Regularized Local Multiband Complex Wavelet Analysis for Piecewise Homogeneous Anisotropic Self-Similar Textures

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Abstract. Texture analysis consists of a classical and everlasting task in image processing, often involved in a broad range of applications, possibly very different in nature. For large classes of textures, scale-free (or fractal) spatial dynamics as well as anisotropy constitute key properties. However, the local (pixelwise) joint estimation of anisotropy and scale-free attributes constitutes a difficult challenge, as both consist of nonlocal properties. Yet accurately detecting variations of these attributes across the image is often crucial. The overarching goal of the present work is thus to propose an inverse problem formulation for the analysis of piecewise homogeneous textures, grounded jointly on scale-free dynamics and anisotropy, and to study its performance for local assessment of textures. The formulation combines several major contributions. First, piecewise homogeneous Gaussian fields are defined as texture mixture models, with prescribed scale-free and anisotropy properties. Second, multiband complex wavelet coefficients, implemented via (nondecimated) dual-tree fast algorithms, are theoretically shown to be sensitive to both scale-free dynamics and anisotropy. Notably, it is shown that the squared-modulus of the wavelet coefficients behaves locally (or pixelwise) approximately as power-laws, with both scaling exponent and intercept jointly sensitive to anisotropy and scale-free dynamics. Thus, a key originality of the present work is to propose to perform local analysis of intrinsically nonlocal properties without having recourse to local averages, which would significantly impair accurate segmentation or accurate local characterization. Third, these local power-law-like behaviors, combined with regularization terms enforcing piecewise homogeneity, are embedded into a minimization procedure, solved by an optimization algorithm, whose convergence conditions are theoretically well-studied, and practically implemented through an efficient proximal algorithm due to strong convexity of the resulting minimization problem. Segmentation performance achieved by the proposed procedure is quantified and compared with respect to the difficulty of the segmentation task, for synthetic piecewise homogeneous Gaussian fields. The potential of the proposed texture analysis tool is also illustrated at work on real-world textures.