engleski

Adaptive Compressive Sensing in Polynomial and Shift-Invariant Spaces

This thesis concerns solving inverse problems, i.e., the problem of reconstructing an unknown signal, image, or shape from a set of measurements. The measurements are obtained from the unknown data by a forward process, which is typically known and non- invertible. Amongst the numerous tasks that fit under this framework, the thesis focuses particularly on compressive sensing (CS) and the inverse obstacle scattering problem. Compressive sensing is a sampling and reconstruction technique that relies on the assumption that the natural signals and images are sparse in some basis or dictionary to be able to go beyond the Nyquist limit. It is based on the idea that the informational content of the observed signal can be encoded in a reduced set of linear measurements, whose number is significantly smaller than the amount of measurements required in classical sampling, and can be successfully recovered by solving an optimization problem. The CS framework has shown effectiveness in many applications where the full set of measurements is not available due to the memory or time requirements (e.g., magnetic resonance imaging), and where high-resolution imaging technologies are impractical or very expensive to implement (e.g., imaging in the non-visible spectrum). Compressive sensing is a well-studied paradigm with numerous theoretical papers. However, the vast majority of the papers concern discrete-domain CS with only a few extending the ideas to the analog domain. This thesis deals with closing the gap between continuous-domain CS and its discretization. We propose using shift-invariant (SI) function spaces and Chebyshev polynomial vector spaces to address the analog nature of the continuous-domain CS measurement procedure and the observed signal. Specifically, we generalize the typical analog CS acquisition systems to a wide class of SI subspaces, which leads to a new reconstruction method and exact recasting of the inherently continuous-domain inverse problem into a finite-dimensional CS problem. The proposed framework yields reconstruction performance gains compared to the conventional CS framework and its acquisition procedure and signal modeling. Furthermore, we propose a continuous block-based Chebyshev signal model that joins the blocks in a spline-like fashion. The model equates values and the first few derivatives on the blocks' boundaries, leaving enough degrees of freedom to fit into the CS framework. Thus, the proposed model inherently avoids the blocking artifacts and mitigates the Runge phenomenon caused by the interpolation on equidistant nodes. The parametric signal model can be recovered directly from a reduced set of CS measurements and offers several benefits when utilized in various signal processing and analysis tasks. Deep learning models are currently impacting reconstruction methods in various imaging tasks and applications ranging from medical, radar, geophysical and scientific imaging. Recently, learning-based approaches achieved great success in leveraging large training datasets to: i) directly compute regularized reconstructions, and ii) train deep generative models that regularize inverse problems by constraining their solutions to remain on learned manifolds. In this perspective, we propose a multilevel strategy for subsampling of principal component projections for CS. Principal component analysis is used to extract additional structure beyond the sparsity in a training dataset, and the multilevel strategy makes the measurement matrix composed of the principal components perform well with the conventional l1 decoder. In addition, the thesis tackles with the inverse obstacle scattering problem. The scattering object is an obstacle with provided boundary conditions and the objective is to determine the obstacle shape from observations of the scattered waves. We propose an implicit neural representation-based framework for inverse acoustic obstacle scattering. The obstacle shape is represented as a zero-level set of a fully-connected neural network which offers several benefits - it easily handles complicated shapes and their perturbations, analytical computation of gradients and higher-order derivatives that are independent of grid resolutions, the optimization problem can be solved directly in the level-set framework, and the problem is more tractable with a few parameters to optimize. On top of that, we employ deep generative models of neural obstacle shape representations to effectively regularize solutions of the inverse scattering problem.

B-spline ; Chebyshev polynomials ; compressive sensing ; compressive imaging ; deep generative models ; finite rate of innovation ; generalized sampling theorem ; adversarial generation of implicit shape representations ; image sampling and reconstruction ; implicit neural representation ; inverse problem ; machine learning ; polynomial representation ; random demodulator ; shift-invariant spaces ; single-pixel camera ; sparsity ; statistical compressive sensing

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