{"title":"Cone-beam consistency conditions for planar trajectories with parallel and perpendicular detectors","authors":"Hung Nguyen, R. Clackdoyle, L. Desbat","doi":"10.1088/1361-6420/ad3fe3","DOIUrl":null,"url":null,"abstract":"\n Cone-beam (CB) projections provide a first-order model for x-ray imaging with an area detector. CB consistency conditions (CBCCs), also known as range conditions for the 3D divergent x-ray transform, are equations that express the redundant information in a collection of CB projections. For applications purposes, CBCCs are most suitably expressed in terms of detector coordinates. CBCCs are only known for a few geometrical configurations, which depend on the source and detector trajectories. Here we only consider source trajectories that lie in a plane, and detector orientations that are parallel to the trajectory plane, or perpendicular to it. The parallel detector is stationary, but the vertical detector rotates around the center of the circular trajectory. We unify and generalize the existing known CBCCs for planar trajectories, by creating an intermediate geometry consisting of a parallel, rotating detector, and we develop new CBCCs for this geometry. Our main result is a theorem on CBCCs for a perpendicular detector, which must necessarily move in response to movement of the source. We also provide a theorem for the more difficult situation of a perpendicular detector but without the restriction that the target object be on one side or the other of the trajectory plane. We present a simple numerical simulations for a toy calibration problem to provide an example application of the new CBCCs.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.0000,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Inverse Problems","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1088/1361-6420/ad3fe3","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
Cone-beam (CB) projections provide a first-order model for x-ray imaging with an area detector. CB consistency conditions (CBCCs), also known as range conditions for the 3D divergent x-ray transform, are equations that express the redundant information in a collection of CB projections. For applications purposes, CBCCs are most suitably expressed in terms of detector coordinates. CBCCs are only known for a few geometrical configurations, which depend on the source and detector trajectories. Here we only consider source trajectories that lie in a plane, and detector orientations that are parallel to the trajectory plane, or perpendicular to it. The parallel detector is stationary, but the vertical detector rotates around the center of the circular trajectory. We unify and generalize the existing known CBCCs for planar trajectories, by creating an intermediate geometry consisting of a parallel, rotating detector, and we develop new CBCCs for this geometry. Our main result is a theorem on CBCCs for a perpendicular detector, which must necessarily move in response to movement of the source. We also provide a theorem for the more difficult situation of a perpendicular detector but without the restriction that the target object be on one side or the other of the trajectory plane. We present a simple numerical simulations for a toy calibration problem to provide an example application of the new CBCCs.
期刊介绍:
An interdisciplinary journal combining mathematical and experimental papers on inverse problems with theoretical, numerical and practical approaches to their solution.
As well as applied mathematicians, physical scientists and engineers, the readership includes those working in geophysics, radar, optics, biology, acoustics, communication theory, signal processing and imaging, among others.
The emphasis is on publishing original contributions to methods of solving mathematical, physical and applied problems. To be publishable in this journal, papers must meet the highest standards of scientific quality, contain significant and original new science and should present substantial advancement in the field. Due to the broad scope of the journal, we require that authors provide sufficient introductory material to appeal to the wide readership and that articles which are not explicitly applied include a discussion of possible applications.