Fig. 1. Patches with similar histograms that are perceptually indiscriminable and those with dissimilar histograms that are perceptually different. Here eight filters, consisting of the intensity filter, two local difference filters, two LoG filters, and three Gabor filters, are used to calculate the spectral histogram, and their corresponding histograms are separated by dash lines with filter profiles shown below. Here profiles are scaled for illustration purposes. The size of all the images is 128×128 and pixel values are between 0 and 255. (a, b) Two patches with their corresponding spectral histograms. The spectral histograms are similar. However, the root-mean-square distance between the two patches is large––94.0 per pixel. (c) A Gaussian noise image with its spectral histogram. The root-mean-square distance between this patch and that in (a) is 84.5 per pixel, smaller than the distance between (a) and (b).

Fig. 2. A texture and synthesized images at different sweeps. The size of the image is 128×128. (a) Observed image. (b) A synthesized image using the Gibbs sampler. The error per filter, defined as ∑_α=1^K∑_i=1^L(α)|H^(α)_Isyn(i)−H_obs^(α)(i)|/K, is 0.116. (c) Initial image for sampling. (d)–(f) Synthesized images at sweep 40, 100, and 4000 with the error per filter of 0.237, 0.098, and 0.028 respectively. (g) The error per filter with respect to the number of sweeps.

Fig. 3. Synthesized images for synthetic textures with different micropatterns. In each column, the upper part shows the observed texture and the lower part a synthesized texture at sweep 4000. (a) A texture consisting of regularly arranged hexagons. (b) A texture consisting of pluses. (c) A texture with filled circles. (d) A texture consisting of R’s. (e) A texture consisting of empty circles. (f) An image consisting of two distinct textures.

Fig. 4. Natural texture of cheetah skin. (a) An image containing a cheetah. The size of the image is 648×972. (b) The cheetah skin from the enclosed area in (a). The size of this area is 104×258. (c) A synthesized image of 256×256.

Fig. 5. Ten texture pairs scanned from Malik and Perona (1990). The size of all the scanned images is 154×154.

Fig. 6. The averaged texture gradient for two selected texture pairs in Fig. 5. (a) The texture pair (+ O). (b) The texture gradient averaged along each column for (a). The horizontal axis is the column number and the vertical axis is the gradient. (c) The texture pair (R-mirror-R). (d) The texture gradient for (c).

Fig. 7. Texture discrimination results. Here the horizontal axis corresponds to the order of the texture pairs in Table 1 and the vertical axis the texture discrimination scores. (…) Psychophysical data from Kröse (1986); (– – –) results from Malik and Perona’s model (Malik & Perona, 1990); (–––) results from the spectral histogram model.

Fig. 8. Asymmetry in texture discrimination. (a) A texture region of +’s flanked by those of L’s with the average texture gradient. The discrimination score produced by the spectral histogram model is 0.005 and the size of the image is 154×230. (b) A region of L’s flanked by those of +’s. The discrimination score is 0.018 and the size of the image is 154×223.

Fig. 9. Comparison with the Heeger and Bergen algorithm (Heeger & Bergen, 1995) for synthetic texture synthesis. In each row, the left column shows the observed image, the middle a synthesized texture using their algorithm, and the right a synthesized texture using the sampling algorithm given in Appendix A. Here the same steerable filters are used in both synthesis algorithms. (a) A texture consisting of circles. The difference between the observed histogram and the synthesized one is 0.111 per filter for their algorithm and 0.016 for our algorithm. (b) A texture consisting of pluses. The difference between the observed histogram and the synthesized one is 0.326 per filter for their algorithm and 0.013 for our algorithm.

Fig. 10. The spectral histograms of two regions with an identical mean but different variances. Here the same eight filters as in Fig. 1 are used for illustration. (a) An image consisting of such two regions, which is generated by adding Gaussian noise with different variances to a uniform image. The left region has a variance of 10 and the right 50. The size of the image is 128×128. (b) The spectral histogram of the left region. (c) The spectral histogram of the right region.

Fig. 11. Boundary detection for a natural texture image: (a) input image, whose size is 277×422; (b) texture gradient produced by the spectral histogram model; (c) detected texture boundaries.

Table 1. Texture discrimination scores