Solving inverse problems in imaging using robust and regularized optimization

Digital images play an important role in human life since they allow observing, analyzing, studying and characterizing the world surrounding us. Their use is ubiquitous in many applications such as medicine, biology, astronomy and industrial manufacture. Nonetheless, the desired digital images are often not available and need to be recovered from corrupted, incomplete and/or indirect observations. Determining the unknown image from the available observations is called an inverse problem. In this thesis, we study several inverse problems in imaging and investigate how to solve them using optimization techniques. These techniques are based on an accurate characterization of the physical model and the properties of the image of interest, making them robust to distortions in the acquisition and modeling processes. In the first part of the thesis, we consider a recent tomographic application called optical deflectometric tomography. The inverse problem investigated consists on obtaining, from noisy and incomplete observations, an accurate image describing the spatial distribution of the refractive index of a transparent object. In the second part of the thesis, the 2-D phase unwrapping problem is analyzed. This problem consists of estimating a phase image from its noisy wrapped observation. By working with a forward model in the derivative domain, the problem is relaxed and solved using convex optimization techniques. In the last part of the thesis, the blind deconvolution problem is studied in the context of astronomical imaging. In this problem, both the actual image and the sensing operator (a convolution kernel) are unknown and are simultaneously estimated from noisy observations.