2016/03/16 by Alexander Katsevich, Katsevich, Alexander
Engineering · Mathematics · Medicine · #Advanced Numerical Analysis Techniques #Classical Analysis and ODEs (math.CA) #FOS: Biological sciences #FOS: Mathematics #Mathematical Analysis and Transform Methods #Medical Imaging Techniques and Applications #Quantitative Methods (q-bio.QM)
paper · pdf · doi:10.48550/arxiv.1603.07617
openalex publication_date 2016/03/16 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
Let X and X^* denote a restricted ray transform along curves and a\ncorresponding backprojection operator, respectively. Theoretical analysis of\nreconstruction from the data Xf is usually based on a study of the\ncomposition X^* D X, where D is some local operator (usually a derivative).\nIf X^* is chosen appropriately, then X^* D X is a Fourier Integral Operator\n(FIO) with singular symbol. The singularity of the symbol leads to the\nappearance of artifacts (added singularities) that can be as strong as the\noriginal (or, useful) singularities. By choosing D in a special way one can\nreduce the strength of added singularities, but it is impossible to get rid of\nthem completely.\n In the paper we follow a similar approach, but make two changes. First, we\nreplace D with a nonlocal operator D that integrates Xf along a\ncurve in the data space. The result D Xf resembles the generalized\nRadon transform R of f. The function D Xf is defined on pairs\n(x0,\Θ)\∈ U\× S2, where U\⊂ mathbb R3 is an open set\ncontaining the support of f, and S2 is the unit sphere in mathbb R3.\nSecond, we replace X^* with a backprojection operator R^* that integrates\nwith respect to \Θ over S2. It turns out that if D and R^*\nare appropriately selected, then the composition R^* D X is an\nelliptic pseudodifferential operator of order zero with principal symbol 1.\nThus, we obtain an approximate reconstruction formula that recovers all the\nsingularities correctly and does not produce artifacts. The advantage of our\napproach is that by inserting D we get access to the frequency\nvariable \Θ. In particular, we can incorporate suitable cut-offs in R^*\nto eliminate bad directions \Θ, which lead to added singularities.\n