During the last five years, several convex optimization algorithms have been proposed for solving inverse problems. Most of the time, they allow us to minimize a criterion composed of two terms one of which permits to “stabilize ” the solution. Different choices are possible for the so-called regularization term, which plays a prominent role for solving ill-posed problems. While a total variation regularization introduces staircase effects, a wavelet regularization may bring other kinds of visual artefacts. A compromise can be envisaged combining these regularization functions. In the context of Poisson data, we propose in this paper an algorithm to achieve the minimization of the associated (possibly constrained) convex optimization problem. 1.

Abstract—We introduce an extended family of continuous-domain stochastic models for sparse, piecewise-smooth signals. These are specified as solutions of stochastic differential equations, or, equivalently, in terms of a suitable innovation model; the latter is analogous conceptually to the classical interpretation of a Gaussian stationary process as filtered white noise. The two specific features of our approach are 1) signal generation is driven by a random stream of Dirac impulses (Poisson noise) instead of Gaussian white noise, and 2) the class of admissible whitening operators is considerably larger than what is allowed in the conventional theory of stationary processes. We provide a complete characterization of these finite-rate-of-innovation signals within Gelfand’s framework of generalized stochastic processes. We then focus on the class of scale-invariant whitening operators which correspond to unstable systems. We show that these can be solved by introducing proper boundary conditions, which leads to the specification of random, spline-type signals that are piecewise-smooth. These processes are the Poisson counterpart of fractional Brownian motion; they are nonstationary and have the same-type spectral signature. We prove that the generalized Poisson processes have a sparse representation in a wavelet-like basis subject to some mild matching condition. We also present a limit example of sparse process that yields a MAP signal estimator that is equivalent to the popular TV-denoising algorithm. Index Terms—Fractals, innovation models, Poisson processes, sparsity, splines, stochastic differential equations, stochastic processes,

Wavelet-domain ℓ1-regularization is a powerful approach for solving inverse problems. In their 2004 landmark paper, Daubechies et al. proved that one could solve such linear inverse problems by means of a “thresholded Landweber ” (TL) algorithm [1]. While this iterative procedure is simple to implement, it is known to converge slowly. Here, we present a multilevel version of the algorithm that is inspired from the multigrid techniques used for solving PDEs, but with one important difference: instead of cycling through coarser versions of the problem (REDUCE part of multigrid), the multilevel algorithm cycles through the successive wavelet subspaces. The method works with arbitrary wavelet representations; it typically yields a 10-fold speed increase over the standard TL algorithm, while providing the same restoration quality. We illustrate the applicability of the method to three biomedical image reconstruction problems: the deconvolution of 3D fluorescence micrographs [2], the global reconstruction of dynamic PET from time measurements [3], and the reconstruction of magnetic resonance images from arbitrary (non-uniform) k-space trajectories. We present experimental results with real data sets in all three cases.

Traditional wavelets have a number of vanishing moments that corresponds to their equivalent order of the derivation. They offer good energy compaction for piecewise smooth signals, but are less appropriate for more complex signals such as those originating in functional imaging; e.g., the hemodynamic response after brain activation in functional magnetic resonance imaging (fMRI) and time activity curves (TACs) in positron emission tomography (PET). The framework of exponential-spline wavelets [1] allows us to design new wavelet bases that act like a given differential operator; i.e., they can be tuned to the characteristics of a system and yield a sparse representation of some corresponding class of signals. We show two examples. For fMRI, the wavelets are tuned according to the hemodynamic response of the system. The combination with a ℓ1-regularization constraint reveals brain activation patterns without the knowledge of a stimulation paradigm [2]. For dynamic PET, the wavelets are tailored to the compartmental description of the dynamics of the tracer distribution. The ℓ1-regularization constraint can then