We consider the problem of filling in missing or damaged wavelet coe#cients due to lossy image transmission or communication. The task is closely related to classical inpainting problems, but also remarkably di#ers in that the inpainting regions are in the wavelet domain. New challenges include that the resulting inpainting regions in the pixel domain are usually not well defined, as well as that degradation is often spatially inhomogeneous. Two novel variational models are proposed to meet such challenges, which combine the total variation (TV) minimization technique with wavelet representations. The associated Euler-Lagrange equations lead to nonlinear partial di#erential equations (PDE's) in the wavelet domain, and proper numerical algorithms and schemes are designed to handle their computation. The proposed models can have e#ective and automatic control over geometric features of the inpainted images, including the sharpness and curvature information of edges.

To recover a sharp image from its blurry observation is the problem known as image deblurring. It frequently arises in imaging sciences and technologies, including optical, medical, and astronomical applications, and is crucial for allowing to detect important features and patterns such as those of a distant planet or some microscopic tissue. Mathematically, image deblurring is intimately connected to backward diffusion processes (e.g., inverting the heat equation), which are notoriously unstable. As inverse problem solvers, deblurring models therefore crucially depend upon proper regularizers or conditioners that help secure stability, often at the necessary cost of losing certain highfrequency details in the original images. Such regularization techniques can ensure the existence, uniqueness, or stability of deblurred images. The present work follows closely the general framework described in our recent monograph [18], but also contains more updated views and approaches to image deblurring, including, e.g., more discussion on stochastic signals, the Bayesian/Tikhonov approach to Wiener filtering, and the iterated-shrinkage algorithm of Daubechies et al. [30,31] for wavelet-based deblurring. The work thus contributes to the development of generic, systematic, and unified frameworks in contemporary image processing.

We propose and study quantitative measures of smoothness f ↦ → A(f) which are adapted to anisotropic features such as edges in images or shocks in PDE’s. These quantities govern the rate of approximation by adaptive finite elements, when no constraint is imposed on the aspect ratio of the triangles, the simplest example being Ap(f) = ‖ p |det(d2f)|‖Lτ which appears when approximating in the L p norm by piecewise linear elements when 1 1 = +1. The quantities A(f) are not semi-norms, τ p and therefore cannot be used to define linear function spaces. We show that these quantities can be well defined by mollification when f has jump discontinuities along piecewise smooth curves. This motivates for using them in image processing as an alternative to the frequently used total variation semi-norm which does not account for the smoothness of the edges. 1