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Image Denoising by Statistical Area Thresholding

2005-04-14
David Coupier , Agnès Desolneux , Bernard Ycart
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A survey of exemplar-based texture synthesis

2018-01-01
Lara Raad , Axel Davy , Agnès Desolneux +1 more
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 … 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 combines ideas from both approaches to avoid their hurdles. Current methods, including the recent CNN approaches, are able to produce impressive synthesis on various kinds of textures. Nevertheless, 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.
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Crossings of smooth shot noise processes

2012-11-23
Hermine Biermé , Agnès Desolneux
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 … 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.
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Exact sampling of determinantal point processes without eigendecomposition

2020-11-23
Claire Launay , Bruno Galerne , Agnès Desolneux
Abstract Determinantal point processes (DPPs) enable the modeling of repulsion: they provide diverse sets of points. The repulsion is encoded in a kernel K that can be seen, in a … Abstract Determinantal point processes (DPPs) enable the modeling of repulsion: they provide diverse sets of points. The repulsion is encoded in a kernel K that can be seen, in a discrete setting, as a matrix storing the similarity between points. The main exact algorithm to sample DPPs uses the spectral decomposition of K , a computation that becomes costly when dealing with a high number of points. Here we present an alternative exact algorithm to sample in discrete spaces that avoids the eigenvalues and the eigenvectors computation. The method used here is innovative, and numerical experiments show competitive results with respect to the initial algorithm.
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A Wasserstein-Type Distance in the Space of Gaussian Mixture Models

2020-01-01
Julie Delon , Agnès Desolneux
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 … 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.
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A Fourier Approach for the Level Crossings of Shot Noise Processes with Jumps

2012-03-01
Hermine Biermé , Agnès Desolneux
We use a change-of-variable formula in the framework of functions of bounded variation to derive an explicit formula for the Fourier transform of the level crossing function of shot noise … We use a change-of-variable formula in the framework of functions of bounded variation to derive an explicit formula for the Fourier transform of the level crossing function of shot noise processes with jumps. We illustrate the result in some examples and give some applications. In particular, it allows us to study the asymptotic behavior of the mean number of level crossings as the intensity of the Poisson point process of the shot noise process goes to infinity.
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On the perimeter of excursion sets of shot noise random fields

2016-01-01
Hermine Biermé , Agnès Desolneux
In this paper, we use the framework of functions of bounded variation and the coarea formula to give an explicit computation for the expectation of the perimeter of excursion sets … In this paper, we use the framework of functions of bounded variation and the coarea formula to give an explicit computation for the expectation of the perimeter of excursion sets of shot noise random fields in dimension $n\geq1$. This will then allow us to derive the asymptotic behavior of these mean perimeters as the intensity of the underlying homogeneous Poisson point process goes to infinity. In particular, we show that two cases occur: we have a Gaussian asymptotic behavior when the kernel function of the shot noise has no jump part, whereas the asymptotic is non-Gaussian when there are jumps.
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A compact representation of random phase and Gaussian textures

2012-03-01
Agnès Desolneux , Lionel Moisan , Samuel Ronsin
In this paper, we are interested in the mathematical analysis of the micro-textures that have the property to be perceptually invariant under the randomization of the phases of their Fourier … In this paper, we are interested in the mathematical analysis of the micro-textures that have the property to be perceptually invariant under the randomization of the phases of their Fourier Transform. We propose a compact representation of these textures by considering a special instance of them: the one that has identically null phases, and we call it "texton". We show that this texton has many interesting properties, and in particular it is concentrated around the spatial origin. It appears to be a simple and useful tool for texture analysis and texture synthesis, and its definition can be extended to the case of color micro-textures.
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Gromov–Wasserstein distances between Gaussian distributions

