Journal of Mathematical Imaging and Vision

In this paper, a new variational image denoising model is proposed. The new model could be seen to be a two-step method. In the first step, structure tensor analysis is used to infer something about the local geometry. The eigenvectors and the eigenvalues of the structure tensor are used in the construction of the denoising energy. In the second step, the actual variational denoising takes place. The steps are coupled in the sense that the energy expression is built using the underlying image, not the data. Two variable exponents are incorporated into the regularizer in order to reduce the staircasing effect, which is often present in the methods based on the first-order partial derivatives, and to increase smoothing along the image boundaries. In addition, two pointwise weight functions try to help to preserve small-scale details. In the theoretical part, the existence of a minimizer of a weak form of the original energy is considered. In the numerical part, an algorithm based on iterative minimization is presented and the numerical experiments demonstrate the possible advantages of the new model over some existing variational and partial differential equations methods.

We present a new method, based on curve evolution, for the reconstruction of a 3D curve from two different projections. It is based on the minimization of an energy functional. Following the work on geodesic active contours by Caselles et al. (in Int. Conf. on Pattern Recognition, 1996, Vol. 43, pp. 693–737), we then transform the problem of minimizing the functional into a problem of geodesic computation in a Riemann space. The Euler-Lagrange equation of this new functional is derived and its associated PDE is solved using the level set formulation, giving the existence and uniqueness results. We apply the model to the reconstruction of a vessel from a biplane angiography.

Sparse modeling can be used to characterize outlier type noise. Thanks to sparse recovery theory, it was shown that 1-norm super-resolution is robust to outliers if enough images are captured. Moreover, sparse modeling of signals is a way to overcome ill-posedness of under-determined problems. This naturally leads to this question: does an added sparsity assumption on the signal improve the robustness to outliers of the 1-norm super-resolution, and if yes, how strong should this assumption be? In this article, we review and extend results of the literature to the robustness to outliers of overdetermined signal recovery problems under sparse regularization, with a convex variational formulation. We then apply them to general random matrices, and show how the regularization parameter acts on the robustness to outliers. Finally, we show that in the case of multi-image processing, the structure of the support of signal and noise must be studied precisely. We show that the sparsity assumption improves robustness if outliers do not overlap with signal jumps, and determine how the regularization parameter can be chosen.

This paper presents a novel descriptor based on the two-dimensional discrete cosine transform (2D DCT) to fight camouflage. The 2D DCT gained popularity in image and video analysis owing to its wide use in signal compression. The 2D DCT is a well-established example to evaluate new techniques in sparse representation and is widely used for block and texture description, mainly due to its simplicity and its ability to condense information in a few coefficients. A common approach, for different applications, is to select a subset of these coefficients, which is fixed for every analyzed signal. In this paper, we question this approach and propose a novel method to select a signal-dependent subset of relevant coefficients, which is the basis for the proposed R-DCT and sR-DCT descriptors. As we propose to describe each pixel with a different set of coefficients, each associated to a particular basis function, in order to compare any two so-obtained descriptors a distance function is required: we propose a novel metric to cope with this situation. The presented experiments over the change detection dataset show that the proposed descriptors notably reduce the likelihood of camouflage respect to other popular descriptors: 92% respect to the pixel luminance, 82% respect to the RGB values, and 65% respect to the best performing LBP configuration.

Critical kernels constitute a general framework in the category of abstract complexes for the study of parallel homotopic thinning in any dimension. In this article, we present new results linking critical kernels to minimal non-simple sets (MNS) and P-simple points, which are notions conceived to study parallel thinning in discrete grids. We show that these two previously introduced notions can be retrieved, better understood and enriched in the framework of critical kernels. In particular, we propose new characterizations which hold in dimensions 2, 3 and 4, and which lead to efficient algorithms for detecting P-simple points and minimal non-simple sets.

This paper deals with the problem of reconstructing the locations of five points in space from two different images taken by calibrated cameras. Equivalently, the problem can be formulated as finding the possible relative locations and orientations, in three-dimensional Euclidean space, of two labeled stars, of five lines each, such that corresponding lines intersect. The problem was first treated by Kruppa more than 50 years ago. He found that there were at most eleven solutions. Later Demazure and also Maybank showed that there were actually ten solutions. In this article will be given another proof of this theorem based on a different parameterisation of the problem neither using the epipoles nor the essential matrix. This is within the same point of view as direct structure recovery in the uncalibrated case. Instead of the essential matrix we use the kinetic depth vectors, which has shown to be were useful in the uncalibrated case. We will also present an algorithm that in most cases calculates the ten different solutions, although some may be complex and some may not be physically realisable. The algorithm is based on a homotopy method and tracks solutions on the so called Chasles' manifold. One of the major contributions of this paper is to bridge the gap between reconstruction methods for calibrated and uncalibrated cameras. Furthermore, we show that the twisted pair solutions are natural in this context because the kinetic depths are the same for both components.