A MULTISCALE IMAGE REPRESENTATION USING HIERARCHICAL (BV,L2) DECOMPOSITIONS
A MULTISCALE IMAGE REPRESENTATION USING HIERARCHICAL (BV,L2) DECOMPOSITIONS
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Date
2004
Authors
TADMOR, EITAN
NEZZAR, SUZANNE
VESE, LUMINITA
Advisor
Citation
E. Tadmor, S. Nezzar & L. Vese (2004). A multiscale image representation using hierarchical (BV,L2) decompositions. Multiscale Modeling & Simulation 2 (4) (2004) 554-579.
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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, λ0; X, Y ) = infu+v=f u X + λ0 v p
Y . 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, λ02j ), 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.