2022-08-18
Julie Delon , Agnès Desolneux , Antoine Salmona
Abstract Gromov–Wasserstein distances were proposed a few years ago to compare distributions which do not lie in the same space. In particular, they offer an interesting alternative to the Wasserstein … Abstract Gromov–Wasserstein distances were proposed a few years ago to compare distributions which do not lie in the same space. In particular, they offer an interesting alternative to the Wasserstein distances for comparing probability measures living on Euclidean spaces of different dimensions. We focus on the Gromov–Wasserstein distance with a ground cost defined as the squared Euclidean distance, and we study the form of the optimal plan between Gaussian distributions. We show that when the optimal plan is restricted to Gaussian distributions, the problem has a very simple linear solution, which is also a solution of the linear Gromov–Monge problem. We also study the problem without restriction on the optimal plan, and provide lower and upper bounds for the value of the Gromov–Wasserstein distance between Gaussian distributions.
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Significant edges in the case of non-stationary Gaussian noise

2007-03-13
I. Abraham , Romain Abraham , Agnès Desolneux +1 more
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Determinantal Point Processes for Image Processing

2021-01-01
Claire Launay , Agnès Desolneux , Bruno Galerne
Determinantal point processes (DPPs) are probabilistic models of configurations that favor diversity or repulsion. They have recently gained influence in the machine learning community, mainly because of their ability to … Determinantal point processes (DPPs) are probabilistic models of configurations that favor diversity or repulsion. They have recently gained influence in the machine learning community, mainly because of their ability to elegantly and efficiently subsample large sets of data. In this paper, we consider DPPs from an image processing perspective, meaning that the data we want to subsample are pixels or patches of a given image. To this end, our framework is discrete and finite. First, we adapt their basic definition and properties to DPPs defined on the pixels of an image, that we call determinantal pixel processes (DPixPs). We are mainly interested in the repulsion properties of such a process and we apply DPixPs to texture synthesis using shot noise models. Finally, we study DPPs on the set of patches of an image. Because of their repulsive property, DPPs provide a strong tool to subsample discrete distributions such as that of image patches.
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Self-organized percolation power laws with and without fractal geometry in the etching of random solids

2004-01-01
Agnès Desolneux , B. Sapoval , Andrea Baldassarri
Classically, percolation critical exponents are linked to the power laws that characterize percolation cluster fractal properties. It is found here that the gradient percolation power laws are conserved even for … Classically, percolation critical exponents are linked to the power laws that characterize percolation cluster fractal properties. It is found here that the gradient percolation power laws are conserved even for extreme gra- dient values for which the frontier of the infinite cluster is no more fractal. In particular the exponent 7/4 which was recently demonstrated to be the exact value for the dimension of the so-called hull or external perimeter of the incipient percolation cluster, controls the width and length of gradient percolation frontiers whatever the gradient magnitude. This behavior is extended to previous model studies of etching by a finite volume of etching solution in contact with a disordered solid. In such a model, the dynamics stop spontaneously on an equilibrium self-similar surface similar to the fractal frontier of gradient percolation. It is shown that the power laws describing the system geometry involves also the fractal dimension of the percolation hull, whatever the value of the dynamically generated gradient, i.e. even for a non-fractal frontier. The comparison between numerical results and the exact results that can be obtained analytically for extreme values of the gradient suggests that there exist a unique power law valid from the smallest possible scale up to infinity. These results suggest the possible existence of an underlying conservation law, relating the length and the statistical width of percolation gradient frontiers.
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Mean geometry for 2D random fields: Level perimeter and level total curvature integrals

2020-04-01
Hermine Biermé , Agnès Desolneux
We introduce the level perimeter integral and the total curvature integral associated with a real-valued function $f$ defined on the plane $\mathbb{R}^{2}$, as integrals allowing to compute the perimeter of … We introduce the level perimeter integral and the total curvature integral associated with a real-valued function $f$ defined on the plane $\mathbb{R}^{2}$, as integrals allowing to compute the perimeter of the excursion set of $f$ above level $t$ and the total (signed) curvature of its boundary for almost every level $t$. Thanks to the Gauss–Bonnet theorem, the total curvature is directly related to the Euler characteristic of the excursion set. We show that the level perimeter and the total curvature integrals can be 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: for piecewise constant shot noise random fields, we give some examples of completely explicit formulas, and for smooth shot noise random fields the provided examples are only partly explicit, since the formulas are given under the form of integrals of some special functions.
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The effect of discretization on the mean geometry of a 2D random field

