DIGITAL IMAGE PROCESSING 4th phần 9 docx

VISUAL TEXTURE DISCRIMINATION 547 16.5. VISUAL TEXTURE DISCRIMINATION A discrete stochastic field is an array of numbers that are randomly distributed in amplitude and governed by some joint probability density (12,13). When converted to light intensities, such fields can be made to approximate natural textures surpris- ingly well by control of the generating probability density. This technique is useful for generating realistic appearing artificial scenes for applications such as airplane flight simulators. Stochastic texture fields are also an extremely useful tool for investigating human perception of texture as a guide to the development of texture feature extraction methods. In the early 1960s, Julesz (14) attempted to determine the parameters of stochas- tic texture fields of perceptual importance. This study was extended later by Julesz et al. (15–17). Further extensions of Julesz’s work have been made by Pollack (18), FIGURE 16.4-2. Brodatz texture fields. ( a ) Sand ( b ) Grass ( c ) Wool ( d ) Raffia 548 IMAGE FEATURE EXTRACTION Purks and Richards (19) and Pratt et al. (13,20). These studies have provided valu- able insight into the mechanism of human visual perception and have led to some useful quantitative texture measurement methods. Figure 16.5-1 is a model for stochastic texture generation. In this model, an array of independent, identically distributed random variables passes through a linear or nonlinear spatial operator to produce a stochastic texture array . By controlling the form of the generating probability density and the spatial operator, it is possible to create texture fields with specified statistical proper- ties. Consider a continuous amplitude pixel at some coordinate in . Let the set denote neighboring pixels but not necessarily nearest geo- metric neighbors, raster scanned in a conventional top-to-bottom, left-to-right fash- ion. The conditional probability density of conditioned on the state of its neighbors is given by (16.5-1) The first-order density employs no conditioning, the second-order density implies that J = 1, the third-order density implies that J = 2, and so on. 16.5.1. Julesz Texture Fields In his pioneering texture discrimination experiments, Julesz utilized Markov process state methods to create stochastic texture arrays independently along rows of the array. The family of Julesz stochastic arrays are defined below (13). 1. Notation. Let denote a row neighbor of pixel and let P(m), for m = 1, 2, , M, denote a desired probability generating function. 2. First-order process. Set for a desired probability function P(m). The resulting pixel probability is (16.5-2) FIGURE 16.5-1. Stochastic texture field generation model. Wjk,() O · {} Fjk,() pW() x 0 jk,() Fjk,() z 1 z 2 … z J ,, ,{} x 0 px 0 z 1 … z J ,,() px 0 z 1 … z J ,, ,() pz 1 … z J ,,() = px 0 () px 0 z 1 () x n Fjk n–,()= x 0 x 0 m= Px 0 () Px 0 m=()Pm()== VISUAL TEXTURE DISCRIMINATION 549 3. Second-order process. Set for , and set , where the modulus function for integers p and q. This gives a first-order probability (16.5-3a) and a transition probability (16.5-3b) 4. Third-order process. Set for , and set for . Choose to satisfy . The governing probabilities then become (16.5-4a) (16.5-4b) (16.5-4c) This process has the interesting property that pixel pairs along a row are inde- pendent, and consequently, the process is spatially uncorrelated. Figure 16.5-2 contains several examples of Julesz texture field discrimination tests performed by Pratt et al. (20). In these tests, the textures were generated according to the presentation format of Figure 16.5-3. In these and subsequent visual texture discrimination tests, the perceptual differences are often small. Proper discrimination testing should be performed using high-quality photographic trans- parencies, prints or