Solving Inverse Problems with Piecewise Linear Estimators: From Gaussian Mixture Models to Structured Sparsity

A general framework for solving image inverse problems is introduced in this paper. The approach is based on Gaussian mixture models, estimated via a computationally efficient MAP-EM algorithm. A dual mathematical interpretation of the proposed framework with structured sparse estimation is described, which shows that the resulting piecewise linear estimate stabilizes the estimation when compared to traditional sparse inverse problem techniques. This interpretation also suggests an effective dictionary motivated initialization for the MAP-EM algorithm. We demonstrate that in a number of image inverse problems, including inpainting, zooming, and deblurring, the same algorithm produces either equal, often significantly better, or very small margin worse results than the best published ones, at a lower computational cost.Comment: 30 page

Learning with Limited Labeled Data in Biomedical Domain by Disentanglement and Semi-Supervised Learning

In this dissertation, we are interested in improving the generalization of deep neural networks for biomedical data (e.g., electrocardiogram signal, x-ray images, etc). Although deep neural networks have attained state-of-the-art performance and, thus, deployment across a variety of domains, similar performance in the clinical setting remains challenging due to its ineptness to generalize across unseen data (e.g., new patient cohort). We address this challenge of generalization in the deep neural network from two perspectives: 1) learning disentangled representations from the deep network, and 2) developing efficient semi-supervised learning (SSL) algorithms using the deep network. In the former, we are interested in designing specific architectures and objective functions to learn representations, where variations in the data are well separated, i.e., disentangled. In the latter, we are interested in designing regularizers that encourage the underlying neural function\u27s behavior toward a common inductive bias to avoid over-fitting the function to small labeled data. Our end goal is to improve the generalization of the deep network for the diagnostic model in both of these approaches. In disentangled representations, this translates to appropriately learning latent representations from the data, capturing the observed input\u27s underlying explanatory factors in an independent and interpretable way. With data\u27s expository factors well separated, such disentangled latent space can then be useful for a large variety of tasks and domains within data distribution even with a small amount of labeled data, thus improving generalization. In developing efficient semi-supervised algorithms, this translates to utilizing a large volume of the unlabelled dataset to assist the learning from the limited labeled dataset, commonly encountered situation in the biomedical domain. By drawing ideas from different areas within deep learning like representation learning (e.g., autoencoder), variational inference (e.g., variational autoencoder), Bayesian nonparametric (e.g., beta-Bernoulli process), learning theory (e.g., analytical learning theory), function smoothing (Lipschitz Smoothness), etc., we propose several leaning algorithms to improve generalization in the associated task. We test our algorithms on real-world clinical data and show that our approach yields significant improvement over existing methods. Moreover, we demonstrate the efficacy of the proposed models in the benchmark data and simulated data to understand different aspects of the proposed learning methods. We conclude by identifying some of the limitations of the proposed methods, areas of further improvement, and broader future directions for the successful adoption of AI models in the clinical environment

Plug-and-Play Methods for Integrating Physical and Learned Models in Computational Imaging

Plug-and-Play Priors (PnP) is one of the most widely-used frameworks for solving computational imaging problems through the integration of physical models and learned models. PnP leverages high-fidelity physical sensor models and powerful machine learning methods for prior modeling of data to provide state-of-the-art reconstruction algorithms. PnP algorithms alternate between minimizing a data-fidelity term to promote data consistency and imposing a learned regularizer in the form of an image denoiser. Recent highly-successful applications of PnP algorithms include bio-microscopy, computerized tomography, magnetic resonance imaging, and joint ptycho-tomography. This article presents a unified and principled review of PnP by tracing its roots, describing its major variations, summarizing main results, and discussing applications in computational imaging. We also point the way towards further developments by discussing recent results on equilibrium equations that formulate the problem associated with PnP algorithms