The problem of reconstructing a wavelet-sparse signal from its partial Fourier information has received a lot of attention since the emergence of compressive sensing (CS). The latest theory within the CS framework analyzes the local coherence between the Fourier and wavelet bases, and recover the signal from frequencies randomly selected according to a variable density profile. Unlike these developments, we adopt a new approach that does not need to analyze the (local) coherence. We show that the problem can be tackled by recovering the wavelet coefficients from the finest to the coarse scale, and only a small set of frequencies are needed to recover the coefficients exactly. As long as the scaling function satisfies a mild condition, the reconstruction is exact. Moreover the frequency set can be deterministically pre-selected and does not need to change even if the wavelet basis changes.

The theory of Finite Rate of Innovation (FRI) can be applied to sampling and reconstructing certain classes of parametric signals. The objective of this paper is to have a sub-Nyquist sampling scheme for continuous-time wavelet-sparse signals within the general framework of FRI theory. Though the signal has a parametric representation in the wavelet basis, it is not possible to recover the signal merely from its low-pass samples, which makes the problem different from the conventional FRI settings. The need for the Fourier coefficients at frequencies widely spread over the spectrum puts challenges on the design of the sampling kernel. This paper presents a new family of sampling kernels that are able to stably reproduce exponentials over a wide range of frequencies and gives numerical examples on applying the new kernel to sampling wavelet-sparse signals.

This paper presents new methods for computing the step sizes of the subband-adaptive iterative shrinkage-thresholding algorithms proposed by Bayram & Selesnick and Vonesch & Unser. The method yields tighter wavelet-domain bounds of the system matrix, thus leading to improved convergence speeds. It is directly applicable to non-redundant wavelet bases, and we also adapt it for cases of redundant frames. It turns out that the simplest and most intuitive setting for the step sizes that ignores subband aliasing is often satisfactory in practice. We show that our methods can be used to advantage with reweighted least squares penalty functions as well as L1 penalties. We emphasize that the algorithms presented here are suitable for performing inverse filtering on very large datasets, including 3D data, since inversions are applied only to diagonal matrices and fast transforms are used to achieve all matrix-vector products.

Image convolution is conventionally approximated by the LTI discrete model. It is well recognized that the higher the sampling rate, the better is the approximation. However sometimes images or 3D data are only available at a lower sampling rate due to physical constraints of the imaging system. In this paper, we model the under-sampled observation as the result of combining convolution and subsampling. Because the wavelet coefficients of piecewise smooth images tend to be sparse and well modelled by tree-like structures, we propose the L0 reweighted-L2 minimization (L0RL2) algorithm to solve this problem. This promotes model-based sparsity by minimizing the reweighted L2 norm, which approximates the L0 norm, and by enforcing a tree model over the weights. We test the algorithm on 3 examples: a simple ring, the cameraman image and a 3D microscope dataset; and show that good results can be obtained.

This paper proposes to use an extended Gaussian Scale Mixtures (GSM) model instead of the conventional l1norm to approximate the sparseness constraint in the wavelet domain. We combine this new constraint with subband-dependent minimization to formulate an iterative algorithm on two shift-invariant wavelet transforms, the Shannon wavelet transform and dual-tree complex wavelet transform (DTCWT). This extented GSM model introduces spatially varying information into the deconvolution process and thus enables the algorithm to achieve better results with fewer iterations in our experiments.