1. Data Consistency Conditions for Cone-Beam Projections on a Circular Trajectory
- Author
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Jerome Lesaint, Laurent Desbat, Simon Rit, Rolf Clackdoyle, Gestes Medico-chirurgicaux Assistés par Ordinateur (TIMC-IMAG-GMCAO), Techniques de l'Ingénierie Médicale et de la Complexité - Informatique, Mathématiques et Applications, Grenoble - UMR 5525 (TIMC-IMAG), Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-VetAgro Sup - Institut national d'enseignement supérieur et de recherche en alimentation, santé animale, sciences agronomiques et de l'environnement (VAS)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-VetAgro Sup - Institut national d'enseignement supérieur et de recherche en alimentation, santé animale, sciences agronomiques et de l'environnement (VAS)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), Imagerie Tomographique et Radiothérapie, Centre de Recherche en Acquisition et Traitement de l'Image pour la Santé (CREATIS), Université Jean Monnet [Saint-Étienne] (UJM)-Hospices Civils de Lyon (HCL)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Université de Lyon-Institut National des Sciences Appliquées (INSA)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Centre National de la Recherche Scientifique (CNRS)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Université Jean Monnet [Saint-Étienne] (UJM)-Hospices Civils de Lyon (HCL)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), and Université de Lyon-Centre National de la Recherche Scientifique (CNRS)-Institut National de la Santé et de la Recherche Médicale (INSERM)
- Subjects
Parallel projection ,Geometry ,Trigonometric polynomial ,01 natural sciences ,Labex PRIMES ,030218 nuclear medicine & medical imaging ,03 medical and health sciences ,0302 clinical medicine ,[INFO.INFO-TS]Computer Science [cs]/Signal and Image Processing ,Projection (mathematics) ,0103 physical sciences ,categₛt2i ,Imagerie tomographique et thérapie par rayonnement ,Electrical and Electronic Engineering ,Trajectory (fluid mechanics) ,ComputingMilieux_MISCELLANEOUS ,Mathematics ,Generalized function ,010308 nuclear & particles physics ,Plane (geometry) ,Applied Mathematics ,Mathematical analysis ,Function (mathematics) ,reseau_international ,Signal Processing ,Gravitational singularity - Abstract
The well-known Helgason–Ludwig data consistency conditions (DCCs) for parallel projections of a two-dimensional function take the form of homogeneous polynomials in $\cos \phi$ and $\sin \phi$ , where $\phi$ is the angle of the parallel projection. In this note, we establish necessary DCCs for cone-beam (CB) projections of a three-dimensional function taken along a circular trajectory. Our DCCs take the form of homogeneous polynomials in $\cos \lambda$ and $\sin \lambda$ , where $\lambda$ is the angular position of the CB projection. This trigonometric polynomial format for the DCCs is particularly convenient for medical imaging applications, and these new DCCs for the standard CB geometry will potentially lead to new DCC applications in X-ray computed tomography (CT) and pinhole single photon emission computed tomography (SPECT) where CB projections are measured. If, as is usually the case, the object intersects the trajectory plane, then singularities appear in the DCC formulas. We describe how to interpret and handle these singularities as generalized functions. Numerical simulations are provided for illustration.
- Published
- 2016