Texture synthesis and nonparametric resampling of random fields
Elizaveta Levina and Peter J. Bickel
Source: Ann. Statist. Volume 34, Number 4
(2006), 1751-1773.
Abstract
This paper introduces a nonparametric algorithm for bootstrapping a stationary random field and proves certain consistency properties of the algorithm for the case of mixing random fields. The motivation for this paper comes from relating a heuristic texture synthesis algorithm popular in computer vision to general nonparametric bootstrapping of stationary random fields. We give a formal resampling scheme for the heuristic texture algorithm and prove that it produces a consistent estimate of the joint distribution of pixels in a window of certain size under mixing and regularity conditions on the random field. The joint distribution of pixels is the quantity of interest here because theories of human perception of texture suggest that two textures with the same joint distribution of pixel values in a suitably chosen window will appear similar to a human. Thus we provide theoretical justification for an algorithm that has already been very successful in practice, and suggest an explanation for its perceptually good results.
First Page:
Show
Hide
Keywords: Bootstrap; Markov random fields; Markov mesh models; mixing random fields; consistency; texture
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aos/1162567632
Digital Object Identifier: doi:10.1214/009053606000000588
Mathematical Reviews number (MathSciNet): MR2283716
References
Abend, K., Harley, T. J. and Kanal, L. N. (1965). Classification of binary random patterns. IEEE Trans. Inform. Theory IT--11 538--544.
Mathematical Reviews (MathSciNet): MR0191728
Digital Object Identifier: doi:10.1109/TIT.1965.1053827
Zentralblatt MATH: 0129.11801
Besag, J. E. (1974). Spatial interaction and the statistical analysis of lattice systems (with discussion). J. Roy. Statist. Soc. Ser. B 36 192--236.
Mathematical Reviews (MathSciNet): MR0373208
JSTOR: links.jstor.org
Cressie, N. and Davidson, J. L. (1998). Image analysis with partially ordered Markov models. Comput. Statist. Data Anal. 29 1--26.
Mathematical Reviews (MathSciNet): MR1665674
Cross, G. C. and Jain, A. K. (1983). Markov random field texture models. IEEE Trans. Pattern Analysis Machine Intelligence 5 25--39.
De Bonet, J. S. (1997). Multiresolution sampling procedure for analysis and synthesis of texture images. In Proc. 24th Annual Conference on Computer Graphics and Interactive Techniques 361--368. ACM Press, New York.
De Valois, R. L., Albrecht, D. G. and Thorell, L. (1982). Spatial-frequency selectivity of cells in macaque visual cortex. Vision Research 22 545--559.
Doukhan, P. (1994). Mixing: Properties and Examples. Lecture Notes in Statist. 85. Springer, New York.
Mathematical Reviews (MathSciNet): MR1312160
Efros, A. A. and Freeman, W. T. (2001). Image quilting for texture synthesis and transfer. In Proc. 28th Annual Conference on Computer Graphics and Interactive Techniques 341--346. ACM Press, New York.
Efros, A. A. and Leung, T. (1999). Texture synthesis by non-parametric sampling. In Proc. IEEE International Conference on Computer Vision 2 1033--1038. IEEE Computer Soc., Washington.
Heeger, D. J. and Bergen, J. R. (1995). Pyramid-based texture analysis/synthesis. In Proc. 22nd Annual Conference on Computer Graphics and Interactive Techniques 229--238. ACM Press, New York.
Hubel, D. H. and Wiesel, T. N. (1965). Binocular interaction in striate cortex of kittens reared with artificial squint. J. Neurophysiology 28 1041--1059.
Julesz, B. (1962). Visual pattern discrimination. IRE Trans. Information Theory 8 84--92.
Julesz, B. (1981). Textons, the elements of texture perception, and their interactions. Nature 290 91--97.
Julesz, B., Gilbert, E. N., Shepp, L. A. and Frish, H. L. (1973). Inability of humans to discriminate between visual textures that agree in second-order statistics---revisited. Perception 2 391--405.
Künsch, H. R. (1989). The jackknife and the bootstrap for general stationary observations. Ann. Statist. 17 1217--1241.
Mathematical Reviews (MathSciNet): MR1015147
Digital Object Identifier: doi:10.1214/aos/1176347265
Project Euclid: euclid.aos/1176347265
Zentralblatt MATH: 0684.62035
Levina, E. and Bickel, P. J. (2001). The Earth Mover's distance is the Mallows distance: Some insights from statistics. In Proc. IEEE International Conference on Computer Vision 2 251--256. IEEE Computer Soc., Washington.
Liang, L., Liu, C., Xu, Y., Guo, B. and Shum, H.-Y. (2001). Real-time texture synthesis by patch-based sampling. Technical Report MSR-TR-2001-40, Microsoft Research.
Liu, R. Y. and Singh, K. (1992). Moving blocks jackknife and bootstrap capture weak dependence. In Exploring the Limits of Bootstrap (R. LePage and L. Billard, eds.) 225--248. Wiley, New York.
Mathematical Reviews (MathSciNet): MR1197787
Zentralblatt MATH: 0838.62036
Malik, J. and Perona, P. (1990). Preattentive texture discrimination with early vision mechanism. J. Optical Society America A 7 923--932.
Paparoditis, E. and Politis, D. (2002). The local bootstrap for Markov processes. J. Statist. Plann. Inference 108 301--328.
Mathematical Reviews (MathSciNet): MR1947405
Digital Object Identifier: doi:10.1016/S0378-3758(02)00315-4
Zentralblatt MATH: 1016.62041
Politis, D. N. and Romano, J. P. (1993). Nonparametric resampling for homogeneous strong mixing random fields. J. Multivariate Anal. 47 301--328.
Mathematical Reviews (MathSciNet): MR1247380
Digital Object Identifier: doi:10.1006/jmva.1993.1085
Zentralblatt MATH: 0795.62087
Popat, K. and Picard, R. (1993). Novel cluster-based probability model for texture synthesis, classification, and compression. In Proc. SPIE Visual Communications and Image Processing 2094 756--768. SPIE, Bellingham, WA.
Portilla, J. and Simoncelli, E. P. (2000). A parametric texture model based on joint statistics of complex wavelet coefficients. Internat. J. Computer Vision 40 49--70.
Rajarshi, M. (1990). Bootstrap in \mboxMarkov sequences based on estimates of transition density. Ann. Inst. Math. Statist. 42 253--268.
Mathematical Reviews (MathSciNet): MR1064787
Digital Object Identifier: doi:10.1007/BF00050835
Zentralblatt MATH: 0714.62036
Simoncelli, E. and Portilla, J. (1998). Texture characterization via joint statistics of wavelet coefficient magnitudes. In Proc. Fifth IEEE International Conference on Image Processing 1 62--66. IEEE Computer Soc., Washington.
Wei, L.-Y. and Levoy, M. (2000). Fast texture synthesis using tree-structured vector quantization. In Proc. 27th Annual Conference on Computer Graphics and Interactive Techniques 479--488. ACM Press, New York.
Wu, Y. N., Zhu, S. C. and Liu, X. (2000). Equivalence of Julesz ensembles and FRAME models. Internat. J. Computer Vision 38 247--265.
Zhu, S. C., Wu, Y. N. and Mumford, D. (1998). Filters, random fields and maximum entropy (FRAME)---towards a unified theory for texture modeling. Internat. J. Computer Vision 27 107--126.
