1. Introduction As, unlike outwear, women's foundation garments are delicate clothes, they may cause loss of body form and make women uncomfortable when their size is not suitable. Especially, as to the brassiere, consumers strongly demand comfort and functions of arranging breast shape as well as the essential functions of clothes [1]. Consumers cannot purchase fully satisfying brassieres, since they cannot try on them in a short period of time and compare their characteristics. It is no surprises that women find it difficult to select a suitable brassiere. Therefore, there is an increasing demand for a system that can predict the three-dimensional (3D) figure wearing the chosen brassiere in the design process and at shops. Simulation of body shape wearing foundation garments is entirely distinct from apparel simulation in that naked body shape largely varies by clothing pressure caused by interaction between the foundation garments and the body. At present, however, it is very difficult to predict breast behavior when wearing a brassiere, because structural detail and the dynamic characteristics of the breasts are not well known. Therefore, there are very few trial researches on the brassiere-wearing simulation that uses a biomechanical method and the results are off from practical use [2]. On the other hand, the authors have already shown that we can simulate brassiere-wearing figures by combining the multi-regression analysis and a 3D human body shape model [2, 3, 4, 5]. In the present paper we improve our previous method of simulating brassiere-wearing figures by introducing

Improving Prediction Power in Simulation of Brassiere-Wearing Figures

Choi Dong-Eun* , Nakamura Kensuke** and Kurokawa Takao**

* Venture Laboratory, Kyoto Institute of Technology**Graduate School of Science and Technology, Kyoto Institute of Technology

*, **Matsugasaki, Sakyo-ku, Kyoto 606-8585, Japan

Abstract: This paper proposes a new method of simulation that combines a three-dimensional shape model of a human body with the genetic algorithm for predicting breast shape changed by wearing a brassiere. The model is a B-spline surface, can describe trunk shape of any female with its 750 control points by fitting the surface to 3D points measured on her trunk surface, and can reconstruct the original body shape with an error less than 1.54 mm. Since the model structure is made common among women, their body shapes can be statistically analyzed by using the control points. Based on the locality of the B-spline surface the breast region is described with 49 specific control points. First, we related 230 naked breasts of 115 Japanese women with those wearing a full-cup brassiere through the multi-regression analysis. Through the genetic algorithmIn during this process we searched the best regression formulas among a plenty of combinations of terms, or coordinates of the 49 control points on the naked models. Second, the regression formulas obtained above were applied to other 22 naked breasts to predict their brassiere-wearing shape. Comparing visually and numerically the simulated results with the actually brassiere-wearing breasts we found that this method could precisely predict brassiere-wearing breast shape based on naked ones. Keywords: three-dimensional body shape model, brassiere-wearing simulation, genetic algorithm, regression model.

the genetic algorithm into search for suitable regression formulas. As a result we could establish simulation predicting brassiere-wearing breast shape more precisely than ever.

2. 3D Human Body Shape Model2.1 Body Shape Description by Body Shape Model In our method a 3D human body model developed by Kurokawa [6] et al. plays an important role for describing and analyzing body shape, and simulating wearing figure. The model is a periodical cylinder-like surface of the bi-cubic B-spline,

m+3 n

P (u, v) = ΣΣ Ni, 3(u) Nj, 3(v) Vi, j (1) i=1 j=1

where Np,q is a B-spline basal function and Vi,j are control points. i and j are the indices of the segments forming the B-spline surface and we make m = 30 and n = 25 for high precision modeling of the female trunk with error less than 1.54 mm. Because the spline (1) is periodical with V1,j = Vm+1, j , V2,j = Vm+2, j and V3,j = Vm+3, j, any body shape can be determined and reconstructed by the 750 control points. Moreover, the B-spline surface has the locality, that is, surface transformation induced by displacement of control points within a small region remains local. Based on this property, we can expect that the shape of any local area on the trunk surface can be expressed with a small subset of control points. In addition, the control points have the same meaning among different modeled women.

