A Multiscale Image Representation Using Hierarchical ( BV , L 2 ) 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 = u 0 + v 0 , where [u 0 ,v 0 ] is the minimizer of a J-functional, J(f,λ_0; X,Y) = infu+v=f \{\|u\|_X + λ_0\|v\|_Y^p\}. 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, u 0 should capture "essential features" of f which are to be separated from the spurious components absorbed by v 0 , and λ₀ is a fixed threshold which dictates separation of scales. To proceed, we iterate the refinement step [u j+1 ,v j+1 ] = {{ arginf}} J(v j , λ 0 2 j ), leading to the hierarchical decomposition, f = ∑ⱼ₌₀^ku j + v k . We focus our attention on the particular case of (X,Y)=(BV,L 2 ) decomposition. The resulting hierarchical decomposition, f ~ ∑_ju j , 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.