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A Klein-bottle-based dictionary for texture representation. (English) Zbl 1328.68279
Summary: A natural object of study in texture representation and material classification is the probability density function, in pixel-value space, underlying the set of small patches from the given image. Inspired by the fact that small \(n\times n\) high-contrast patches from natural images in gray-scale accumulate with high density around a surface \(\mathcal K\subset\mathbb R^{n^2}\) with the topology of a Klein bottle, we present in this paper a novel framework for the estimation and representation of distributions around \(\mathcal K\), of patches from texture images. More specifically, we show that most \(n\times n\) patches from a given image can be projected onto \(\mathcal K\) yielding a finite sample \(S\subset\mathcal K\), whose underlying probability density function can be represented in terms of Fourier-like coefficients, which in turn, can be estimated from \(S\). We show that image rotation acts as a linear transformation at the level of the estimated coefficients, and use this to define a multiscale rotation-invariant descriptor. We test it by classifying the materials in three popular data sets: The CUReT, UIUCTex and KTH-TIPS texture databases.

68U10 Computing methodologies for image processing
68T10 Pattern recognition, speech recognition
62H30 Classification and discrimination; cluster analysis (statistical aspects)
62H35 Image analysis in multivariate analysis
62G07 Density estimation
Full Text: DOI
[1] Aherne, F. J., Thacker, N. A., & Rockett, P. I. (1998). The bhattacharyya metric as an absolute similarity measure for frequency coded data. Kybernetika, 34(4), 363–368. · Zbl 1274.62058
[2] Bell, A. J., & Sejnowski, T. J. (1997). The ”independent components” of natural scenes are edge filters. Vision Research, 37(23), 3327. · doi:10.1016/S0042-6989(97)00121-1
[3] Beyer, K., Goldstein, J., Ramakrishnan, R., and Shaft, U. (1999). When is ”nearest neighbor” meaningful? Database Theory-ICDT’99 (pp. 217–235).
[4] Broadhurst, R. E. (2005). Statistical estimation of histogram variation for texture classification. In Proc. Intl. Workshop on texture analysis and synthesis. (pp. 25–30).
[5] Brodatz, P. (1966). Textures: A photographic album for artists and designers (Vol. 66). New York: Dover.
[6] Carlsson, G. (2009). Topology and data. Bulletin of the American Mathematical Society, 46(2), 255. · Zbl 1172.62002 · doi:10.1090/S0273-0979-09-01249-X
[7] Carlsson, G., Ishkhanov, T., De Silva, V., & Zomorodian, A. (2008). On the local behavior of spaces of natural images. International Journal of Computer Vision, 76(1), 1–12. · Zbl 05322191 · doi:10.1007/s11263-007-0056-x
[8] Crosier, M., & Griffin, L. D. (2010). Using basic image features for texture classification. International Journal of Computer Vision, 88(3), 447–460. · Zbl 06023103 · doi:10.1007/s11263-009-0315-0
[9] Dana, K. J., Van Ginneken, B., Nayar, S. K., & Koenderink, J. J. (1999). Reflectance and texture of real-world surfaces. ACM Transactions on Graphics (TOG), 18(1), 1–34. · doi:10.1145/300776.300778
[10] De Silva, V., Morozov, D., & Vejdemo-Johansson, M. (2011). Persistent cohomology and circular coordinates. Discrete and Computational Geometry, 45(4), 737–759. · Zbl 1216.68322 · doi:10.1007/s00454-011-9344-x
[11] De Wit, T. D., & Floriani, E. (1998). Estimating probability densities from short samples: A parametric maximum likelihood approach. Physical Review E, 58(4), 5115. · doi:10.1103/PhysRevE.58.5115
[12] Edelman, A., & Murakami, H. (1995). Polynomial roots from companion matrix eigenvalues. Mathematics of Computation, 64(210), 763–776. · Zbl 0833.65041 · doi:10.1090/S0025-5718-1995-1262279-2
[13] Franzoni, G. (2012). The klein bottle: Variations on a theme. Notices of the AMS, 59(8), 1094–1099. · Zbl 1284.57002
[14] Lewis, D. (2005). Feature classes for 1D, 2nd order image structure arise from natural image maximum likelihood statistics. Network, 16(2–3), 301–320. · doi:10.1080/09548980500289874
[15] Griffin, L. D. (2007). The second order local-image-structure solid. IEEE Transactions on Pattern Analysis and Machine Intelligence, 29(8), 1355–1366. · Zbl 05340921 · doi:10.1109/TPAMI.2007.1066
[16] Harris, C. & Stephens, M. (1988) A combined corner and edge detector. In Proc. of Fourth Alvey Vision Conference. (pp. 147–151).
