Multiscale Besov Space Smoothing of Images

Applied and Computational Mathematics Seminar
Monday, October 10, 2011 - 14:00
1 hour (actually 50 minutes)
Skiles 006
Purdue University, Department of Mathematics
We consider a variant of Rudin--Osher--Fatemi variational image smoothing that replaces the BV semi-norm in the penalty term with the B^1_\infty(L_1) Besov space semi-norm.  The space B^1_\infty(L_1$ differs from BV  in a number of ways:  It is somewhat larger than BV, so functions inB^1_\infty(L_1) can exhibit more general singularities than exhibited  by functions in BV, and, in contrast to BV, affine functions are assigned no penalty in B^1_\infty(L_1).   We provide a discrete model that uses a result of Ditzian and Ivanov to compute reliably with moduli of smoothness; we also incorporate some ``geometrical'' considerations into this model. We then present a convergent iterative method for solving the discrete variational problem.  The resulting algorithms are multiscale, in that  as the amount of smoothing increases, the results are computed using differences over increasingly large pixel distances.  Some computational results will be presented.  This is joint work with Greg Buzzard, Antonin Chambolle, and Stacey Levine.