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

Secured and progressive transmission of compressed images on the Internet: application to telemedicine

International audienceWithin the framework of telemedicine, the amount of images leads first to use efficient lossless compression methods for the aim of storing information. Furthermore, multiresolution scheme including Region of Interest (ROI) processing is an important feature for a remote access to medical images. What is more, the securization of sensitive data (e.g. metadata from DICOM images) constitutes one more expected functionality: indeed the lost of IP packets could have tragic effects on a given diagnosis. For this purpose, we present in this paper an original scalable image compression technique (LAR method) used in association with a channel coding method based on the Mojette Transform, so that a hierarchical priority encoding system is elaborated. This system provides a solution for secured transmission of medical images through low-bandwidth networks such as the Internet

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

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

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

Le code à effacement Mojette : Applications dans les réseaux et dans le Cloud

Dans ce travail, je présente l'intérêt du code correcteur à effacement Mojette pour des architectures de stockage distribuées tolérantes aux pannes. De manière générale, l'approche par code permet de réduire d'un facteur 2 le volume de données stockées par rapport à l'approche standard par réplication qui consiste à copier la donnée en autant de fois que l'on suppose de pannes. De manière spécifique, le code à effacement Mojette présente les performances requises pour la lecture et l'écriture de données chaudes i.e très régulièrement sollicitées. Ces performances en entrées/sorties permettent par exemple l'exécution de machines virtuelles sur des données distribuées par le système de fichier RozoFS. En outre, j'effectue un rappel de mes contributions dans le domaine des réseaux auto-organisés de type P2P et ad hoc mobile en présentant respectivement les protocoles P2PWeb et MP-OLSR. L'ensemble de ce travail est le fruit de 5 encadrements doctoraux et de 3 projets collaboratifs majeurs

Consideraciones acerca de la viabilidad de un sensor plenóptico en dispositivos de consumo

Doctorado en Ingeniería IndustrialPassive distance measurement of the objects in an image gives place to interesting applications that have the potential to revolutionize the field of photography. In this thesis a prototype of plenoptic camera for mobile devices was created and studied. This technique has two main disadvantages: the need for modifying the camera module and the loss of resolution. Because of this, the prototype was discarded in order to utilize another technique: depth from focus. In this technique the capture method consists in taking several images while varying the focus distance. The set of images is called focal-stack. Different focus operators are studied, which give a measure of defocus per pixel and plane of the focal-stack. The curvelet based focus operator is chosen as the most adequate. It is computationally more intensive than other operators but it is capable of decomposing natural images using few coefficients. In order to make viable its usage in mobile devices a new curvelet transform based on the discrete Radon transform is built. The discrete Radon transform has logarithmic complexity, does not use the Fourier transform and uses only integer sums. Lastly, different versions of the Radon transform are analyzed with the goal of achieving an even faster transform. These transforms are implemented to be executed on mobile devices. Additionally, an application of the Radon transform is presented. It consists in the detection of bar-codes that have any orientation in an image.La medida pasiva de distancia a los objetos en una imagen da lugar a interesantes aplicaciones con capacidad para revolucionar la fotografía. En esta tesis se creó y estudió un prototipo de cámara plenóptica para dispositivos móviles. Esta técnica presenta dos inconvenientes: la necesidad de modificar el módulo de cámara y la pérdida de resolución. Por ello, el prototipo fue descartado para utilizar otra técnica: la profundidad a partir del desenfoque. En esta técnica el método de captura consiste en tomar varias imagenes variando la distancia de enfoque. El conjunto de imágenes se denomina focal-stack. Se estudian distintos operadores de desenfoque, que dan una medida de desenfoque por pixel y por plano del focal-stack. Siendo elegido como óptimo el operador de desenfoque curvelet, que es computacionalmente más intensivo que otros operadores pero es capaz de descomponer imagenes naturales utilizando muy pocos coeficientes. Para hacer posible su uso en dispositivos móviles se construye una nueva transformada curvelet basada en la transformada discreta de Radon. La transformada discreta de Radon tiene complejidad linearítmica, no utiliza la transformada de Fourier y usa sólo sumas de enteros. Por último, se analizan distintas versiones de la transformada de Radon con el objetivo de conseguir una transformada aún más rápida y se implementan para ser ejecutadas en dispositivos móviles. Además se presenta una aplicación de la transformada de Radon consistente en la detección de códigos de barras con cualquier orientación en una imagen