A Klein-Bottle-Based Dictionary for Texture Representation

Abstract

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 $$n\times n$$ n × n high-contrast patches from natural images in gray-scale accumulate with high density around a surface $$\fancyscript{K}\subset {\mathbb {R}}^{n^2}$$ K ? 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 $$\fancyscript{K}$$ K , of patches from texture images. More specifically, we show that most $$n\times n$$ n × n patches from a given image can be projected onto $$\fancyscript{K}$$ K yielding a finite sample $$S\subset \fancyscript{K}$$ S ? K , whose underlying probability density function can be represented in terms of Fourier-like coefficients, which in turn, can be estimated from $$S$$ 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.