interwoven cloth whose reflectance properties are significantly affected by the weaving patterns. Although there are several systems that support editing of weaving patterns, they lack an inverse design tool that automatically determines the spatially varying bidirectional reflectance distribution function (BRDF) from the weaving patterns required to make the cloth display the desired appearance. We propose a method for computing the cloth BRDFs that can be used to display the desired image provided by the user. We formulate this problem as a cost minimization and solve it by computing the shortest path of a graph. We demonstrate the effectiveness of the method with several examples.

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## Acknowledgements

This work was supported by JSPS KAKENHI Grant Number JP15H05924.

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## Appendix A: extension to multiple target images

### Appendix A: extension to multiple target images

The method described in this paper focuses on the display of a single target image. This section describes its extension to multiple target images. Let us explain the case where the user provides *two* target images, \(I^0\) and \(I^1\), with corresponding viewing directions, \({\mathbf {v}}^0\) and \({\mathbf {v}}^1\). The number of intensity levels of the two images is assumed to be *L*. Notice that \({\mathbf {v}}^0\), \({\mathbf {v}}^1\), and *L* are carefully selected, since the BRDFs of weaving patterns have limitation in their expression ability (see below for detailed discussion).

Now, our goal is to find the optimal pattern \({\mathbf {w}}_{i,j}\) that provides intensities \((I^0_i, I^1_j)\) when viewed from \({\mathbf {v}}^0\) and \({\mathbf {v}}^1\), respectively, where \(I^k_i\) is the *i*th intensity of \(I^k\). We modify the minimization problem in Eq. 3 to take into account two target images,

where \({\mathcal {W}}=\{{\mathbf {w}}_{i,j} \; | \; i, j \in (0,\ldots ,L-1)\}\) is a set of weaving patterns, \(I_{c}({\mathbf {w}}_{i,j}, {\mathbf {v}}^k)\) is the intensity of the cloth with weaving pattern \({\mathbf {w}}_{i,j}\) viewed from direction \({\mathbf {v}}^k\), and \({\mathcal {N}}\) is a set of neighboring patterns defined as \({\mathcal {N}} = \{((i,j),(i+1,j)),((i,j), (i,j+1)) \;|\; i,j \in (0,\ldots ,L-2)\}\).

Similar to the single target image case, we first compute the intensity \(I_{c}({\mathbf {w}}, {\mathbf {v}}^k)\) for all \(N_w\) weaving patterns from the two viewing directions, \({\mathbf {v}}^0\) and \({\mathbf {v}}^1\). Next, the weaving patterns are classified into \(L \times L\) groups. For a pattern \({\mathbf {w}}\), we search the set (*i*, *j*) that minimizes \(d_{i,j} = |I^0_i - I_{c}({\mathbf {w}}, {\mathbf {v}}^0)| + |I^1_j - I_{c}({\mathbf {w}}, {\mathbf {v}}^1)|\). The pattern \({\mathbf {w}}\) is classified into the (*i*, *j*)th group if \(d_{i,j}< \epsilon \) or is not classified to any group otherwise. \(\epsilon \) is a user parameter adjusted to balance the intensity error and the number of patterns in each group. We then choose the optimal set of the weaving patterns that minimize the objective function shown in Eq. 9.

Notice that, for the case of multiple target images, we have no efficient methods, such as the dynamic programming. One simple approach is a greedy search algorithm. We initialize a solution (i.e., weaving patterns in all groups) by randomly choosing a weaving pattern in each group. We next visit all groups in the raster-scan order and select the best pattern at each group such that it minimizes Eq. 9 only considering the neighboring patterns.

*Expression ability of weaving patterns* To deal with two target images, the optimal set of weaving patterns \({\mathcal {W}}^*\) should have enough expression ability; we have to have a weaving pattern that provides intensities \(I^0_i\) and \(I^1_j\) when viewed from \({\mathbf {v}}^0\) and \({\mathbf {v}}^1\), respectively, for all (*i*, *j*) pairs. Although we have the large number of weaving patterns, i.e., \(N_w = 22,874\) for \(m = 4\), it does not mean that we can always find the optimal pattern that provides a desired intensity pair.

To analyze the expression ability of BRDFs of weaving patterns, we experiment with three different conditions where we vary viewing directions and thread types as in Fig. 13 (top). We plot intensity pairs of all \(N_w\) weaving patterns in Fig. 13 (bottom), where horizontal and vertical axes represent the intensities of patterns \({\mathbf {w}}\) viewed from \({\mathbf {v}}_0\) and \({\mathbf {v}}_1\), respectively. The distribution of each plot visualizes the possible intensity pairs of weaving patterns in each condition; the larger distribution is, the higher expression ability is. Figure 13a, b, d, e can be considered as failure cases, where the same type of thread is used for the weft and warp. It is impossible to find a pattern that provides high/low intensities viewed from \(\mathbf{v}_0/ \mathbf{v}_1\) (see the red circle in Fig. 13e). On the other hand, the relatively uniform distribution is observed when using different thread types as shown in Fig. 13c, f where various intensity pairs exist. Although the intensity ranges are limited as \(I^0_i \in [0.1,0.25]\) and \(I^1_j\in [0.4,0.8]\), we can configure possible \(L \times L\) intensity pairs within the intensity range (see the red rectangle in Fig. 13f).

The plots of three different conditions show that the BRDFs of weaving patterns have inherent limitations in their expression ability. The analysis suggests that it is important to carefully select viewing directions, intensity range of images, and thread types to improve the rendering quality for two (or more) target images. Searching such a desired set of viewing directions and thread types is an interesting future work.

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Dobashi, Y., Iwasaki, K., Okabe, M. *et al.* Inverse appearance modeling of interwoven cloth.
*Vis Comput* **35, **175–190 (2019). https://doi.org/10.1007/s00371-017-1455-9

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### Keywords

- Cloth rendering
- Inverse approach
- BRDF