International Journal of Computer Vision

, Volume 107, Issue 1, pp 75–97 | Cite as

A Klein-Bottle-Based Dictionary for Texture Representation

  • Jose A. PereaEmail author
  • Gunnar Carlsson


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 \(\fancyscript{K}\subset {\mathbb {R}}^{n^2}\) with the topology of a Klein bottle (Carlsson et al. International Journal of Computer Vision 76(1):1–12, 2008), we present in this paper a novel framework for the estimation and representation of distributions around \(\fancyscript{K}\), of patches from texture images. More specifically, we show that most \(n\times n\) patches from a given image can be projected onto \(\fancyscript{K}\) yielding a finite sample \(S\subset \fancyscript{K}\), 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 multi-scale rotation-invariant descriptor. We test it by classifying the materials in three popular data sets: The CUReT, UIUCTex and KTH-TIPS texture databases.


Texture representation Texture classification Klein bottle Fourier coefficients Patch distribution Density estimation 



Jose Perea was partially supported by the National Science Foundation (NSF) through grant DMS 0905823. Gunnar Carlsson was supported by the NSF through grants DMS 0905823 and DMS 096422, by the Air Force Office of Scientific Research through grants FA9550-09-1-0643 and FA9550-09-1-0531, and by the National Institutes of Health through grant I-U54-ca149145-01.


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsDuke UniversityDurhamUSA
  2. 2.Department of MathematicsStanford UniversityStanfordUSA

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