Keith Dillon

Techniques for finding regularized solutions to underdetermined linear systems can be viewed as imposing prior knowledge on the unknown vector. The success of modern techniques, which can impose priors such as sparsity and non-negativity, is the result of advances in optimization algorithms to solve problems which lack closed-form solutions. Techniques for characterization and analysis of the system to determine when information is recoverable, however, still typically rely on closed-form solution techniques such as singular value decomposition or a filter cutoff estimate. In this letter we propose optimization approaches to broaden the approach to system characterization.

We start by deriving conditions for when each unknown element of a system admits a unique solution, subject to a broad class of types of prior knowledge. With this approach we can pose a convex optimization problem to find “how unique” each element of the solution is, which may be viewed as a generalization of resolution to incorporate prior knowledge. We find that the result varies with the unknown vector itself, i.e., it is data-dependent, such as when the sparsity of the solution improves the chance it can be uniquely reconstructed. The approach can be used to analyze systems on a case-by-case basis, estimate the amount of important information present in the data, and quantitatively understand the degree to which the regularized solution may be trusted.

K. Dillon and Y. Fainman, “Element-wise uniqueness, prior knowledge, and data-dependent resolution,” SIViP, pp. 1–8, Apr. 2016. (pdf)

Formulating the inverse problem as an optimization problem, we find conditions for which a system’s measurements can produce a bounded result for both the linear case and the non-negative case (e.g., intensity imaging). We also consider the problem of selecting measurements to yield the most bounded results. Finally we simulate examples of the application of bounded estimation to different two-dimensional multiview systems.

K. Dillon and Y. Fainman, “Bounding pixels in computational imaging,” Appl. Opt., vol. 52, no. 10, pp. D55–D63, Apr. 2013. (pdf)

We derive an approach for imaging attenuative sample parameters with a confocal scanning system. The technique employs computational processing to form the estimate in a pixel-by-pixel manner from measurements at the Fourier plane, rather than detecting a focused point at a pinhole. While conventional imaging system analysis and design assumes an independent scatterer at each point in the sample, attenuation must be treated with a tomographic approach. We show that a simple estimator may be derived that requires minimal computation and compare it to the conventional pinhole estimate.

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We introduce and experimentally validate a computational imaging technique that employs confocal scanning and coherent detection in the Fourier domain. We show how this method may be used to tomographically reconstruct attenuation, aberration, and even occlusion. We also show how these image parameters may be combined with the conventional confocal image reconstruction of the object reflectivity. We demonstrate the method experimentally by imaging a sample consisting of an occlusion above a mirror of varying reflectivity.

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Truncated expansions such as Zernike polynomials provide a powerful approach for describing wavefront data. However, many simple calculations with data in this form can require significant computational effort. Important examples include recentering, renormalizing, and translating the wavefront data. This paper describes a technique whereby these operations and many others can be performed with a simple matrix approach using monomials. The technique may be applied to other expansions by reordering the data and applying transformations. The key is the use of the vectorization operator to convert data between vector and matrix descriptions. With this conversion, one-dimensional polynomial techniques may be employed to perform separable operations. Examples are also given for differentiation and integration of wavefronts.

K. Dillon, “Bilinear wavefront transformation,” J. Opt. Soc. Am. A, vol. 26, no. 8, pp. 1839–1846, 2009. (pdf)

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