Date of Award
Doctor of Philosophy (PhD)
The single-scatter approximation is fundamental for many tomographic imaging problems. This class broadly includes x-ray scattering imaging and optical scatter imaging for certain media. In all cases, noisy measurements are affected by both local events and nonlocal attenuation. Related applications typically focus on reconstructing one of two images: scatter density or total attenuation. However, both images are media specific. Both images are useful for object identification. Knowledge of one image aides estimation of the other, especially when estimating images from noisy data.Joint image recovery has been demonstrated analytically in the context of the broken ray transform (BRT) for attenuation and scatter-density images. The BRT summarizes the nonlocal affects of attenuation in single-scatter measurement geometries. We find BRT analysis particularly interesting as joint image recovery has been demonstrated analytically using only two scatter angles. Limiting observations to two scatter angles is significant because it supports joint reconstruction in two dimensions for anisotropic scatter modalities (e.g. Bragg, Compton). However, all analytic inversion strategies share two fundamental assumptions limiting their utility: nonzero scatter everywhere, and a deterministic data model.There are two themes to our work. First, we consider the BRT in a purely deterministic setting. We are the first to recognize the BRT as a linear shift-invariant operator. This linear-systems perspective motivates frequency-domain analysis both of the data and operator. Frequency-domain representations provide new insights on the operator and a common framework for contrasting recent inversion formulas. New algorithms are presented for regularized inversion of the BRT in addition to fast forward and adjoint operators. Second, we incorporate the BRT in a stochastic data model. Approximating the detectors as photon counting processes, we model the data as Poisson distributed. Our iterative algorithm, alternating scatter and attenuation image updates, guarantees monotonic reduction of the regularized log-likelihood function of the data. We are the first to consider joint image estimation from noisy data. Our results demonstrate a significant improvement over analytic methods for data sets with missing data (regions with zero scatter). In addition to joint image estimation, our approach can be specialized for single image estimation. With known attenuation, we can improve the quality of scatter image estimates. Similarly, with known scatter, we can improve the quality of attenuation image estimates.Through analysis and simulations, we highlight challenges for attenuation image estimation from BRT data, and ambiguity in the joint image recovery problem. Performance will vary with scaling of the problem. Total attenuation, detected counts, and scatter angle all affect the quality of image estimates. We are the first to incorporate both scatter density and attenuation in noisy data models. Our results demonstrate the benefits of accounting for both images, and should inform design of future measurement systems.
Joseph A. O'Sullivan
R. Martin Arthur, Ulugbek S. Kamilov, David G. Politte, Yuan-Chuan Tai,