electronic displays. The following moments were used as simple indicators of differences between generating distributions and densities of the sto- chastic fields. (16.5-5a) (16.5-5b) (16.5-5c) (16.5-5d) Fj1,() m= Pm() 1 M⁄= x 0 x 1 m+()MOD M{}= p MOD q{}≡ pqpq÷()×[]– Px 0 () 1 M = px 0 x 1 ()Px 0 x 1 m+()MOD M{}=[]Pm()== Fj1,()m= Pm() 1 M⁄= Fj2,()n= Pn() 1 M⁄= x 0 2x 0 x 1 x 2 m++()=MODM{} Px 0 () 1 M = px 0 x 1 () 1 M = px 0 x 1 x 2 ,()P 2x 0 x 1 x 2 m++()MOD M{}=[]Pm()== η Ex 0 {}= σ 2 Ex 0 η–[] 2 {}= α Ex 0 η–[]x 1 η–[]{} σ 2 = θ Ex 0 η–[]x 1 η–[]x 2 η–[]{} σ 3 = 550 IMAGE FEATURE EXTRACTION The examples of Figure 16.5-2a and b indicate that texture field pairs differing in their first- and second-order distributions can be discriminated. The example of Figure 16.5-2c supports the conjecture, attributed to Julesz, that differences in third-order, and presumably, higher-order distribution texture fields cannot be perceived provided that their first- and second-order distributions are pairwise identical. FIGURE 16.5-2. Field comparison of Julesz stochastic fields; . ( a ) Different first order s A = 0.289, s B = 0.204 ( b ) Different second order s A = 0.289, s B = 0.289 a A = 0.250, a B =− 0.250 ( c ) Different third order s A = 0.289, s B = 0.289 a A = 0.000, a B = 0.000 q A = 0.058, q B =− 0.058 η A η B 0.500== VISUAL TEXTURE DISCRIMINATION 551 16.5.2. Pratt, Faugeras and Gagalowicz Texture Fields Pratt et al. (20) have extended the work of Julesz et al. (14–17) in an attempt to study the discrimination ability of spatially correlated stochastic texture fields. A class of Gaussian fields was generated according to the conditional probability density (16.5-6a) where (16.5-6b) (16.5-6c) The covariance matrix of Eq. 16.5-6a is of the parametric form FIGURE 16.5-3. Presentation format for visual texture discrimination experiments. px 0 z 1 … z J ,,() 2π() J 1+ K J 1+ 1–2⁄ 1 2 – v J 1+ η J 1+ –() T K J 1+ () 1– v J 1+ η J 1+ –() ⎩⎭ ⎨⎬ ⎧⎫ exp 2π() J K J 1–2⁄ 1 2 – v J η J –() T K J () 1– v J η J –() ⎩⎭ ⎨⎬ ⎧⎫ exp = v J z 1 z J = … v J 1+ x 0 v J = 552 IMAGE FEATURE EXTRACTION (16.5-7) where denote correlation lag terms. Figure 16.5-4 presents an example of the row correlation functions used in the texture field comparison tests described below. Figures 16.5-5 and 16.5-6 contain examples of Gaussian texture field comparison tests. In Figure 16.5-5, the first-order densities are set equal, but the second-order nearest neighbor conditional densities differ according to the covariance function plot of Figure 16.5-4a. Visual discrimination can be made in Figure 16.5-5, in which the correlation parameter differs by 20%. Visual discrimination has been found to be marginal when the correlation factor differs by less than 10% (20). The first- and second-order densities of each field are fixed in Figure 16.5-6, and the third-order FIGURE 16.5-4. Row correlation factors for stochastic field generation. Dashed line, field A; solid line, field B. ( b ) Constrained third-order density ( a ) Constrained second-order density K J 1+ 1 α β γ … α βσ 2– K J γ = … αβγ…,,, VISUAL TEXTURE DISCRIMINATION 553 conditional densities differ according to the plan of Figure 16.5-4b. Visual dis- crimination is possible. The test of Figure 16.5-6 seemingly provides a counter- example to the Julesz conjecture. In this test, and , but . However, the general second-order density pairs and are not necessarily equal for an arbitrary neighbor , and therefore the conditions necessary to disprove Julesz’s conjecture are violated. To test the Julesz conjecture for realistically appearing texture fields, it is nec- essary to generate a pair of fields with identical first-order densities, identical FIGURE 16.5-5. Field comparison of Gaussian stochastic fields with different second-order nearest neighbor densities; . FIGURE 16.5-6. Field comparison of Gaussian stochastic