Parametric Level Set Methods for Inverse Problems

In this paper, a parametric level set method for reconstruction of obstacles in general inverse problems is considered. General evolution equations for the reconstruction of unknown obstacles are derived in terms of the underlying level set parameters. We show that using the appropriate form of parameterizing the level set function results a significantly lower dimensional problem, which bypasses many difficulties with traditional level set methods, such as regularization, re-initialization and use of signed distance function. Moreover, we show that from a computational point of view, low order representation of the problem paves the path for easier use of Newton and quasi-Newton methods. Specifically for the purposes of this paper, we parameterize the level set function in terms of adaptive compactly supported radial basis functions, which used in the proposed manner provides flexibility in presenting a larger class of shapes with fewer terms. Also they provide a "narrow-banding" advantage which can further reduce the number of active unknowns at each step of the evolution. The performance of the proposed approach is examined in three examples of inverse problems, i.e., electrical resistance tomography, X-ray computed tomography and diffuse optical tomography

End-to-End Optimization of Metasurfaces for Imaging with Compressed Sensing

We present a framework for the end-to-end optimization of metasurface imaging systems that reconstruct targets using compressed sensing, a technique for solving underdetermined imaging problems when the target object exhibits sparsity (i.e. the object can be described by a small number of non-zero values, but the positions of these values are unknown). We nest an iterative, unapproximated compressed sensing reconstruction algorithm into our end-to-end optimization pipeline, resulting in an interpretable, data-efficient method for maximally leveraging metaoptics to exploit object sparsity. We apply our framework to super-resolution imaging and high-resolution depth imaging with a phase-change material: in both situations, our end-to-end framework computationally discovers optimal metasurface structures for compressed sensing recovery, automatically balancing a number of complicated design considerations. The optimized metasurface imaging systems are robust to noise, significantly improving over random scattering surfaces and approaching the ideal compressed sensing performance of a Gaussian matrix, showing how a physical metasurface system can demonstrably approach the mathematical limits of compressed sensing

Proceedings of the FEniCS Conference 2017

Proceedings of the FEniCS Conference 2017 that took place 12-14 June 2017 at the University of Luxembourg, Luxembourg