A Convex Variational Model for Learning Convolutional Image Atoms from Incomplete Data.

A variational model for learning convolutional image atoms from corrupted and/or incomplete data is introduced and analyzed both in function space and numerically. Building on lifting and relaxation strategies, the proposed approach is convex and allows for simultaneous image reconstruction and atom learning in a general, inverse problems context. Further, motivated by an improved numerical performance, also a semi-convex variant is included in the analysis and the experiments of the paper. For both settings, fundamental analytical properties allowing in particular to ensure well-posedness and stability results for inverse problems are proven in a continuous setting. Exploiting convexity, globally optimal solutions are further computed numerically for applications with incomplete, noisy and blurry data and numerical results are shown.

Interpreting translation-invariant wavelet shrinkage as a new image smoothing scale space.

Coifman and Donoho (1995) suggested translation-invariant wavelet shrinkage as a way to remove noise from images. Basically, their technique applies wavelet shrinkage to a two-dimensional (2-D) version of the semi-discrete wavelet representation of Mallat and Zhong (1992), Coifman and Donoho also showed how the method could be implemented in O(Nlog N) operations, where there are N pixels. In this paper, we provide a mathematical framework for iterated translation-invariant wavelet shrinkage, and show, using a theorem of Kato and Masuda (1978), that with orthogonal wavelets it is equivalent to gradient descent in L (2)(I) along the semi-norm for the Besov space B(1) (1)(L(1)(I)), which, in turn, can be interpreted as a new nonlinear wavelet-based image smoothing scale space. Unlike many other scale spaces, the characterization is not in terms of a nonlinear partial differential equation.