Eric Mason

Eric Mason, Birsen Yazici

Passive synthetic aperture radar (SAR) uses existing signals of opportunity such as communication and broadcasting signals. In our prior work, we have developed a low-rank matrix recovery (LRMR) method that can reconstruct scenes with extended and densely distributed point targets, overcoming shortcomings of conventional methods. The approach is based on correlating two sets of bistatic measurements, which results in a linear mapping of the tensor product of the scene reflectivity with itself. Recognizing this tensor product as a rank-one positive semi-definite (PSD) operator, we pose passive SAR image reconstruction as a LRMR problem with convex relaxation. In this paper, we present a performance analysis of the convex LRMR-based passive SAR image reconstruction method. We use the restricted isometry property (RIP) and show that exact reconstruction is guaranteed under the condition that the pixel spacing or resolution satisfies a certain lower bound. We show that for sufficiently large center frequencies, our method provides superior resolution than that of Fourier based methods, making it a super-resolution technique. Additionally, we show that phaseless imaging is a special case of our passive SAR imaging method. We present extensive numerical simulation to validate our analysis.

Eric Mason, Hrushikesh Mhaskar, Adam Guo

A recent paper (Neural Networks, {\bf 132} (2020), 253-268) introduces a straightforward and simple kernel based approximation for manifold learning that does not require the knowledge of anything about the manifold, except for its dimension. In this paper, we examine how the pointwise error in approximation using least squares optimization based on similarly localized kernels depends upon the data characteristics and deteriorates as one goes away from the training data. The theory is presented with an abstract localized kernel, which can utilize any prior knowledge about the data being located on an unknown sub-manifold of a known manifold. We demonstrate the performance of our approach using a publicly available micro-Doppler data set, and investigate the use of different preprocessing measures, kernels, and manifold dimensions. Specifically, it is shown that the localized kernel introduced in the above mentioned paper when used with PCA components leads to a near-competitive performance to deep neural networks, and offers significant improvements in training speed and memory requirements. To demonstrate the fact that our methods are agnostic to the domain knowledge, we examine the classification problem in a simple video data set.

Bariscan Yonel, Eric Mason, Birsen Yazıcı

We introduce a deep learning (DL) framework for inverse problems in imaging, and demonstrate the advantages and applicability of this approach in passive synthetic aperture radar (SAR) image reconstruction. We interpret image recon- struction as a machine learning task and utilize deep networks as forward and inverse solvers for imaging. Specifically, we design a recurrent neural network (RNN) architecture as an inverse solver based on the iterations of proximal gradient descent optimization methods. We further adapt the RNN architecture to image reconstruction problems by transforming the network into a recurrent auto-encoder, thereby allowing for unsupervised training. Our DL based inverse solver is particularly suitable for a class of image formation problems in which the forward model is only partially known. The ability to learn forward models and hyper parameters combined with unsupervised training approach establish our recurrent auto-encoder suitable for real world applications. We demonstrate the performance of our method in passive SAR image reconstruction. In this regime a source of opportunity, with unknown location and transmitted waveform, is used to illuminate a scene of interest. We investigate recurrent auto- encoder architecture based on the 1 and 0 constrained least- squares problem. We present a projected stochastic gradient descent based training scheme which incorporates constraints of the unknown model parameters. We demonstrate through extensive numerical simulations that our DL based approach out performs conventional sparse coding methods in terms of computation and reconstructed image quality, specifically, when no information about the transmitter is available.