fields with different third-order nearest neighbor densities; . ( a ) a A = 0.750, a B = 0.900 ( b ) a A = 0.500, a B = 0.600 η A η B 0.500 σ A ,σ B 0.167== == p A x 0 () p B x 0 ()=[] p A x 0 x 1 ,()p B x 0 x 1 ,()= p A x 0 x 1 x 2 ,,()p B x 0 x 1 x 2 ,,()≠ p A x 0 z j ,() p B x 0 z j ,() z j ( a ) b A = 0.563, b B = 0.600 ( b ) b A = 0.563, b B = 0.400 η A η B 0.500 σ A ,σ B 0.167 α A ,α B 0.750== == == 554 IMAGE FEATURE EXTRACTION Markovian type second-order densities, and differing third-order densities for every pair of similar observation points in both fields. An example of such a pair of fields is presented in Figure 16.5-7 for a non-Gaussian generating process (19). In this example, the texture appears identical in both fields, thus supporting the Julesz conjecture. Gagalowicz has succeeded in generating a pair of texture fields that disprove the Julesz conjecture (21). However, the counterexample, shown in Figure 16.5-8, is not very realistic in appearance. Thus, it seems likely that if a statistically based FIGURE 16.5-7. Field comparison of correlated Julesz stochastic fields with identical first- and second-order densities, but different third-order densities. FIGURE 16.5-8. Gagalowicz counterexample. h A = 0.500, h B = 0.500 s A = 0.167, s B = 0.167 a A = 0.850, a B = 0.850 q A = 0.040, q B =− 0.027 TEXTURE FEATURES 555 texture measure can be developed, it need not utilize statistics greater than second-order. Because a human viewer is sensitive to differences in the mean, variance and autocorrelation function of the texture pairs, it is reasonable to investigate the sufficiency of these parameters in terms of texture representation. Figure 16.5-9 pre- sents examples of the comparison of texture fields with identical means, variances and autocorrelation functions, but different nth-order probability densities. Visual discrimination is readily accomplished between the fields. This leads to the conclu- sion that these low-order moment measurements, by themselves, are not always suf- ficient to distinguish texture fields. 16.6. TEXTURE FEATURES As noted in Section 16.4, there is no commonly accepted quantitative definition of visual texture. As a consequence, researchers seeking a quantitative texture measure have been forced to search intuitively for texture features, and then attempt to evalu- ate their performance by techniques such as those presented in Section 16.1. The following subsections describe several texture features of historical and practical importance. References 22 to 24 provide surveys on image texture feature extrac- tion. Randen and Husoy (25) have performed a comprehensive study of many tex- ture feature extraction methods. FIGURE 16.5-9. Field comparison of correlated stochastic fields with identical means, variances and autocorrelation functions, but different nth-order probability densities gener- ated by different processing of the same input field. Input array consists of uniform random variables raised to the 256th power. Moments are computed. h A = 0.413, h B = 0.412 s A = 0.078, s B = 0.078 a A = 0.915, a B = 0.917 q A = 1.512, q B = 0.006 556 IMAGE FEATURE EXTRACTION 16.6.1. Fourier Spectra Methods Several studies (8,26,27) have considered textural analysis based on the Fourier spectrum of an image region, as discussed in Section 16.2. Because the degree of texture coarseness is proportional to its spatial period, a region of coarse texture should have its Fourier spectral energy concentrated at low spatial frequencies. Con- versely, regions of fine texture should exhibit a concentration of spectral energy at high spatial frequencies. Although this correspondence exists to some degree, diffi- culties often arise because of spatial changes in the period and phase of texture pat- tern repetitions. Experiments (10) have shown that there is considerable spectral overlap of