Tests of isotropy for rough textures of trended images

Abstract

Abstract:In this paper, we propose a statistical methodology to test whether the texture of an image is isotropic or not. This methodology is based on the well-known quadratic variations defined as averages of square image increments. Specific to our approach, these variations are computed in different directions using grid-preserving image rotations. We study these variations asymptotically in a framework of intrinsic random fields allowing us to take into account the presence of polynomial trends in images. We establish a convergence result linking variation and scale logarithms through an asymptotic Gaussian linear model. This model involves direction-dependent intercepts which are equal when textures are isotropic. Hence, we test the texture isotropy using Fisher tests that check the validity of the assumption of the intercept equality. These tests are validated using 6000 realizations of anisotropic fractional Brownian fields simulated on a grid of size 100 × 100. Results show that more than 70% of anisotropic cases can be detected with less than 1% of misclassified isotropic cases. Eventually, we apply our methodology to the segmentation of some biological microscopic images.Key words and phrases: texture analysis, image processing, isotropy, anisotropy, intrinsic random field, statistical test, fractional Brownian field, quadratic variations, biological imaging, microscopy.

Some anisotropy indices for the characterization of Brownian textures and their application to breast images

“…holds for any pair of M -increments Z λ,x and Z µ,y of Z. They have a spectral representation [29,31,30,13] which extends the Bochner representation of stationary field covariances. For a large class of M -IRF, this representation is 85 characterized by a spectral density f .…”

“…For instance, an AFBF cannot describe images with anisotropic textures and isotropic trends. Hence, following [13], we will make the assumption that images are sampled from an M -IRF whose spectral density f fulfills the condition…”

“…This method is an implementation of the characterization of anisotropic Besov space by hyperbolic wavelet transforms [12]. In this paper, we propose some original features, called anisotropy indices, which aims at characterizing the field anisotropy within a framework of extended AFBF (see [13] and Section 2). 25 For the definition of these indices, we set some requirements.…”

“…By contrast, it is possible to accurately estimate some direc- 35 tional quantities which are indirectly related to parameters of AFBF. In [13], such quantities were brought out during the construction of isotropy tests (see Section 2 for details). They are attached to the use of multi-oriented quadratic variations, and accurately estimated using linear regression techniques.…”

“…Besides, the Hölder irregularity of a M -IRF can be characterized from the asymptotic behavior of its spectral density at high-frequencies [33,13]. Let Z be a field with a density f satisfying Condition (6).…”