The V-line transform with some generalizations and cone differentiation

Ambartsoumian, Gaik; Jebelli, Mohammad Javad Latifi

doi:10.1088/1361-6420/aafcf3

Cited by 16 publications

(29 citation statements)

References 46 publications

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“…The scatter update expression (23) requires no changes accounting for transmission measurements. All additional terms in (8) due to transmission measurements are constant with respect to α. However, for tracking reduction in the objective (9), it is important to incorporate transmission measurements and (34) when computing (8).…”

Section: Incorporating Transmission Measurementsmentioning

confidence: 99%

“…Choosing Y to index both d and α aides separability when updating scatter image estimates. Differentiating (8) with respect to α(y) we obtain separable functions for each y ∈ Y. Further, ( 8) is convex with respect to α for µ fixed.…”

Section: Appendix a Scatter Image Fidelitymentioning

confidence: 99%

“…The I-divergence(8) is convex over α ∈ S when there exists at least one i ∈ I such that d i (y) > 0.Proof. For convenience we define ġi (y : µ) := ∂ ∂α(y) g i (y : α, µ) (40)= I 0 (y) exp − x∈X h i (y|x) µ(x)…”

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confidence: 99%

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Iterative Algorithms for Joint Scatter and Attenuation Estimation From Broken Ray Transform Data

Walker,

O'Sullivan

2020

Preprint

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The single-scatter approximation is fundamental in many tomographic imaging problems including x-ray scatter imaging and optical scatter imaging for certain media. In all cases, noisy measurements are affected by both local scatter events and nonlocal attenuation. Prior works focus on reconstructing one of two images: scatter density or total attenuation. However, both images are media specific and useful for object identification.Nonlocal effects of the attenuation image on the data are summarized by the broken ray transform (BRT). While analytic inversion formulas exist, poor conditioning of the inverse problem is only exacerbated by noisy measurements and sampling errors. This has motivated interest in the related star transforms incorporating BRT measurements from multiple source-detector pairs. However, all analytic methods operate on the log of the data. For media comprising regions with no scatter a new approach is required.We are the first to present a joint estimation algorithm based on Poisson data models for a single-scatter measurement geometry. Monotonic reduction of the log-likelihood function is guaranteed for our iterative algorithm while alternating image updates. We also present a fast algorithm for computing the discrete BRT forward operator. Our generalized approach can incorporate both transmission and scatter measurements from multiple source-detector pairs. Transmission measurements resolve low-frequency ambiguity in the joint image estimation problem, while multiple scatter measurements resolve the attenuation image. The benefits of joint estimation, over single-image estimation, vary with problem scaling. Our results quantify these benefits and should inform design of future acquisition systems.

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Section: Incorporating Transmission Measurementsmentioning

confidence: 99%

Section: Appendix a Scatter Image Fidelitymentioning

confidence: 99%

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Iterative Algorithms for Joint Scatter and Attenuation Estimation From Broken Ray Transform Data

Walker,

O'Sullivan

2020

Preprint

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“…Such operators appear in mathematical models of single-scattering optical tomography and singlescattering X-ray CT [9,10,11,15,21]. Particular attention has been drawn to the transforms with only two rays, usually called either broken ray transforms or V-line transforms [1,3,5,6,8,11,12,14,15,17,18,20]. In addition to that, multiple authors have studied various generalizations of these transforms to higher dimensions (see the review article [2] and the references there).…”

Section: Introductionmentioning

confidence: 99%

Inversion and Symmetries of the Star Transform

Ambartsoumian¹,

Jebelli²

2020

Preprint

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The star transform is a generalized Radon transform mapping a function of two variables to its integrals along "star-shaped" trajectories, which consist of a finite number of rays emanating from a common vertex. Such operators appear in mathematical models of various imaging modalities based on scattering of elementary particles. The paper presents a comprehensive study of the inversion of the star transform. We describe the necessary and sufficient conditions for invertibility of the star transform, introduce a new inversion formula, discuss its stability properties and demonstrate its efficiency on numerical examples. As an unexpected bonus of our approach, we prove a long-standing conjecture from algebraic geometry about the zero sets of elementary symmetric polynomials.

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“…Here we generalize the Radon transforms of [40] and analyze their stability microlocally. Our theory also has applications of interest in gamma ray source imaging in CST, specifically towards the broken-ray transforms of [1,2,5,9,10,11,27,36], and the cone Radon transforms of [4,13,15,20,21,23,24,25,26,28,35,37,41].…”

Section: Introductionmentioning

confidence: 99%

Microlocal analysis of generalized Radon transforms from scattering tomography

Webber¹,

Quinto²

2020

Preprint

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Here we present a novel microlocal analysis of generalized Radon transforms which describe the integrals of L 2 functions of compact support over surfaces of revolution of C ∞ curves q. We show that the Radon transforms are elliptic Fourier Integral Operators (FIO) and provide an analysis of the left projections Π L . Our main theorem shows that Π L satisfies the semi-global Bolker assumption if and only if g = q /q is an immersion. An analysis of the visible singularities is presented, after which we derive novel Sobolev smoothness estimates for the Radon FIO. Our theory has specific applications of interest in Compton Scattering Tomography (CST) and Bragg Scattering Tomography (BST). We show that the CST and BST integration curves satisfy the Bolker assumption and provide simulated reconstructions from CST and BST data. Additionally we give example "sinusoidal" integration curves which do not satisfy Bolker and provide simulations of the image artefacts. The observed artefacts in reconstruction are shown to align exactly with our predictions.

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The V-line transform with some generalizations and cone differentiation

Cited by 16 publications

References 46 publications

Iterative Algorithms for Joint Scatter and Attenuation Estimation From Broken Ray Transform Data

Iterative Algorithms for Joint Scatter and Attenuation Estimation From Broken Ray Transform Data

Inversion and Symmetries of the Star Transform

Microlocal analysis of generalized Radon transforms from scattering tomography

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