For instance, the point V12,9 relates to the shape of the small area including the nipple in all models.2.2 Normalization of Posture Our idea for wearing simulation is to relate the shape of the naked breast to that of the same breast in a brassiere with suitable regression formulas given through analysis of body shape of many female subjects and to apply the formulas to prediction of body shape of any woman wearing the same type brassiere employed in the analysis. In order to analyze breast shape variation by wearing a brassiere we need body surface data measured in the same posture before and after putting on the brassiere. However, posture control is very difficult without restraining the body. We solved this problem by normalizing postures of the body shape models derived from the measurements. The normalization is done by rotating the models on the three orthogonal axes [4, 5]. After posture normalization the origin of the models coordinate is settled at the point defined near the breasts on the median plane. 2.3 Extraction of Breast Control Points The control points that describe the breast shape are located in the breast area and its circumference part. We already found that the area transformed by wearing a brassiere was described by the 7 × 7 control points shown in Fig. 1 [2]. This fact means that 49 points × 3D = 147 coordinate values fully determine shape of a single breast, and we can use these control points to analyze the change of the breast shape before and after putting on a brassiere. According to the arrangement of the 49 control points they are sequentially numbered from right to left, and from top to bottom as i (i = 1,..., 49). The 3D coordinate of the breast control points before and after putting on the brassiere is denoted as Vi (xi, yi, zi) and V'i (x'i, y'i, z'i), respectively.

3. Body Measurement Measured subjects were 142 Japanese women with the age range from 20s to 50s. First, they were measured naked by an optical 3D measurement device. The obtained data consisted of about 160 thousand 3D

points. Softness index (I: soft, II: medial, III: stiff ) was assigned to each breast by means of touch examination by specialists [2, 5]. Next, they wear requested to wear a full cup brassiere made by Wacoal Corp. shown in Fig. 2. The size of the brassiere was determined by fitters based on Martin measurements. Their cup size ranged between A70 and E70. Then their 3D shape was optically measured again. The data reflected the shape and thickness of the brassiere itself. After completing the measurement, both data were transformed to the 3D body shape model by determining the location of the control points through fitting Eq. (1) to the data. The profile of the measured breasts is shown in Table 1(a).

4. Analysis of Breast Shape Change When Putting on a Brassiere 4.1 Previous Method As stated in 2.2 the key in this study is whether we can find regression formulas having a larger correlation coefficient. With a large coefficient the formula might predict the coordinate values of the corresponding control point for any brassiere-wearing woman with high probability. In [5], the authors proposed 15 types of regression formulas and selected one with the largest correlation coefficient for each of the 147 coordinates. Using these formulas we could simulate brassiere-wearing figures with mean sagittal error of 5 to 6 mm. This method, however, has the limitation that higher simulation accuracy cannot be expected due to narrow variability of the formula types. At this stage where we have achieved a small prediction error of 5 mm, it is extremely difficult to slightly reduce the error for a lot of unspecified

Table 1 Number of breasts measured for analysis and simulation

Softness Index I: soft II: medial III: stiff(a) Number of breasts 124 128 32

(b) Analysis 114 116 -(c) Simulation 10 12 -

Fig. 2 A full-cup brassiere used for measurement

Fig.1 An example of distribution of the 49 breast control points that describe the right breast

women. We need an innovative technique of simulation in order to decrease the prediction error. In this paper we improve this method of simulation by selecting a suitable regression formula out of actually innumerable combinations of variables. For this purpose we employ the genetic algorithm.4.2 Establishment of Regression Model Our idea of simulation is depicted in Fig. 3. The simulation process is divided into two phases; Analysis Phase and Simulation Phase. In Analysis Phase, breast shape change before and after putting on a brassiere is expressed by a regression model given by adopting more than a certain number of subjects. And the prediction (simulation) of the breast shape when wearing the same kind of brassiere as used for establishing the regression model is conducted based upon the model. In the regression model fi is expressed by Eq. (2), where we denote by V and V' the sets of control point coordinate values before and after putting on the brassiere.

v'i = fi(v), v⊂ V, v'i∈ V' (2)

Here v'i (i = 1,..., 147) is one of the control point coordinate values when wearing, and v is a subset of V. In this paper, it is assumed that fi is a linear regression formula with a constant number of terms. That is, when we use α coordinate values before wearing, then we can express as

α v'i = fi(v) = a0 + Σ akvk, vk∈ v (3) k=1

The coefficients ak ( k = 0,...,α ) are determined through the regression analysis, in which the squared error

2 ^ ε = Σ (v'i − v'i)2 (4)

