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The subject of this thesis is image restoration, that is, deconvolution and denoising. Our work is motivated by applications in the rapidly expanding field of biological imaging and in particular fluorescence microscopy. Inverse problems in this area typically involve regularly structured but high-dimensional data sets, thus requiring a specific tradeoff between restoration quality, computational complexity and user interaction. Our interest lies in novel algorithmic solutions for addressing these challenges. The common point of our developments is that they rely centrally on wavelets and multilevel concepts. The thesis is organized in three main parts. In the first part, we provide a general overview of fluorescence microscopy in the context of biological research. We then review the current state of the art in deconvolution microscopy, within a unifying variational framework. The second part is concerned with deconvolution using a non-quadratic (typically ℓ1) wavelet-domain regularization. In the context of deconvolution microscopy, this approach is novel. It amounts to imposing a sparsity constraint on the object to be recovered, a concept that has recently attracted considerable interest in signal and image processing. This type of regularization was shown to belong to the most effective restoration methods, but it involves a computationally challenging optimization problem. We propose a multilevel version of the standard "thresholded Landweber" algorithm that yields a substantial acceleration. This allows us to apply wavelet regularization to large-scale deconvolution problems encountered in deconvolution microscopy. In the third part, we present some applications of risk estimation techniques for automating the process of image restoration. We first propose a denoising method that is specifically designed for very low light intensities. The algorithm is designed from the beginning for complete self-adjustment of its parameters and for low computational complexity. It also performs competitively with most existing approaches for Poisson intensity estimation. Our second contribution is a general method for estimating the signal-to-noise ratio during the execution of iterative image-restoration procedures; it is applicable to most algorithms emanating from variational formulations, in particular those involving wavelet regularization.

Acoustic tomography aims at recovering the unknown parameters that describe a field of interest by studying the physical characteristics of sound propagating through the considered field. The tomographic approach is appealing in that it is non-invasive and allows to obtain a significantly larger amount of data compared to the classical one-sensor one-measurement setup. It has, however, two major drawbacks which may limit its applicability in a practical setting: the methods by which the tomographic data are acquired and then converted to the field values are computationally intensive and often ill-conditioned. This thesis specifically addresses these two shortcomings by proposing novel acoustic tomography algorithms for signal acquisition and field reconstruction. The first part of our exposition deals with some theoretical aspects of the tomographic sampling problems and associated reconstruction schemes for scalar and vector tomography. We show that the classical time-of-flight measurements are not sufficient for full vector field reconstruction. As a solution, an additional set of measurements is proposed. The main advantage of the proposed set is that it can be directly computed from acoustic measurements. It thus avoids the need for extra measuring devices. We then describe three novel reconstruction methods that are conceptually quite different. The first one is based on quadratic optimization and does not require any a priori information. The second method builds upon the notion of sparsity in order to increase the reconstruction accuracy when little data is available. The third approach views tomographic reconstruction as a parametric estimation problem and solves it using recent sampling results on non-bandlimited signals. The proposed methods are compared and their respective advantages are outlined. The second part of our work is dedicated to the application of the proposed algorithms to three practical problems: breast cancer detection, thermal therapy monitoring, and temperature monitoring in the atmosphere. We address the problem of breast cancer detection by computing a map of sound speed in breast tissue. A noteworthy contribution of this thesis is the development of a signal processing technique that significantly reduces the artifacts that arise in very inhomogeneous and absorbent tissue. Temperature monitoring during thermal therapies is then considered. We show how some of our algorithms allow for an increased spatial resolution and propose ways to reduce the computational complexity. Finally, we demonstrate the feasibility of tomographic temperature monitoring in the atmosphere using a custom-built laboratory-scale experiment. In particular, we discuss various practical aspects of time-of-flight measurement using cheap, off-the-shelf sensing devices.