{"title":"带有平行和垂直探测器的平面轨迹的锥形光束一致性条件","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":"{\"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}","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
摘要
锥形光束(CB)投影为使用区域探测器进行 X 射线成像提供了一阶模型。CB 一致性条件(CBCCs)也称为三维发散 X 射线变换的范围条件,是表达 CB 投影集合中冗余信息的方程。在应用中,CBCC 最适合用探测器坐标来表示。目前只知道一些几何配置的 CBCC,这些配置取决于光源和探测器的轨迹。在这里,我们只考虑位于一个平面内的光源轨迹,以及与轨迹平面平行或垂直的探测器方向。平行探测器是静止的,但垂直探测器会围绕圆形轨迹的中心旋转。通过创建一个由平行、旋转探测器组成的中间几何体,我们统一并推广了现有已知的平面轨迹 CBCC,并为这个几何体开发了新的 CBCC。我们的主要成果是一个垂直探测器的 CBCC 理论,该探测器必须随着光源的移动而移动。我们还为更困难的垂直探测器情况提供了一个定理,但没有目标物体位于轨迹平面一侧或另一侧的限制。我们针对一个玩具校准问题进行了简单的数值模拟,为新的 CBCCs 提供了一个应用实例。
Cone-beam consistency conditions for planar trajectories with parallel and perpendicular detectors
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.