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PROGRAM | Electrical & Computer Engineering

Smoothness on Rank-Order Path Graphs for Compressive Spectral Imaging and Inverse Problems

By: Juan Felipe Florez Ospina Chair: Gonzalo Arce


This work studies the problem of signal recovery from noisy undersampled linear measurements under the premise that the signal of interest is sufficiently smooth on a pre-specified graph. We analyze fundamental aspects of a signal estimate such as uniqueness, accuracy, and stability in this context, leading to bounds on the estimation error in addition to desirable properties of the pre-specified graph. Of course, there is no universal graph that is well suited to every application. We adopt, however, a relatively general strategy to select the graph which can be adapted to different scenarios. Specifically, we start with a simplifying assumption that the graph belongs to the family of path graphs, and ask the question: what is the graph on which the signal of interest is the smoothest over the set of path graphs? We demonstrate that such a graph is indeed the path graph constructed on the permutation of the vertex set that sorts the entries of the signal in ascending order. Such a permutation is related to the rank-order statistics of the signal, and therefore we refer to the graph as the rank-order path graph induced by the signal.

Rank-order path graphs possess structural properties such as connectedness and sparsity, which are fundamental for efficient and stable signal recovery. Our goal is thus to show that smoothness on rank-order path graphs can be used effectively to tackle complex inverse problems and particularly the problem of compressive spectral imaging (CSI) with side information (SI). CSI with SI poses an ideal scenario, where the rank-order statistics of the side information can be used to construct rank-order path graphs on which the signal of interest is sufficiently smooth. This standpoint leads to new efficient spectral image estimation algorithms that depend only on fast (iterative) inversion of highly sparse matrices. The algorithms can operate on both global and local (or patch-based) regimes by leveraging a suitably designed collection of graphs. Patch-based algorithms lead to significant speedups in spectral image estimation, and also we observe they enjoy a certain level of robustness against non-uniform illumination and specular reflections. Nonetheless, unlike the global approach, the proposed patch-based algorithms are susceptible to instabilities when the patch size is not large enough. We thus propose strategies to overcome these difficulties.

The proposed methods are evaluated with simulated and real measurements. For real data, we implement a compressive spectral imaging system with two different side information cameras; detailed descriptions of hardware design and calibration procedures are provided. We compare the proposed methods with traditional and state-of-the-art approaches. Graphs inferred from side information using popular graph learning methods can also be suitable for CSI, so we explore alternatives to rank-order path graphs. Our numerical and real experiments demonstrate the advantages of our approach.

Lastly, we elaborate on the role of rank-order path graphs relative to the emerging field of learned-based regularization. We propose alternatives to overcome the current limitations of the method and present new avenues for future research. These include adaptive CSI systems, regularization by denoising on rank-order path graphs using the DCT, and learning to rank for inverse problems.

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