Browsing by Author "Leong, Oscar"

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    Learned Generative Priors for Imaging Inverse Problems
    (2021-04-30) Leong, Oscar; Hicks, Illya; Hand, Paul
    A ubiquitous and fundamental task across the natural sciences is an imaging inverse problem, where the goal is to reconstruct a desired image from a small number of noisy measurements. Due to the ill-posed nature of such problems, it is desirable to enforce that the reconstructed image obeys particular structural properties believed to be obeyed by the image of interest. To minimize the amount of measurements required, the desired properties often have a low-dimensional structure. Such properties are known as priors and the dominant paradigm over the last two decades or so has been to exploit the sparsity of natural images in a hand-crafted basis. In recent years, however, the field of machine learning, and deep learning in particular, has demonstrated the effectiveness of data-driven priors in the form of generative models. These models represent signals as lying on an explicitly parameterized low-dimensional manifold, and have shown to generate highly realistic, yet synthetic images from a number of complex image classes, ranging from human faces to proteins. This dissertation proposes a novel framework for image recovery by exploiting these data-driven priors, and offers three main contributions. First, we rigorously prove that these learned models can help recover images from fewer nonlinear measurements than traditional hand-crafted techniques in the challenging inverse problem, phase retrieval. We additionally discuss how our developed theory has a broader applicability to more general settings without structural information on the image. Finally, we present a method using invertible generative models to overcome dataset biases and representational issues in previous generative prior-based approaches, and theoretically analyze the method’s recovery performance in compressive sensing. This thesis, more broadly, offers a new paradigm for image recovery under deep generative priors and gives concrete empirical and theoretical evidence towards the benefits of utilizing such learned priors in a variety of inverse problems.
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    Phase Retrieval Under a Generative Prior
    (2019-04-11) Leong, Oscar; Hicks, Illya; Hand, Paul
    The phase retrieval problem, arising from X-ray crystallography and medical imaging, asks to recover a signal given intensity-only measurements. When the number of measurements is less than the dimensionality of the signal, solving the problem requires additional assumptions, or priors, on its structure in order to guarantee recovery. Many techniques enforce a sparsity prior, meaning that the signal has very few non-zero entries. However, these methods have seen various computational bottlenecks. We sidestep this issue by enforcing a generative prior: the assumption that the signal is in the range of a generative neural network. By formulating an empirical risk minimization problem and directly optimizing over the domain of the generator, we show that the objective’s energy landscape exhibits favorable global geometry for gradient descent with information theoretically optimal sample complexity. Based on this geometric result, we introduce a gradient descent algorithm to converge to the true solution. We corroborate these results with experiments showing that exploiting generative models in phase retrieval tasks outperforms sparse phase retrieval methods.