S

is minimized, where S is the number of the breasts for the

analysis and v'i means the actual value of the function fi(v) determined through the regression analysis. The correlation coefficient r between {v'i}and {v'i} at this time can be used to evaluate the regression formula fi. It is clear that we cannot include all of the 147 coordinate values in Eq. (3) under the limited number of subjects. Therefore, we have to find a good regression formula fi., or a linear combination of the coordinate values, having larger r. On the basis of preliminary considerations, we decide to take α = 1, 2, 3, 4, 5 and 10. In order to search a good combination of terms (control point coordinates) in Analysis Phase we employ the genetic algorithm as illustrated in Fig. 3.4.3 Genetic Algorithm The genetic algorithm (GA) is an optimization method developed as a model for the evolution, or the natural selection of inheritance of living creature, and is generally shown by the flow chart in Fig. 4 [7]. In this algorithm, out of an assumed pool of genes, a combination of one or more genes is chosen and is considered as an individual. Through the appropriate selection process, excellent

Fig. 3 Analysis Phase and Simulation Phase in our method

Fig. 4 The genetic algorithm

gene combinations are found and survive. The evolving process is facilitated by mutation and gene crossover when creating a new generation. In our case the gene pool is V, and a gene is each individual coordinate value vi. A combination of genes, or an individual in the population, corresponds to v in Eq. (3). The algorithm generates a large population at each generation. Individuals are evaluated in terms of the correlation coefficient r after the regression analysis with Eq. (4) and the population gradually evolves toward one having larger r. In this study in order to raise searching ability, we use an improved GA other than usual one [8]. In Fig. 4, we repeat generations until r becomes large enough and converges. As a result we get 147 regression formulas fi(v) with a larger correlation coefficient. They are used to predict brassiere-wearing breast shape in Simulation Phase.

5. Results of Simulation5.1 Regression Analysis through Genetic Algorithm The 3D-measurements of the subjects and their softness indices obtained in Chapter 4 were used in order to derive suitable regression formulas and predict brassiere-wearing shape of the breasts along the story in Chapter 4 and Fig. 3. However, the class of “stiff” breasts was precluded from the data for this study because of paucity of its data as shown in Table 1(a). As there is no reason to treat right and left breasts separately, we mixed both breasts into a set of right breasts. This was done by transform coordinates of the left breast into right ones. Thus we had two sets of breasts; 124 “soft” and 128 “medial” breasts. Then they were randomly split into two; one used for Analysis Phase and another for Simulation Phase. The numbers are shown in Table 1(b) and (c).

As expected, when the number of variables increased, the correlation coefficients became larger. Figure 5 describes the distribution of the correlation coefficients obtained under α = 1, 5 and 10. The mean correlation coefficients over all the regression formulas in the soft and medial classes are 0.90 and 0.80, respectively. In any case of α = 1, 2, 3, 4, 5 and 10 variables, we had regression formulas having the mean correlation coefficients exceeding 0.75.5.2 Simulation of Brassier-Wearing We applied the regression formulas obtained in 5.1 to brassiere-wearing simulation. At first, the location of the breast control points of a brassiere wearing woman, whose data were not adopted for the analysis in 5.1, was predicted by substituting her control point coordinate values to the left side values of the regression formulas. Then the breast control points of her naked body shape model were replaced by the predicted ones. All the control points of this new model were used to represent the predicted, or simulated, brassiere-wearing body on the screen. Figure 6 shows one of the results in the case of α = 4 for each of the classes “soft” (a) and “medial” (b). At a glance we can say that the simulated figures resemble the actually brassiere-wearing figures in shape of the breasts and their surroundings. Especially, the effects of wearing the brassiere on the size and shape of the breasts and the location of the nipples are prominent. Since the simulated breast shape includes the thickness and design characteristics of the target brassiere, simple mapping of the brassiere texture on the breasts area yields a realistic image as shown in Fig. 7.5.3 Evaluation of Simulation Method When people see some shape, many of them recognize it and make its impression based on its contour.

