Multiscale Texture Extraction with Hierarchical (BV,Gp,L2) Decomposition

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 /2 i +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.