Abstract. 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.

This paper presents a new algorithm for an efficient computation of morphological operations for gray images and its specific hardware. The method is based on a new recursive morphological decomposition method of 8-convex structuring elements by only causal twopixel structuring elements (2PSE). Whatever the element size, erosion or/and dilation can then be performed during a unique raster-like image scan involving a fixed reduced analysis neighborhood. The resulting process offers low computation complexity combined with easy description of the element form. The dedicated hardware is generic and fully regular, built from elementary interconnected stages. It has been synthesized into an FPGA and achieves high frequency performances for any shape and size of structuring element.

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International audienceThe 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 = pil -- 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

Abstract-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. The FRT is also Maximum Distance Separable (MDS). Incorporation of a known level of redundancy into data and projection spaces enables the use of forward error correction to recover the exact, original data when network packets are lost or corrupted during data transmission. By writing the above transforms 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 "rowsolving" 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.

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International audienceIterative methods are now recognized as powerful tools to solve inverse problems such as tomographic reconstruction. In this paper, the main goal is to present a new reconstruction algorithm made from two components. An iterative algorithm, namely the Conjugate Gradient (CG) method, is used to solve the tomographic problem in the least square (LS) sense for our specific discrete Mojette geometry. The results are compared (with the same geometry) to the corresponding Mojette Filtered Back Projection (FBP) method. In the fist part of the paper, we recall the discrete geometry used to define the projection M and backprojection M* operators. In the second part, the CG algorithm is presented within the context of the Mojette geometry. Noise is then added onto these Mojette projections with respect to the sampling and reconstructions are performed. Finally the Toeplitz block Toeplitz (TBT) character of M*M is demonstrated