[17] Hatcher, A. (2002). Algebraic topology. Cambridge: Cambridge University Press. · Zbl 1044.55001
[18] Hayman, E., Caputo, B., Fritz, M., & Eklundh, J. O. (2004). On the significance of real-world conditions for material classification. Computer Vision ECCV, 3024, 253–266. · Zbl 1098.68776
[19] Hubel, D. H., & Wiesel, T. N. (1959). Receptive fields of single neurones in the cat’s striate cortex. The Journal of Physiology, 148(3), 574–591. · doi:10.1113/jphysiol.1959.sp006308
[20] Hubel, D. H., & Wiesel, T. N. (1968). Receptive fields and functional architecture of monkey striate cortex. The Journal of Physiology, 195(1), 215–243. · doi:10.1113/jphysiol.1968.sp008455
[21] Jurie, F. and Triggs, B. (2005). Creating efficient codebooks for visual recognition. In Computer Vision, 2005. ICCV 2005. Tenth IEEE International Conference on, IEEE. (Vol. 1, pp. 604–610).
[22] Koenderink, J. J. (1984). The structure of images. Biological Cybernetics, 50(5), 363–370. · Zbl 0537.92011 · doi:10.1007/BF00336961
[23] Lazebnik, S., Schmid, C., & Ponce, J. (2005). A sparse texture representation using local affine regions. IEEE Transactions on Pattern Analysis and Machine Intelligence, 27(8), 1265–1278. · Zbl 05112026 · doi:10.1109/TPAMI.2005.151
[24] Lee, A. B., Pedersen, K. S., & Mumford, D. (2003). The nonlinear statistics of high-contrast patches in natural images. International Journal of Computer Vision, 54(1), 83–103. · Zbl 1070.68661
[25] Leung, T., & Malik, J. (2001). Representing and recognizing the visual appearance of materials using three-dimensional textons. International Journal of Computer Vision, 43(1), 29–44. · Zbl 0972.68606 · doi:10.1023/A:1011126920638
[26] Moler, C. (1991). Cleve’s corner: Roots-of polynomials, that is. The MathWorks Newsletter, 5(1), 8–9.
[27] Pedersen, Kim S and Lee, Ann B. (2002). Toward a full probability model of edges in natural images. In Computer Vision-ECCV 2002, Springer (pp. 328–342). · Zbl 1034.68651
[28] Reed, M., & Simon, B. (1972). Methods of modern mathematical physics: Functional analysis (Vol. 1). New York: Academic Press. · Zbl 0242.46001
[29] Rubner, Y., Tomasi, C., & Guibas, L. J. (2000). The earth mover’s distance as a metric for image retrieval. International Journal of Computer Vision, 40(2), 99–121. · Zbl 1012.68705 · doi:10.1023/A:1026543900054
[30] Silverman, B. W. (1986). Density estimation for statistics and data analysis (Vol. 26). London: Chapman & Hall.
[31] van Hateren, J. H., & van der Schaaf, A. (1998). Independent component filters of natural images compared with simple cells in primary visual cortex. In Proceedings of the Royal Society of London. Series B: Biological Sciences, (Vol. 265(1394) pp. 359–366).
[32] Varma, M. and Ray, D. (2007). Learning the discriminative power-invariance trade-off. In Computer Vision, 2007. ICCV 2007. IEEE 11th International Conference on, IEEE. (pp. 1–8).
[33] Varma, M., & Zisserman, A. (2004). Unifying statistical texture classification frameworks. Image and Vision Computing, 22(14), 1175–1183. · doi:10.1016/j.imavis.2004.03.012
[34] Varma, M., & Zisserman, A. (2005). A statistical approach to texture classification from single images. International Journal of Computer Vision, 62(1), 61–81. · doi:10.1007/s11263-005-4635-4
[35] Varma, M., & Zisserman, A. (2009). A statistical approach to material classification using image patch exemplars. IEEE Transactions on Pattern Analysis and Machine Intelligence, 31(11), 2032–2047. · doi:10.1109/TPAMI.2008.182
[36] Watson, G. S. (1969). Density estimation by orthogonal series. The Annals of Mathematical Statistics, 40(4), 1496–1498. · Zbl 0188.50602 · doi:10.1214/aoms/1177697523
[37] Weinberger, K. Q., & Saul, L. K. (2009). Distance metric learning for large margin nearest neighbor classification. The Journal of Machine Learning Research, 10, 207–244. · Zbl 1235.68204
[38] Zhang, J., Marszalek, M., Lazebnik, S., & Schmid, C. (2007). Local features and kernels for classification of texture and object categories: A comprehensive study. International Journal of Computer Vision, 73(2), 213–238. · Zbl 05146333 · doi:10.1007/s11263-006-9794-4
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