2021-09-22
Hermine Biermé , Agnès Desolneux
The study of the geometry of excursion sets of 2D random fields is a question of interest from both the theoretical and the applied viewpoints. In this paper we are … The study of the geometry of excursion sets of 2D random fields is a question of interest from both the theoretical and the applied viewpoints. In this paper we are interested in the relationship between the perimeter (resp. the total curvature, related to the Euler characteristic by Gauss–Bonnet Theorem) of the excursion sets of a function and the ones of its discretization. Our approach is a weak framework in which we consider the functions that map the level of the excursion set to the perimeter (resp. the total curvature) of the excursion set. We will be also interested in a stochastic framework in which the sets are the excursion sets of 2D random fields. We show in particular that, under some stationarity and isotropy conditions on the random field, in expectation, the perimeter is always biased (with a 4/π factor), whereas the total curvature is not. We illustrate all our results on different examples of random fields.
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Gromov-Wasserstein Distances between Gaussian Distributions

2021-01-01
Antoine Salmona , Julie Delon , Agnès Desolneux
The Gromov-Wasserstein distances were proposed a few years ago to compare distributions which do not lie in the same space. In particular, they offer an interesting alternative to the Wasserstein … The Gromov-Wasserstein distances were proposed a few years ago to compare distributions which do not lie in the same space. In particular, they offer an interesting alternative to the Wasserstein distances for comparing probability measures living on Euclidean spaces of different dimensions. In this paper, we focus on the Gromov-Wasserstein distance with a ground cost defined as the squared Euclidean distance and we study the form of the optimal plan between Gaussian distributions. We show that when the optimal plan is restricted to Gaussian distributions, the problem has a very simple linear solution, which is also solution of the linear Gromov-Monge problem. We also study the problem without restriction on the optimal plan, and provide lower and upper bounds for the value of the Gromov-Wasserstein distance between Gaussian distributions.
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Can Push-forward Generative Models Fit Multimodal Distributions?

2022-01-01
Antoine Salmona , Valentin De Bortoli , Agnès Desolneux +1 more
Many generative models synthesize data by transforming a standard Gaussian random variable using a deterministic neural network. Among these models are the Variational Autoencoders and the Generative Adversarial Networks. In … Many generative models synthesize data by transforming a standard Gaussian random variable using a deterministic neural network. Among these models are the Variational Autoencoders and the Generative Adversarial Networks. In this work, we call them "push-forward" models and study their expressivity. We show that the Lipschitz constant of these generative networks has to be large in order to fit multimodal distributions. More precisely, we show that the total variation distance and the Kullback-Leibler divergence between the generated and the data distribution are bounded from below by a constant depending on the mode separation and the Lipschitz constant. Since constraining the Lipschitz constants of neural networks is a common way to stabilize generative models, there is a provable trade-off between the ability of push-forward models to approximate multimodal distributions and the stability of their training. We validate our findings on one-dimensional and image datasets and empirically show that generative models consisting of stacked networks with stochastic input at each step, such as diffusion models do not suffer of such limitations.
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Patch Redundancy in Images: A Statistical Testing Framework and Some Applications

2019-01-01
Valentin De Bortoli , Agnès Desolneux , Bruno Galerne +1 more
In this work we introduce a statistical framework in order to analyze the spatial redundancy in natural images. This notion of spatial redundancy must be defined locally. To do so, … In this work we introduce a statistical framework in order to analyze the spatial redundancy in natural images. This notion of spatial redundancy must be defined locally. To do so, we define an auto-similarity function which, given one image, computes a dissimilarity measurement between patches. To derive a criterion for taking a decision on the similarity between two patches, we present an a contrario model. Namely, two patches are said to be similar if the associated dissimilarity measurement is unlikely to happen in a background model. Choosing Gaussian random fields as background models, we derive nonasymptotic expressions for the probability distribution function of similarity measurements. We present an algorithm in order to assess redundancy in natural images and discuss applications in denoising, periodicity analysis, and texture ranking.
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Macrocanonical Models for Texture Synthesis

