Deep learning and optimization based interferometric and phaseless imaging

Kazemi, Samia, Binte
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Ji, Qiang
Chen, Tianyi
Lai, Rongjie
Yazici, Birsen
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Electrical engineering
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This electronic version is a licensed copy owned by Rensselaer Polytechnic Institute (RPI), Troy, NY. Copyright of original work retained by author.
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Interferometric imaging involves reconstructing an image from the cross-correlations of the scattered waves reflected from the illuminated medium. Phaseless imaging on the other hand involves reconstructing an image from the auto-correlated scattered signal measurements. The objective of this thesis is to address these two inverse problems in imaging in a unified mathematical framework and develop theory, methods and algorithms. Specifically, our objective is to design provably convergent and exact deep learning (DL) based techniques for the interferometric and phaseless image reconstruction problems. Towards this end, we first introduced a DL-based phaseless imaging approach, which we refer to as DL-Wirtinger flow or DL-WF, and theoretically established sufficient conditions on the network parameters to guarantee exact recovery under deterministic imaging geometries. We designed a corresponding deep imaging network composed of three DL-based network elements: an encoder that produces an optimal initial encoded representation from the spectral estimation, a recurrent neural network (RNN) that generates the encoded image estimation, and a decoder that produces the final image from the RNN output. The RNN is designed using the unrolling technique from the non-convex optimization based Wirtinger flow (WF) algorithm, while the learned decoding operator helps integrate prior information during the reconstruction process by restricting the recovered signals to its range space. Recovery in the lower dimensional encoded representation space leads to improved sample complexity, and all three trained network components jointly contribute towards the faster convergence rate and improved reconstruction quality compared to the optimization-based methods. Furthermore, we extended the existing mathematical tools used to study the WF algorithm under general deterministic forward maps to account for our encoded representation and decoding prior based inversion approach. We additionally designed a computationally efficient decoding prior based phaseless imaging approach that does not require prior information of the forward map unlike DLWF, and instead acquires this knowledge indirectly during a supervised training phase. The improvement in computational efficiency is accomplished, firstly, by a simple DL-based initialization step that replaces the computationally expensive spectral initialization used in DL-WF, and secondly, by addressing a modified optimization problem using the kernel PCA concept and a DL-based sample processing function that simplifies the gradient calculation at each iterative update stage. This new optimization problem necessitates an alternative mathematical tool compared to DL-WF, that accounts for the network architecture and the training set sizes while establishing theoretical performance guarantee of this approach. Later, we generalized the nature of learned prior information integrated during the recovery process by adopting imaging schemes that apply general DL-based prior information in the Bayesian sense for interferometric and phaseless imaging problems. Additionally, unlike the previous two DNs that are designed to implement a least square minimization type algorithm whose convergence to the ground truth relies heavily on the accuracy of the initial point, we instead devised a denoising scheme for the steps of the power method with the goal of optimally searching for the leading eigenvector of the back-projected measurements. The leading eigenvector of this matrix resembles the true unknown for sampling vectors offering high redundancies in the quadratic measurements. Moreover, we introduced a novel strategy for designing a denoiser formulation under certain assumptions on the unknown signals that leads to theoretical recovery guarantees. We further presented a DL-based implementation of this denoiser that offers improved computational cost, necessary for its application to high dimensional signal recovery problems, as well as enables learned regularization for attaining faster convergence. Finally, we extended our quadratic inversion technique using deep decoding prior tothe challenging wave-based imaging configuration for which the Born approximation related linearization of the sampling vectors are not valid. We introduced an RNN type network which is empirically observed to offer improved reconstruction quality and faster convergence rate compared to the state-of-the-art.
School of Engineering
Dept. of Electrical, Computer, and Systems Engineering
Rensselaer Polytechnic Institute, Troy, NY
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