Sphere Packings, Lattices and Groups

Consider a discrete source producing a sequence of message letters from a finite alphabet. A single-letter distortion measure is given by a non-negative matrix (d ij ). The entryd ij measures the ?cost? or ?distortion? if letteriis reproduced at the receiver as letterj. The average distortion of a communications system (source-coder-noisy channel-decoder) is taken to bed= ? i.j P ij d ij whereP ij is the probability ofibeing reproduced asj. It is shown that there is a functionR(d) that measures the ?equivalent rate? of the source for a given level of distortion. For coding purposes where a leveldof distortion can be tolerated, the source acts like one with information rateR(d). Methods are given for calculatingR(d), and various properties discussed. Finally, generalizations to ergodic sources, to continuous sources, and to distortion measures involving blocks of letters are developed. In this paper a study is made of the problem of coding a discrete source of information, given afidelity criterionor ameasure of the distortionof the final recovered message at the receiving point relative to the actual transmitted message. In a particular case there might be a certain tolerable level of distortion as determined by this measure. It is desired to so encode the information that the maximum possible signaling rate is obtained without exceeding the tolerable distortion level. This work is an expansion and detailed elaboration of ideas presented earlier [1], with particular reference to the discrete case. We shall show that for a wide class of distortion measures and discrete sources of information there exists a functionR(d) (depending on the particular distortion measure and source) which measures, in a sense, the equivalent rateRof the source (in bits per letter produced) whendis the allowed distortion level. Methods will be given for evaluatingR(d) explicitly in certain simple cases and for evaluatingR(d) by a limiting process in more complex cases. The basic results are roughly that it is impossible to signal at a rate faster thanC/R(d) (source letters per second) over a memoryless channel of capacityC(bits per second) with a distortion measure less than or equal tod. On the other hand, by sufficiently long block codes it is possible to approach as closely as desired the rateC/R(d) with distortion leveld. Finally, some particular examples, using error probability per letter of message and other simple distortion measures, are worked out in detail.

Coding Theorems for a Discrete Source With a Fidelity CriterionInstitute of Radio Engineers, International Convention Record, vol. 7, 1959.

A graph is just a bunch of points with lines between some of them, like a map of cities linked by roads. A rather simple notion. Nevertheless, the theory of graphs has broad and important applications, because so many things can be modeled by graphs. For example, planar graphs — graphs in which none of the lines cross are— important in designing computer chips and other electronic circuits. Also, various puzzles and games are solved easily if a little graph theory is applied.

A Theory of Graphs

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Comptes Rendus de l'Académie des Sciences.

Survey paper: A survey on routing algorithms for wireless Ad-Hoc and mesh networks

This paper presents a new method to solve the relative pose between two images, using three pairs of homologous points and the knowledge of the vertical direction. The vertical direction can be determined in two ways: The first requires direct physical measurements such as the ones provided by an IMU (inertial measurement unit). The other uses the automatic extraction of the vanishing point corresponding to the vertical direction in an image. This knowledge of the vertical direction solves two unknowns among the three parameters of the relative rotation, so that only three homologous couples of points are requested to position a couple of images. Rewriting the coplanarity equations thus leads to a much simpler solution. The remaining unknowns resolution is performed by "hiding a variable" approach. The elements necessary to build a specific algebraic solver are given in this paper, allowing for a real-time implementation. The results on real and synthetic data show the efficiency of this method.

https://hal.archives-ouvertes.fr/hal-00632261/document

A New Solution to the Relative Orientation Problem Using Only 3 Points and the Vertical Direction

In this paper the Mojette transforms class is described. After recalling the birth of the Mojette transform, the Dirac Mojette transform is recalled with its basic properties. Generalizations to spline transform and to nD Mojette transform are also recalled. Applications of the Mojette transform demonstrate the power of frame description instead of basis in order to match different goals ranging from image coding, watermarking, discrete tomography, transmission and distributed storage. Finally, new insights for the future trends of the Mojette transform are sketched.

/pdf/the-mojette-transform-the-first-ten-years-168f1wtaga.pdf

The mojette transform: the first ten years

The first part of the book gives the basics of the Mojette transform both mathematically and the corresponding optimal algorithms. The second part exemplifies its use threough different fields: image representation, watermarking, medical imaging, distributed storage, information and cryptography.

The Mojette Transform: Theory and Applications

The goal of this paper is to describe a new fully-reversible image transform specifically designed for an efficient (pseudo-critical) coding while preserving a psychovisual Fourier domain description. There is now strong evidence for the presence of directional and angular sensitivity in the cells of the human visual cortex and the representation proposed here has for main objective to respect this human like filter bank. The decomposition is performed using a discrete Radon transform for the angular patches and by splitting each projection with a 1D spline wavelet for the radial part. Consequently, the whole algorithm is performed in the spatial domain. Finally, we show that the transform is both well-suited for psychovisual quantization and channel adapted coding.© (1995) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Psychovisual image coding via an exact discrete Radon transform

The goal of this paper is to describe a new fully-reversible image transform specifically designed for imagecoding and transmission in a context of possible loss of information as encountered in ATM networks. The so-calledMojette transform, is based on a discrete exact Radon transform which allows for a natural redundancy of the initialimage information. The inverse Mojette transform is particularly flexible and efficient : any portion of the codedimage can be replaced by another one if transmission problems occur. Futhermore, the transform is also well suitedto the two-layers paradigm used for video transmission on ATM networks by using a multiscale and/or quadtreedecomposition into the transform domain. Finally, we discuss the merits of the transform both for image storage andtransmission in order to show its ability to operate as the core representation of digital images.Keywords exact discrete Radon transform, high-speed networks, ATM, wavelet decomposition, image coding,source-channel adapted coding.1. INTRODUCTION

Jean-Pierre Guédon

Papers

A New Solution to the Relative Orientation Problem Using Only 3 Points and the Vertical Direction

The mojette transform: the first ten years

The Mojette Transform: Theory and Applications

Psychovisual image coding via an exact discrete Radon transform

Controlled redundancy for image coding and high-speed transmission