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"J'en ai lu des livres sur l'Afrique, depuis le temps que je l'aime et que je m'acharne à la comprendre. Peu d'entre eux m'ont autant éclairé."Érik Orsenna C'est en Afrique, désormais, que se jouent la démographie et la croissance de notre planète. Parmi les premières régions touchées par le réchauffement climatique, ce continent porte aussi en germe les drames humanitaires et écologiques de demain. À rebours des idées reçues, cet ouvrage dresse un panorama complet de la réalité africaine, pour comprendre le présent et appréhender l'avenir. Cette nouvelle édition mise à jour aborde successivement les questions de politique, d'économie et de culture, sans omettre les sujets d'actualité que sont le terrorisme, la pauvreté, l'aide au développement... Conçu par un ancien diplomate ayant consacré l'essentiel de sa carrière à l'Afrique, il constitue un outil unique de découverte et d'étude, agrémenté d'une douzaine de cartes.

"J'en ai lu des livres sur l'Afrique, depuis le temps que je l'aime et que je m'acharne à la comprendre. Peu d'entre eux m'ont autant éclairé." Érik Orsenna On ne peut plus négliger l'Afrique, car c'est là que se jouent la démographie et la croissance ou, sinon, les drames humanitaires et écologiques de demain. À rebours des idées reçues, cet ouvrage dresse un panorama complet de la réalité africaine, pour comprendre le présent et appréhender l'avenir. Il aborde successivement les questions de politique, d'économie et de culture, sans omettre les sujets d'actualité que sont le terrorisme, la pauvreté, l'aide au développement... Conçu par un ancien diplomate ayant consacré l'essentiel de sa carrière à l'Afrique, il constitue un outil unique de découverte et d'étude, agrémenté d'une douzaine de cartes.

There is a continuous effort to develop the theory and methods for computing digital distance functions, and to lower the rotational dependency of distance functions. Working on the digital space, e.g., on the square grid,digital distance functions are defined by minimal costpaths, which can be processed (back-tracked etc.) without any errors or approximations. Recently, digital distance functions defined by weight sequences, which is a concept allowing multiple types of weighted steps combined with neighborhood sequences, were developed. With appropriate weight sequences, the distance between points on the perimeter of a square and the center of the square (i.e., for squares of a given size the weight sequence can be easily computed) are exactly the Euclidean distance for these distances based on weight sequences. However, distances based on weight sequences may not fulfill the triangular inequality. In this paper, continuing the research, we provide a sufficient condition for weight sequences to provide metric distance. Further, we present an algorithm to compute the distance transform based on these distances. Optimization results are also shown for the approximation of the Euclidean distance inside the given square.

In this paper, we present a general framework for digital distance functions, defined as minimal cost paths, on the square grid. Each path is a sequence of pixels, where any two consecutive pixels are adjacent and associated with a weight. The allowed weights between any two adjacent pixels along a path are given by a weight sequence, which can hold an arbitrary number of weights. We build on our previous results, where only two or three unique weights are considered, and present a framework that allows any number of weights. We show that the rotational dependency can be very low when as few as three or four unique weights are used. Moreover, by using n weights, the Euclidean distance can be perfectly obtained on the perimeter of a square with side length 2n. A sufficient condition for weight sequences to provide metrics is proven.

We describe an algorithm that computes a “translated” 2D Neighborhood-Sequence Distance Transform (DT) using a look up table approach. It requires a single raster scan of the input image and produces one line of output for every line of input. The neighborhood sequence is specified either by providing one period of some integer periodic sequence or by providing the rate of appearance of neighborhoods. The full algorithm optionally derives the regular (centered) DT from the “translated” DT, providing the result image on-the-fly, with a minimal delay, before the input image is fully processed. Its efficiency can benefit all applications that use neighborhood- sequence distances, particularly when pipelined processing architectures are involved, or when the size of objects in the source image is limited.

