Inversion and Symmetries of the Star Transform

Ambartsoumian, Gaik; Latifi, Mohammad J.

doi:10.1007/s12220-021-00680-7

Cited by 3 publications

(6 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remark 10. The reconstruction artifacts, located on the edges of the support of the vector field, are similar to those that appear in the numerical inversions of the star transform on scalar fields, and their discussion can be found in [5].…”

Section: Recovery Of Solenoidal and Potential Vector Fieldssupporting

confidence: 52%

“…To generate Lf and T f , we evaluate numerically the divergent beam transforms X u of projections f , u , f , u ⊥ , and X v of f , v , f , v ⊥ , and combine these quantities according to formulas (4) and (5). The data for If and J f are generated in a similar fashion by numerical evaluation of X 1 u and X 1 v of the appropriate projections and combining the resulting quantities according to formulas (6) and (7).…”

Section: Data Formationmentioning

confidence: 99%

“…These transforms map functions to their integrals along paths or surfaces with a "vertex" inside their support, e.g. along broken rays (also called V-lines) [2,4,7,9,18,19,22,25,49,52] and stars [5,54] in R 2 , or over various conical surfaces [4,21,22,43,50] in R 3 and higher dimensions. Such operators appear in mathematical models of several imaging techniques based on scattered particles, including single scattering optical tomography [17,20], single scattering X-ray tomography [34,53], fluorescence imaging [19], Compton scattering emission tomography [40], and Compton camera imaging [51].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Generalized V-line transforms in 2D vector tomography

Ambartsoumian¹,

Jebelli²,

Mishra³

2020

Inverse Problems

Self Cite

View full text Add to dashboard Cite

We study the inverse problem of recovering a vector field in R 2 from a set of new generalized V-line transforms in three different ways. First, we introduce the longitudinal and transverse V-line transforms for vector fields in R 2 . We then give an explicit characterisation of their respective kernels and show that they are complements of each other. We prove invertibility of each transform modulo their kernels and combine them to reconstruct explicitly the full vector field. In the second method, we combine the longitudinal and transverse V-line transforms with their corresponding first moment transforms and recover the full vector field from either pair. We show that the available data in each of these setups can be used to derive the signed V-line transform of both scalar component of the vector field, and use the known inversion of the latter. The final major result of this paper is the derivation of an exact closed form formula for reconstruction of the full vector field in R 2 from its star transform with weights. We solve this problem by relating the star transform of the vector field to the ordinary Radon transform of the scalar components of the field.

show abstract

Section: Recovery Of Solenoidal and Potential Vector Fieldssupporting

confidence: 52%

Section: Data Formationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Generalized V-line transforms in 2D vector tomography

Ambartsoumian¹,

Jebelli²,

Mishra³

2020

Inverse Problems

Self Cite

View full text Add to dashboard Cite

show abstract

“…the integral of h along the line l = {x ∈ R 2 : x • ξ = s}, where ξ is the unit normal vector to the line, and s is the signed distance from the origin. From the lemmas 1 and 2 in [6] we have the following identity:…”

Section: Full Recovery Of a Tensor Field From Its Star Transformmentioning

confidence: 99%

“…Generalized Radon-type transforms, mapping a scalar function to a family of its integrals along V-lines with a vertex inside the image support, have attracted substantial interest of researchers during the last two decades (see [4] and the references therein). Numerous interesting results have been obtained about the injectivity, range description, inversion, stability and other properties of the V-line transform (VLT), as well as the closely related star transform on scalar fields in R 2 (see [3,5,6,9,10,15,19,20,34,36,37] and references therein).…”

Section: Introductionmentioning

confidence: 99%

V-line 2-tensor tomography in the plane

Ambartsoumian,

Mishra,

Zamindar

2024

Inverse Problems

Self Cite

View full text Add to dashboard Cite

In this article, we introduce and study various V-line transforms (VLTs) defined on symmetric 2-tensor fields in R2. The operators of interest include the longitudinal, transverse, and mixed VLTs, their integral moments, and the star transform. With the exception of the star transform, all these operators are natural generalizations to the broken-ray trajectories of the corresponding well-studied concepts defined for straight-line paths of integration. We characterize the kernels of the VLTs and derive exact formulas for reconstruction of tensor fields from various combinations of these transforms. The star transform on tensor fields is an extension of the corresponding concepts that have been previously studied on vector fields and scalar fields (functions). We describe all injective configurations of the star transform on symmetric 2-tensor fields and derive an exact, closed-form inversion formula for that operator.

show abstract