Crossings of smooth shot noise processes

In this paper, we consider smooth shot noise processes and their expected number of level crossings. When the kernel response function is sufficiently smooth, the mean number of crossings function is obtained through an integral formula. Moreover, as the intensity increases, or equivalently, as the number of shots becomes larger, a normal convergence to the classical Rice's formula for Gaussian processes is obtained. The Gaussian kernel function, that corresponds to many applications in physics, is studied in detail and two different regimes are exhibited.Comment: Published in at http://dx.doi.org/10.1214/11-AAP807 the Annals of Applied Probability ( http://www.imstat.org/aap/ ) by the Institute of Mathematical Statistics (http://www.imstat.org

Image denoising by statistical area thresholding

Area openings and closings are morphological filters which efficiently suppress impulse noise from an image, by removing small connected components of level sets. The problem of an objective choice of threshold for the area remains open. Here, a mathematical model for random images will be considered. Under this model, a Poisson approximation for the probability of appearance of any local pattern can be computed. In particular, the probability of observing a component with size larger than $k$ in pure impulse noise has an explicit form. This permits the definition of a statistical test on the significance of connected components, thus providing an explicit formula for the area threshold of the denoising filter, as a function of the impulse noise probability parameter. Finally, using threshold decomposition, a denoising algorithm for grey level images is proposed

A survey of exemplar-based texture synthesis

Exemplar-based texture synthesis is the process of generating, from an input sample, new texture images of arbitrary size and which are perceptually equivalent to the sample. The two main approaches are statistics-based methods and patch re-arrangement methods. In the first class, a texture is characterized by a statistical signature; then, a random sampling conditioned to this signature produces genuinely different texture images. The second class boils down to a clever "copy-paste" procedure, which stitches together large regions of the sample. Hybrid methods try to combine ideas from both approaches to avoid their hurdles. The recent approaches using convolutional neural networks fit to this classification, some being statistical and others performing patch re-arrangement in the feature space. They produce impressive synthesis on various kinds of textures. Nevertheless, we found that most real textures are organized at multiple scales, with global structures revealed at coarse scales and highly varying details at finer ones. Thus, when confronted with large natural images of textures the results of state-of-the-art methods degrade rapidly, and the problem of modeling them remains wide open.Comment: v2: Added comments and typos fixes. New section added to describe FRAME. New method presented: CNNMR

Significant edges in the case of a non-stationary Gaussian noise

In this paper, we propose an edge detection technique based on some local smoothing of the image followed by a statistical hypothesis testing on the gradient. An edge point being defined as a zero-crossing of the Laplacian, it is said to be a significant edge point if the gradient at this point is larger than a threshold s(\eps) defined by: if the image $I$ is pure noise, then \P(\norm{\nabla I}\geq s(\eps) \bigm| \Delta I = 0) \leq\eps. In other words, a significant edge is an edge which has a very low probability to be there because of noise. We will show that the threshold s(\eps) can be explicitly computed in the case of a stationary Gaussian noise. In images we are interested in, which are obtained by tomographic reconstruction from a radiograph, this method fails since the Gaussian noise is not stationary anymore. But in this case again, we will be able to give the law of the gradient conditionally on the zero-crossing of the Laplacian, and thus compute the threshold s(\eps). We will end this paper with some experiments and compare the results with the ones obtained with some other methods of edge detection

Mean Geometry for 2D random fields: level perimeter and level total curvature integrals

International audienceWe introduce the level perimeter integral and the total curvature integral associated with areal valued function f defined on the plane R^2 as integrals allowing to compute the perimeter of the excursion set of f above level t and the total (signed) curvature of itsboundary for almost every level t. Thanks to the Gauss-Bonnet theorem, the total curvature is directly related to theEuler Characteristic of the excursion set. We show that the level perimeter and the total curvature integrals can be explicitly computed in two different frameworks: smooth (at least C^2)functions and piecewise constant functions (also called here elementary functions). Considering 2D random fields (in particular shot noise random fields), we compute their mean perimeter and total curvature integrals, and this provides new explicit computations of the mean perimeter and Euler Characteristic densities of excursion sets, beyond the Gaussian framework

A Wasserstein-type distance in the space of Gaussian Mixture Models

In this paper we introduce a Wasserstein-type distance on the set of Gaussian mixture models. This distance is defined by restricting the set of possible coupling measures in the optimal transport problem to Gaussian mixture models. We derive a very simple discrete formulation for this distance, which makes it suitable for high dimensional problems. We also study the corresponding multi-marginal and barycenter formulations. We show some properties of this Wasserstein-type distance, and we illustrate its practical use with some examples in image processing