regions of distinctly different natural texture, such as urban, rural and woodland regions extracted from aerial photographs. On the other hand, Fourier spectral analysis has proved successful (28,29) in the detection and classification of coal miner’s black lung disease, which appears as diffuse textural deviations from the norm. 16.6.2. Edge Detection Methods Rosenfeld and Troy (30) have proposed a measure of the number of edges in a neighborhood as a textural measure. As a first step in their process, an edge map array is produced by some edge detector such that for a detected edge and otherwise. Usually, the detection threshold is set lower than the normal setting for the isolation of boundary points. This texture measure is defined as (16.6-1) where is the dimension of the observation window. A variation of this approach is to substitute the edge gradient for the edge map array in Eq. 16.6-1. A generalization of this concept is presented in Section 16.6.4. 16.6.3. Autocorrelation Methods The autocorrelation function has been suggested as the basis of a texture measure (30). Although it has been demonstrated in the preceding section that it is possible to generate visually different stochastic fields with the same autocorrelation function, this does not necessarily rule out the utility of an autocorrelation feature set for nat- ural images. The autocorrelation function is defined as (16.6-2) Ejk,() Ejk,() 1= Ejk,() 0= Tjk,() 1 W 2 Ej mk n+,+() nw–= w ∑ mw–= w ∑ = W 2w 1+= Gjk,() A F mn,() Fjk,()Fj mk n–,–() k ∑ j ∑ = [...]... S, K 9. 80 8.47 Grass – raffia Set 1a Grass – sand Field Pair 8 .98 13 .93 14.43 3.85 4.64 10.61 6. 39 Set 1 8. 89 13.75 14.38 3.76 4. 59 10. 49 6.37 Set 2 7.20 10 .90 12.72 2.74 2.48 8.74 5.61 Set 3 Laplacian Texture Feature 5.88 8.47 10.86 2. 49 2.31 6 .95 4.21 Set 4 12.20 17.28 18.75 6.75 5.62 9. 46 15.34 Set 1 9. 08 11. 19 12.3 6.40 4.05 8.15 12.34 Set 2 7.31 8.24 10.52 5. 39 1.87 6.33 11.48 Set 3 Sobel 5 .99 6.08... 12.60 1. 59 1.04 1.15 Grass – raffia 4.52 Set 2b 4.61 Set 1a Grass – sand Field Pair 4.20 0. 39 8.24 10 .93 1.07 0.51 4.04 Set 3c Whitening 0.88 1.47 2. 19 0.24 0.14 0.52 0.77 Set 4d 3. 59 4. 59 7.73 2.23 2.23 3.48 1. 29 Set 1 3.51 4.43 7.65 2.14 2. 19 3.38 1.28 Set 2 2.17 1.53 7.42 1.57 1.76 0.55 0. 19 Set 3 Laplacian Texture Feature 1.24 3.13 1.40 0.28 0.13 1.87 0.66 Set 4 6.31 7.73 9. 98 5. 09 2 .98 2.20 9. 90 Set... “Analysis of Multichannel Narrow-Band Filters for Image Texture Segmentation,” IEEE Trans Signal Processing, 39, 9, September 199 1, 2025–2043 40 D Gabor, “Theory of Communication,” J Institute of Electrical Engineers, 93 , 194 6, 4 29 457 41 S Grigorescu, N Petkov and P Kruizinga, “Comparison of Texture Features Based on Gabor Filters,” IEEE Trans Image Processing, 11, 10, October 2002, 1160–1167 42 S... 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Markov Random Fields,” IEEE Trans Information Theory, IT-18, 2, March 197 2, 232–240 13 W K Pratt, O D Faugeras and A Gagalowicz, “Applications of Stochastic Texture Field Models to Image Processing, ” Proc IEEE, 69, 5, May 198 1, 542–551 14 B Julesz, “Visual Pattern Discrimination,” IRE Trans Information Theory, IT-8, 1, February 196 2, 84 92 15 B Julesz et al., “Inability of Humans to Discriminate Between . 13.22 12 .98 3.85 3.76 2.74 2. 49 6.75 6.40 5. 39 5.13 Sand – wool 19. 14 19. 08 17.43 15.72 14.43 14.38 12.72 10.86 18.75 12.3 10.52 8. 29 Raffia – wool 13. 29 13.14 10.32 7 .96 13 .93 13.75 10 .90 8.47. sand 9. 80 9. 72 8 .94 7.48 6. 39 6.37 5.61 4.21 15.34 12.34 11.48 10.12 Grass – raffia 8.47 8.34 6.56 4.66 10.61 10. 49 8.74 6 .95 9. 46 8.15 6.33 4. 59 Grass – wool 4.17 4.03 1.87 1.70 4.64 4. 59 2.48. raffia 12.76 12.60 10 .93 0.24 2.23 2.14 1.57 0.28 5. 09 4. 79 3.51 2.30 Sand – wool 12.61 12.55 8.24 2. 19 7.73 7.65 7.42 1.40 9. 98 5.01 1.67 0.56 Raffia – wool 4.20 3.87 0. 39 1.47 4. 59 4.43 1.53 3.13