Fast Mojette Transform for Discrete Tomography

A new algorithm for reconstructing a two dimensional object from a set of one dimensional projected views is presented that is both computationally exact and experimentally practical. The algorithm has a computational complexity of O(n log2 n) with n = N^2 for an NxN image, is robust in the presence of noise and produces no artefacts in the reconstruction process, as is the case with conventional tomographic methods. The reconstruction process is approximation free because the object is assumed to be discrete and utilizes fully discrete Radon transforms. Noise in the projection data can be suppressed further by introducing redundancy in the reconstruction. The number of projections required for exact reconstruction and the response to noise can be controlled without comprising the digital nature of the algorithm. The digital projections are those of the Mojette Transform, a form of discrete linogram. A simple analytical mapping is developed that compacts these projections exactly into symmetric periodic slices within the Discrete Fourier Transform. A new digital angle set is constructed that allows the periodic slices to completely fill all of the objects Discrete Fourier space. Techniques are proposed to acquire these digital projections experimentally to enable fast and robust two dimensional reconstructions.Comment: 22 pages, 13 figures, Submitted to Elsevier Signal Processin

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

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

Lossless Image Compression via Predictive Coding of Discrete Radon Projections

International audienceThis paper investigates predictive coding methods to compress images represented in the Radon domain as a set of projections. Both the correlation within and between discrete Radon projections at similar angles can be exploited to achieve lossless compression. The discrete Radon projections investigated here are those used to define the Mojette transform first presented by Guedon et al. [Psychovisual image coding via an exact discrete Radon transform, in: T.W. Lance (Ed.), Proceedings of the Visual Communications AND Image Processing (VCIP), May 1995, Taipei, Taiwan, pp. 562-572]. This work is further to the preliminary investigation presented by Autrusseau et al. [Lossless compression based on a discrete and exact radon transform: a preliminary study, in: Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), vol. II, May 2006, Toulouse, France, pp. 425-428]. The 1D Mojette projections are re-arranged as two dimensional images, thus allowing the use of 2D image compression techniques onto the projections. Besides the compression capabilities, the Mojette transforms brings an interesting property: a tunable redundancy. As the Mojette transform is able to both compress and add redundancy, the proposed method can be viewed as a joint lossless source-channel coding technique for images. We present here the evolution of the compression ratio depending on the chosen redundancy

Analysis of Mojette Transform Implementation on Reconfigurable Hardware

Inscribing invisible marks (watermarking) into an image has different applications such as copyright, steganography or data integrity checking. Many different techniques have been employed for the last years in different spaces (Fourier, wavelet, Mojette domains, etc.). The presentation outline the development work related to create a functional block scheme of Mojette transform and Inverse Mojette transform using reconfigurable hardware

Redundant Image Representation via Multi-Scale Digital Radon Projection

International audienceA novel ordering of digital Radon projections co-efficients is presented here that enables progressive image reconstruc- tion from low resolution to full resolution. The digital Radon transform applied here is the Mojette transform first defined by Guedon et al. in [1]. The Mojette transform is a natural way to generate redundancy to any specified degree and has been demonstrated to be useful for redundant representation for robust data storage and transmission. Combining this with the wavelet transform facilitates compression, i.e., joint source-channel coding, along with the additional property of scalability

Simulation and Performance Analysis of MP-OLSR for Mobile Ad hoc Networks

Mobile ad hoc networks (MANETs) consist of a collection of wireless mobile nodes which dynamically exchange data without reliance on a fixed base station or a wired backbone network, which makes routing a crucial issue for the design of a ad hoc networks. In this paper we discussed a hybrid multipath routing protocol named MP-OLSR. It is based on the link state algorithm and employs periodic exchange of messages to maintain topology information of the networks. In the mean time, it updates the routing table in an on-demand scheme and forwards the packets in multiple paths which have been determined at the source. If a link failure is detected, the algorithm recovers the route automatically. Concerning the instability of the wireless networks, the redundancy coding is used to improve the delivery ratio. The simulation in NS2 shows that the new protocol can effectively improve the performance of the networks

Projections et distances discrètes

Le travail se situe dans le domaine de la géométrie discrète. La tomographie discrète sera abordée sous l'angle de ses liens avec la théorie de l'information, illustrés par l'application de la transformation Mojette et de la "Finite Radon Transform" au codage redondant d'information pour la transmission et le stockage distribué. Les distances discrètes seront exposées selon les points de vue théorique (avec une nouvelle classe de distances construites par des chemins à poids variables) et algorithmique (transformation en distance, axe médian, granulométrie) en particulier par des méthodes en un balayage d'image (en "streaming"). Le lien avec les séquences d'entiers non-décroissantes et l'inverse de Lambek-Moser sera mis en avant

Tomographie et géométrie discrètes avec la transformée Mojette

We explore through this thesis the insights of discrete tomography over classical tomography in continuous space. We use the Mojette transform, a discrete and exact form of the Radon transform, as a link between classical tomography and discrete tomography. We focus especially on the study of the discrete space induced by the Mojette transform operator through four research axis.Axis 1 focuses on the Mojette space properties in regards to discrete affine transforms of digital images. We provide tools to achieve affine transforms directly from the projections of a digital object, without preliminary tomographic reconstruction. This property is well-known for the continuous Radon transform but non-trivial for its sampled versions.Axis 2 seeks for some links between continuous-sampled projections related to medical imaging acquisition modalities and discrete projections derived by the Mojette transform. We implement interpolation schemes to estimate discrete projections from the continuous ones — on either synthetic or real data — and their reconstruction.In axis 3, we provide an algebraic framework for the description and inversion of the Mojette transform. The input data, the projections as well as the operators are modeled as polynomials. Within this framework, the Mojette projection operator advantageously reduce to a Vandermonde matrix.This thesis has been realized at both IRCCyN Lab and Keosys company within the Quanticardi FUI project. Axis 4 focuses on the design and the implementation of a clinical software for the absolute quantification of myocardial perfusion with positron emission tomography.Dans cette thèse, nous explorons les voies offertes par la tomographie discrète par rapport à la tomographie classique en milieu continu. Nous utilisons la transformée Mojette, version discrète et exacte de la transformée de Radon, que nous présentons comme un lien entre la tomographie classique et la tomographie discrète. Nous nous attachons à l’étude de l’espace sous-jacent à l’opérateur de transformée Mojette. Ce travail se décline suivant quatre axes de recherche.L’axe 1 est consacré au comportement de l’espace Mojette pour les transformations affines discrètes de l’image. Nous montrons qu’il est possible de réaliser certaines transformations affines directement à partir des projections discrètes d’un objet, sans reconstruction préalable.L’axe 2 consiste à examiner les liens entre les projections continues issues de modalités d’acquisitions en imagerie médicale et celles obtenues par transformée Mojette. Nous présentons différentes méthodes d’estimation des projections discrètes à partir de projections continues — réelles ou simulées — et leur reconstruction.L’axe 3 a pour objet l’inversion algébrique de la transformée Mojette. Les données d’entrée, les projections et les opérateurs sont modélisés par des polynômes. Ce formalisme, relevant de la tomographie discrète, permet d’exprimer la matrice de transformation Mojette sous forme Vandermonde.Cette thèse a été réalisée conjointement à l’IRCCyN et à Keosys dans le cadre du projet FUI Quanticardi. L’axe 4 est dédié à la conception et au développement d’un logiciel de quantification absolue de la perfusion myocardique en tomographie par émission de positons