<H1>6. Results</H1>
        We tested our system on four fabric samples, including a knit and the three common weave patterns (plain weave, twill, and satin), and three fiber types (cotton, wool, and synthetic): cotton satin (#4), rayon/spandex knit (#12), cotton denim (#14), and wool/cotton blend (#18). Each fabric was tested with seven deformations (see Section 3.2): for fitting, stretch in X and Y, simple shear, and bending in X and Y; and for evaluation, complex shearing and corner pulling. Figure 6 shows the four individual tension measurements for a typical stretching test, plotted as a function of the overall extension of the cloth. The measurement shows the typical behavior of a woven fabric: a nonlinear curve with increasing stiffness for higher strain, and large hysteresis. The test repeats three times, retracing the same loop each time after the initial extension from rest. We worked with three cloth models built from the components described in Section 4. The Springs model uses the spring membrane model with the spring bending model; the Soft Constraints model uses Baraff and Witkin’s membrane model with the Discrete Shells bending model; and the St. VK model uses the diagonalized St. Venant-Kirchoff membrane model with the Discrete Shells bending model. We fit all the models in four variants: linear (constant stiffness for each deformation mode), isotropic (identical stiffness in warp and weft), linear and isotropic (the simplest variant), and nonlinear orthotropic (the most general variant). The results are too numerous to include in the paper; we refer the reader to the supplementary material, which illustrates the behavior of the nonlinear orthotropic variant of all three models for all four fabrics, and the behavior of the variants of the Soft Constraints model for denim, a largely nonlinear and anisotropic material. Observations. In Figure 7 and Table 2 we present results for fitting the Soft Constraints model to Sample #12. For each test we show a selected frame (near maximum distortion) with renderings illustrating the captured and fitted cloth geometry and forces. To illustrate the fitting residuals more quantitatively, we show a force-displacement plot comparing a summary of the measured forces to the predictions of the fitted model and a vector-field plot illustrating the position error over the geometry of the fitted mesh (see caption for details). Similar tables are shown for all cloth samples and several model variants in the supplementary document. The four selected fabrics span a large range of possible cloth behaviors. In a nutshell, #12 is isotropic and very compliant in stretch and bending; #4 is also isotropic, very stiff in stretch but compliant in bending; #14 is stiff and quite isotropic in stretch, but extremely anisotropic in bending (with 33/1 stiffness ratio in weft and warp); and #18 is anisotropic both in stretch (with 10/1 stiffness ratio) and in bending (with 13/1 stiffness ratio). The maximum stretch stiffness for #4 is 250 times higher than for #12, while #14 is 10 times stiffer in shear than any other fabric. All four fabrics show similar hysteresis behavior, with loading-to-unloading stretch stiffness ratios ranging from 1.4/1 to 1.8/1. Sample #12 is nearly linear in the test deformation range, while all other three fabrics exhibit nonlinearity. Interestingly, nonlinearity may arise in some deformation modes but not in others, with no clear pattern. For stretching, all three cloth models fit nicely to the average of the hysteresis bands, even in highly nonlinear cases. The fitting residual is larger for stiffer fabrics, and the nonlinear orthotropic model variants fit anisotropic fabrics best, as expected, while linear and/or isotropic variants reach a reasonable compromise but are not always able to remain inside the hysteresis band. For shearing, the fitting force residual is larger for #14, the stiffest fabric. Across models, the Soft Constraints and St. VK models fit to the average of the shearing hysteresis band, while the Springs model deviates at times. For bending, no forces are available, and we evaluate the position residual as well as profiles of sample curves orthogonal to the support plane. The fitting residual is similar for all fabrics, but distinctly higher for the Springs model. Often, the residual is dominated by a difference in curl near the edge of the sample, while the overall shape is well fit. The last two rows of Figure 7 show the evaluation tests; these data were not used in fitting the models. The behavior of sample #12, the most linear fabric, is predicted well in all cases, as seen in the force-displacement plots, the buckling behavior in corner pulling, and the (lower) effective shear stiffness of the sheet when allowed to buckle in the complex shear test. In the three other samples, however, the force-displacement plot of the corner pulling test reveals that the stiffness is underestimated for the Soft Constraints and
        c 2012 The Author(s) c 2012 The Eurographics Association and Blackwell Publishing Ltd.
        Miguel et al. / Data-Driven Estimation of Cloth Simulation Models
        Scenario Stretch-X Stretch-Y Shear Bend-X Bend-Y Residual 0.53N 0.62N 0.22N 21.3mm 22.0mm
        
          Table 2: RMS residuals of non-linear orthotropic fitting to Sample #12 with the Soft Constraints model.
        
        St. VK models, and overestimated for the Springs model. Visually, the mismatch is more apparent in the complex shear test, where models with underestimated stiffness exhibit wider folds than the real fabrics. We have also evaluated the fitted models on new test samples of each fabric, to validate their generality. Specifically, we have tested stretching on new samples of rayon/spandex knit (#12.2) and cotton denim (#14.2), and shearing on new samples of cotton satin (#4.2) and wool/cotton blend (#18.2). The force-displacement plots of the real cloth samples, shown in the supplementary document, indicate very similar behavior between fitting and test samples for #12 and #14, and a larger disparity for #4 and #18. The evaluation plots for the simulation models behave similar for the test and fitting cases, but the matching quality depends on the actual disparity across cloth samples. Discussion. While overall force-displacement behavior is nicely matched, the actual folding shapes of simulated cloth may deviate largely from the captured cloth, because even a small change in material properties may lead to distant stable configurations in the L 2 sense. For this reason, the traditional L 2 metric is not appropriate for evaluating error in this case. The discontinuity of stable configurations is also the cause of flickering and twitches in some of our examples. The Springs model exhibits the worst fitting quality in shearing force-displacement curves, and the highest fitting residual for bending. This is probably due to the inherent coupling of stretch and bending deformation components in this model. Nevertheless, the overall deformations in complex shearing fit reasonably well. In contrast to continuum models, complex parameter tuning has often been regarded as a caveat of mass-spring models; but our results indicate that satisfactory parameter estimation is possible by incorporating anisotropy and nonlinearity into the model. The Soft Constraints and St. VK models produce results with very similar quality, which is expected as the models present only subtle differences as described in Section 4.1. At least three effects are missed by the tested models: hysteresis, Poisson effect (due to the diagonalization of the standard StVK model), and cross-modal stiffening (e.g., shear stiffening due to stretching). We indeed identified stretch stiffening in the shearing deformations, therefore we chose clip-parallel forces as objective function to minimize the effect of stretch errors on shear optimization. We conjecture that missing cross-modal stiffening may also be, to a large extent, the reason for stiffness underestimation in the corner pulling test for the Soft Constraints and St. VK models. An extension to the nonlinear model of Wang et al. [WRO11]
 could help alleviate these problems.
        c 2012 The Author(s) c 2012 The Eurographics Association and Blackwell Publishing Ltd.
        Miguel et al. / Data-Driven Estimation of Cloth Simulation Models
        
          
          Figure 7: Fitting results for Sample #12 with the Soft Constraints model. Top to bottom; Stretch-X, Simple shear, Bend-X, Corner pull, Complex shear. Left to right: captured geometry, equilibrium of fitted model, force comparison (thin line: measurement; thick line: model), position residual (vertex position minus corresponding measured position, magnified 5x). “Effective shear” is the shear angle of the best-fit transformation to the motion of the clamped cloth vertices. “Aggregated torque” is the torque about the center of the cloth applied by the clamps, with every other clamp counted negatively. “Stretching force difference” is the difference in the two corner-to-corner stretching forces.
        
      
      
        