A Multiscale Image Representation Using Hierarchical (BV,L2) Decompositions

Abstract

We propose a new multiscale image decomposition which offers a hierarchical, adaptive representation for the different features in general images. The starting point is a variational decomposition of an image, f = u0 + v0 , where [u0 ,v0 ] is the minimizer of a J-functional, J(f,\lambda_0; X,Y) = \inf_{u+v=f} \bigl\{\|u\|_X + \lambda_0\|v\|_Y^p\bigr\}. Such minimizers are standard tools for image manipulations (e.g., denoising, deblurring, compression); see, for example, [M. Mumford and J. Shah, { Proceedings of the IEEE Computer Vision Pattern Recognition Conference, San Francisco, CA, 1985] and [L. Rudin, S. Osher, and E. Fatemi, Phys. D, 60 (1992), pp. 259--268]. Here, u0 should capture "essential features" of f which are to be separated from the spurious components absorbed by v0 , and $\lambda_0$ is a fixed threshold which dictates separation of scales. To proceed, we iterate the refinement step [uj+1 ,vj+1 ] = \mathop{{\rm arginf}} J(vj , \lambda0 2j), leading to the hierarchical decomposition, f = \sum_{j=0}^kuj + vk . We focus our attention on the particular case of (X,Y)=(BV,L2 ) decomposition. The resulting hierarchical decomposition, f \sim \sum_juj , is essentially nonlinear. The questions of convergence, energy decomposition, localization, and adaptivity are discussed. The decomposition is constructed by numerical solution of successive Euler--Lagrange equations. Numerical results illustrate applications of the new decomposition to synthetic and real images. Both greyscale and color images are considered.