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

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, λ0;X,Y ) = infu+v=f { ‖u‖X + λ0‖v‖pY } . 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 λ0 is a fixed threshold which dictates separation of scales. To proceed, we iterate the refinement step [uj+1, vj+1] = arginf J(vj , λ02 j), leading to the hierarchical decomposition, f = ∑k j=0 uj + vk. We focus our attention on the particular case of (X,Y ) = (BV,L2) decomposition. The resulting hierarchical decomposition, f ∼ ∑ j uj , 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.