The Discrete radon transform: A more efficient approach to image reconstruction

The Radon transform and its inversion are the mathematical keys that enable tomography. Radon transforms are defined for continuous objects with continuous projections at all angles in [0,π). In practice, however, we pre-filter discrete projections take

Transformation Mojette en dimension n

La transformation Mojette (déjà présentée en 2D et 3D au Gretsi [1]) représente un signal discret par un ensemble fini d'hyperplans. Cette transformée permet un très grand nombre de choix (directions des hyperplans, nombre, ordre de spline) tout en gardant une complexité opératoire très faible comparable à la FFT. Dans ce papier, la transformation Mojette en dimension n est présentée et les résultats acquis en dimension 2 et 3 sont généralisés. Deux exemples d'applications illustrent l'intérêt pour cette généralisation

Recovering missing slices of the discrete fourier transform using ghosts

The discrete Fourier transform (DFT) underpins the solution to many inverse problems commonly possessing missing or unmeasured frequency information. This incomplete coverage of the Fourier space always produces systematic artifacts called Ghosts. In this paper, a fast and exact method for deconvolving cyclic artifacts caused by missing slices of the DFT using redundant image regions is presented. The slices discussed here originate from the exact partitioning of the Discrete Fourier Transform (DFT) space, under the projective Discrete Radon Transform, called the discrete Fourier slice theorem. The method has a computational complexity of O(n\log-{2}n) (for an n=N\times N image) and is constructed from a new cyclic theory of Ghosts. This theory is also shown to unify several aspects of work done on Ghosts over the past three decades. This paper concludes with an application to fast, exact, non-iterative image reconstruction from a highly asymmetric set of rational angle projections that give rise to sets of sparse slices within the DFT

Quantised Angular Momentum Vectors and Projection Angle Distribution for Discrete Radon Transformations

A quantum mechanics based method is presented to generate sets of digital angles that may be well suited to describe projections on discrete grids. The resulting angle sets are an alternative to those derived using the Farey fractions from number theory

Recovering missing slices of the discrete fourier transform using ghosts

The discrete Fourier transform (DFT) underpins the solution to many inverse problems commonly possessing missing or unmeasured frequency information. This incomplete coverage of the Fourier space always produces systematic artifacts called Ghosts. In this paper, a fast and exact method for deconvolving cyclic artifacts caused by missing slices of the DFT using redundant image regions is presented. The slices discussed here originate from the exact partitioning of the Discrete Fourier Transform (DFT) space, under the projective Discrete Radon Transform, called the discrete Fourier slice theorem. The method has a computational complexity of O(n\log-{2}n) (for an n=N\times N image) and is constructed from a new cyclic theory of Ghosts. This theory is also shown to unify several aspects of work done on Ghosts over the past three decades. This paper concludes with an application to fast, exact, non-iterative image reconstruction from a highly asymmetric set of rational angle projections that give rise to sets of sparse slices within the DFT

Building a bone μ-CT images atlas for micro-architecture recognition

Trabecular bone and its micro-architecture are of prime importance for health. Changes of bone micro-architecture are linked to different pathological situations like osteoporosis and begin now to be understood. In a previous paper, we started to investigate the relationships between bone and vessels and we also proposed to build a Bone Atlas. This study describes how to proceed for the elaboration and use of such an atlas. Here, we restricted the Atlas to legs (tibia, femur) of rats in order to work with well known geometry of the bone micro-architecture. From only 6 acquired bone, 132 trabecular bone volumes were generated using simple mathematical morphology tools. The variety and veracity of the created micro-architecture volumes is presented in this paper. Medical application and final goal would be to determinate bone micro-architecture with some angulated radiographs (3 or 4) and to easily diagnose the bone status (healthy, pathological or healing bone⋯)

Direct inversion of Mojette projections

We present algorithms to reconstruct images from minimal sets of discrete Mojette projections using direct back-projection (DBP) with various forms of correction. This paper extends previous work on discrete projection inversion by Servières et al [1, 2

Mojette tomographic reconstruction for micro-CT: A bone and vessels quality evaluation

Micro-CT represents a modality where the quality of CT reconstruction is very high thanks to the acquisition properties. The goal of this paper is to challenge our proposed Mojette discrete reconstruction scheme from real micro-CT data. A first study wa