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Multiscale Texture Extraction with Hierarchical (BV,G p ,L 2) Decomposition

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Abstract

In this paper, we first present a hierarchical (BV,G p ,L 2) variational decomposition model and then use it to achieve multiscale texture extraction which offers a hierarchical, separated representation of image texture in different scales. The starting point is the use of the variational (BV,G p ,L 2) decomposition; a given image f∈L 2(Ω) is decomposed into a sum of u 0+v 0+r 0, where (u 0,v 0)∈(BV(Ω),G p (Ω)) is the minimizer of an energy functional E(f,λ 0;u,v) and r 0 is the residual (i.e. r 0=f−u 0−v 0). In this decomposition, v 0 represents the fixed scale texture of f, which is measured by the parameter λ 0. To achieve a multiscale representation, we proceed to capture essential textures of f which have been absorbed by the residuals. Such a goal can be achieved by iterating a refinement decomposition to the residual of the previous step, i.e. r i =u i+1+v i+1+r i+1, where (u i+1,v i+1) is the minimizer of E(r i ,λ 0/2i+1;u,v). In this manner, we can obtain a hierarchical representation of f. In addition, we discuss some theoretical properties of the hierarchical (BV,G p ,L 2) decomposition and give its numerical implementation. Finally, we apply this hierarchical decomposition to the multiscale texture extraction. The performance of this method is demonstrated with both synthetic and real images.

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  1. College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, P.R. China

    Liming Tang & Chuanjiang He

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Tang, L., He, C. Multiscale Texture Extraction with Hierarchical (BV,G p ,L 2) Decomposition. J Math Imaging Vis 45, 148–163 (2013). https://doi.org/10.1007/s10851-012-0351-1

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