The discrete Fourier slice theorem is an important tool for signal processing, especially in the context of the exact reconstruction of an image from its projected views. This paper presents a digital reconstruction algorithm to recover a two dimensional

The Mojette transform is a form of discrete Radon transform that maps a 2D image (P ×Q pixels) to a set of I 1D projections. Several fast inversion methods exist that require O(PQI) operations but those methods are ill-conditioned. Several robust (or wel

We present a new, robust discrete back-projection filtration algorithm to reconstruct digital images from close-to-minimal sets of arbitrarily oriented discrete projected views. The discrete projections are in the Mojette format, with either Dirac or Haar pixel sampling. The strong aliasing in the raw image reconstructed by direct back-projection is corrected via a de-convolution using the Fourier transform of the discrete point-spread function (PSF) that was used for the forward projection. The de-convolution is regularised by applying an image-sized digital weighting function to the raw PSF. These weights are obtained from the set of back-projected points that partially tile the image area to be reconstructed. This algorithm produces high quality reconstructions at and even below the Katz sufficiency limit, which defines a minimal criterion for projection sets that permit a unique discrete reconstruction for noisefree data. As the number of input discrete projected views increases, the PSF more fully tiles the discrete region to be reconstructed, the deconvolution and its weighting mask become progressively less important. This algorithm then merges asymptotically with the perfect reconstruction method found by Servières et al in 2004. However the Servières approach, for which the PSF must exactly tile the full area of the reconstructed image, requires O(N2) uniformly distributed projection angles to reconstruct N ×N data. The independence of each (back-) projected view makes our algorithm robust to random, symmetrically distributed noise. We present, as results, images reconstructed from sets of O(N) projected view angles that are either uniformly distributed, randomly selected, or clustered about orthogonal axes.

In this paper, a family of weighted neighborhood sequence distance functions defined on the square grid is presented. With this distance function, the allowed weight between any two adjacent pixels along a path is given by a weight sequence. We build on our previous results, where only two or three unique weights are considered, and present a framework that allows any number of weights. We show that the rotational dependency can be very low when as few as three or four unique weights are used. An algorithm for computing the distance transform (DT) that can be used for image processing applications is also presented.

In recent years, the theory behind distance functions defined by neighbourhood sequences has been developed in the digital geometry community. A neighbourhood sequence is a sequence of integers, where each element defines a neighbourhood. In this paper, we establish the equivalence between the representation of convex digital disks as an intersection of half-planes ( H -representation) and the expression of the distance as a maximum of non-decreasing functions. Both forms can be deduced one from the other by taking advantage of the Lambek-Moser inverse of integer sequences. Examples with finite sequences, cumulative sequences of periodic sequences and (almost) Beatty sequences are given. In each case, closed-form expressions are given for the distance function and H -representation of disks. The results can be used to compute the pair-wise distance between points in constant time and to find optimal parameters for neighbourhood sequences.

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

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.

This paper presents a path-based distance where local displacement costs vary both according to the displacement vector and with the travelled distance. The corresponding distance transform algorithm is similar in its form to classical propagation-based algorithms, but the more variable distance increments are either stored in look-up-tables or computed on-the-fly. These distances and distance transform extend neighborhood-sequence distances, chamfer distances and generalized distances based on Minkowski sums. We introduce algorithms to compute, in , a translated version of a neighborhood sequence distance map with a limited number of neighbors, both for periodic and aperiodic sequences. A method to recover the centered distance map from the translated one is also introduced. Overall, the distance transform can be computed with minimal delay, without the need to wait for the whole input image before beginning to provide the result image.

The Mojette transform and the finite Radon transform (FRT) are discrete data projection methods that are exactly invertible and are computed using simple addition operations. Incorporation of a known level of redundancy into data and projection spaces enables the use of the FRT to recover the exact, original data when network packets are lost during data transmission. The FRT can also be shown to be Maximum Distance Separable (MDS). By writing the FRT transform in Vandermonde form, explicit expressions for discrete projection and inversion as matrix operations have been obtained. A cyclic, prime-sized Vandermonde form for the FRT approach is shown here to yield explicit polynomial expressions for the recovery of image rows from projected data and vice-versa. These polynomial solutions are consistent with the heuristic algorithms for "row-solving" that have been published previously. This formalism also opens the way to link "ghost" projections in FRT space and "anti-images" in data space that may provide a key to an efficient method of encoding and decoding general data sets in a systematic form.

The Mojette transform is an entirely discrete form of the Radon transform developed in 1995. It is exactly invertible with both the forward and inverse transforms requiring only the addition operation. Over the last 10 years it has found many applications including image watermarking and encryption, tomographic reconstruction, robust data transmission and distributed data storage. This paper presents an elegant and efficient algorithm to directly apply the inverse Mojette transform. The method is derived from the inter-dependance of the "rational" projection vectors (pi, qi) which define the direction of projection over the parallel set of lines b = p il - qik. Projection values are acquired by summing the value of image pixels, f(k,l), centered on these lines. The new inversion is up to 5 times faster than previously proposed methods and solves the redundancy issues of these methods.