Fig. 5 Distribution of correlation coefficients of regression formulas for x, y and z coordinates at the 49 control points

According such visual property we thought it better to examine the difference between the silhouettes of the actual and the simulated breast shape for quantitatively evaluating our method of wearing simulation. For this purpose we defined the side silhouette and the horizontal silhouette. A silhouette means a contour dividing the background and the body. The side silhouette is also a contour seen from the right side of the body and the horizontal one is contour seen from above. To find the silhouettes, we projected a vertical lattice made of 30 vertical lines, 15 horizontal lines and their 450 intersecting points onto the breast area on the model surface and the location of the projected points were calculated. Then we found the frontal most intersecting point on each projected horizontal lattice line. Connecting the given 15 points we had the side silhouette. In a similar way we found the horizontal silhouette made of 30 points. For evaluating the simulated shape of breasts, the sagittal (direction of the z axis) difference between the corresponding intersecting points on the brassiere wearing and the simulated breast surfaces was regarded as the error at the points. Based on this definition of point error, the silhouette error E was derived as the mean value of the point errors along the side or horizontal silhouette. When silhouette errors were averaged over all the members of the set for simulation, it was called as the

prediction errors on the side or horizontal silhouette. Figure 8 illustrates randomly chosen examples of side and horizontal silhouettes of actual and simulated brassiere-wearing breast shapes in the “soft’’ and “medial’’ classes when α = 1. We can detect even subtle differences in the two kinds of silhouettes. We found that larger differences appear mainly in the part where the z coordinates value sharply varies. But when both shape images like Fig. 6 were visually compared from any direction, we could not recognize the difference between the two. The overall results are listed in Table 2 in the case that α = 1, 5 and 10 for each of “soft” and “medial” classes. There is a tendency that as the number of variables becomes larger, the silhouette error E increases. We can find that the error decreased by about 1.0 mm in comparison with the previous results [5].5.4 Discussions In order to improve the prediction accuracy of brassiere-wearing breast shape we classified breasts into three classes by their softness. The effects of this classification had been suggested and confirmed in [2, 4, 5]. The same was true in this study. Although simulation was not conducted with “stiff” breasts due to the small number of them, brassiere-wearing shape of “stiff” breasts must be able to be simulated with accuracy higher than that in “soft” or “medial” breasts.

I:Soft II:MedialNumber of breasts 10 12

α = 1 Side 4.9 4.1 Horizontal 4.2 4.5 E α = 5 Side 3.8 4.4 [mm] Horizontal 5.6 5.8 α = 10 Side 5.0 4.5 Horizontal 5.5 5.8

Naked With a brassiere Simulated Naked With a brassiere Simulated (a) Soft (b) Medial

Fig. 6 Simulation of wearing a brassiere ( α = 4 )

Fig. 7 Example of texture mapping of brassiere image

Table 2 Mean prediction error E for the two classes of breast softness

The prediction accuracy of brassiere-wearing breast shape has been remarkably improved in comparison with the previous results [5]. In the previous study in order to accomplish silhouette error of 5 to 6 mm we had to prepare various types of nonlinear regression formulas having many terms. In the present study more accurate simulation was accomplished with linear regression formulas having one to several variables searched through GA. We can expect to raise the prediction power of our method by adding nonlinear variables to the gene pool. The results in this paper were derived from the small number of simulation examples. If simulation is repeated statistically enough number of times, then we may have smaller mean silhouette errors, because the variation of body shape is very large. In this study only one type of full-cup brassieres was used as a target. On the basis of our experience in the former studies [2, 4, 5] we believe that the method proposed here can be applied to any type of brassieres. And we also believe that accumulated data on regression formulas with may kinds of brassieres would certainly play productive roles in design processes and at shops.

6. Conclusions A new method of simulating brassiere-wearing figures was proposed by combining GA and the regression analysis with the 3D human body shape model. In search of suitable regression formulas for prediction of brassiere-wearing breast shape, GA was very effective. As a result the method improved prediction accuracy by 1.0 mm or more in comparison with achievement in our previous study [2, 5]. There is still room to hope for improvement by increasing the number of subjects and introducing nonlinear terms into regression formulas. The proposed method is also expected to apply to manipulating other undergarments such as girdles and body suits.

Acknowledgement: The authors would like to thank the members of Human Science Laboratory, Wacoal Corporation, Kyoto, Japan for their helpful cooperation in measuring many subjects and offering 3D data for analysis.

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Fig. 8 Comparison of bust silhouettes between brassiere-wearing model (×)and simulated brassiere-wearing one (● ).