Limiting Aspects of Nonconvex TV theta Models

作者：Hintermueller Michael; Valkonen Tuomo^*; Wu Tao

来源：SIAM Journal on Imaging Sciences, 2015, 8(4): 2581-2621.

DOI：10.1137/141001457

• total variation
• nonconvex
• regularization
• area-strict convergence
• multiscale analysis

Recently, nonconvex regularization models have been introduced in order to provide a better prior for gradient distributions in real images. They are based on using concave energies. in the total variation-type functional TV phi (u) := integral phi(vertical bar del u(x)vertical bar) dx. In this paper, it is demonstrated that for typical choices of phi, functionals of this type pose several difficulties when extended to the entire space of functions of bounded variation, BV(Omega). In particular, if phi(t) = t(q) for q epsilon (0, 1), and TV phi is defined directly for piecewise constant functions and extended via weak* lower semicontinuous envelopes to BV(Omega), then it still holds that TV phi(u) = infinity for u not piecewise constant. If, on the other hand, TV phi is defined analogously via continuously differentiable functions, then TV phi = 0 (!). We study a way to remedy the models through additional multiscale regularization and area strict convergence, provided that the energy phi(t) = t(q) is linearized for high values. The fact that such energies actually better match reality and improve reconstructions is demonstrated by statistics and numerical experiments.

• 出版日期2015