2019-01-01
Valentin De Bortoli , Agnès Desolneux , Bruno Galerne +1 more
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Maximum Entropy Methods for Texture Synthesis: Theory and Practice

2021-01-01
Valentin De Bortoli , Agnès Desolneux , Alain Durmus +2 more
Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 18 December 2019Accepted: 20 October 2020Published online: 26 January 2021Keywordstexture synthesis, information geometry, … Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 18 December 2019Accepted: 20 October 2020Published online: 26 January 2021Keywordstexture synthesis, information geometry, maximum entropy, Markov chains, Langevin algorithm, convolutional neural networkAMS Subject Headings94A17, 94A20, 62B10, 62L20, 62P99, 60J05, 60J22, 65C05, 65C40, 68U10Publication DataISSN (online): 2577-0187Publisher: Society for Industrial and Applied MathematicsCODEN: sjmdaq
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A Zero-One Law for First-Order Logic on Random Images

2004-01-01
David Coupier , Agnès Desolneux , Bernard Ycart
For an $n\times n$ random image with independent pixels, black with probability $p(n)$ and white with probability $1-p(n)$, the probability of satisfying any given first-order sentence tends to 0 or … For an $n\times n$ random image with independent pixels, black with probability $p(n)$ and white with probability $1-p(n)$, the probability of satisfying any given first-order sentence tends to 0 or 1, provided both $p(n)n^{\frac{2}{k}}$ and $(1-p(n))n^{\frac{2}{k}}$ tend to 0 or $+\infty$, for any integer $k$. The result is proved by computing the threshold function for basic local sentences, and applying Gaifman's theorem.
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Regularity and Crossings of Shot Noise Processes

2011-03-31
Hermine Biermé , Agnès Desolneux
In this paper, we consider shot noise processes and their expected number of level crossings. When the kernel response function is sufficiently smooth, the crossings mean number function is obtained … In this paper, we consider shot noise processes and their expected number of level crossings. When the kernel response function is sufficiently smooth, the crossings mean number 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 is studied in detail and two different regimes are exhibited.
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Detection of Small Anomalies on Moving Background

2019-08-26
Axel Davy , Agnès Desolneux , Jean‐Michel Morel
We consider the problem of detecting small targets in videos where the textured background is also possibly moving. The proposed method is based on a two steps statistical framework. In … We consider the problem of detecting small targets in videos where the textured background is also possibly moving. The proposed method is based on a two steps statistical framework. In a first step, the optical flow is computed using a pyramidal scheme incorporating statistical tests for a result with reliability guarantees. In the second step, the detection of targets is changed into a problem of anomaly detection in noise, and statistical testing ensures a control of the number of false detections.

A survey of exemplar-based texture synthesis

2017-01-01
Lara Raad , Axel Davy , Agnès Desolneux +1 more
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 … 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.
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Redundancy in Gaussian random fields

2020-01-01
Valentin De Bortoli , Agnès Desolneux , Bruno Galerne +1 more
In this paper, we introduce a notion of spatial redundancy in Gaussian random fields. This study is motivated by applications of the a contrario method in image processing. We define … In this paper, we introduce a notion of spatial redundancy in Gaussian random fields. This study is motivated by applications of the a contrario method in image processing. We define similarity functions on local windows in random fields over discrete or continuous domains. We derive explicit Gaussian asymptotics for the distribution of similarity functions when computed on Gaussian random fields. Moreover, for the special case of the squared L 2 norm, we give non-asymptotic expressions in both discrete and continuous periodic settings. Finally, we present fast and accurate approximations of these non-asymptotic expressions using moment methods and matrix projections.
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Maximum entropy methods for texture synthesis: theory and practice

2019-12-03
Valentin De Bortoli , Agnès Desolneux , Alain Durmus +2 more
Recent years have seen the rise of convolutional neural network techniques in exemplar-based image synthesis. These methods often rely on the minimization of some variational formulation on the image space … Recent years have seen the rise of convolutional neural network techniques in exemplar-based image synthesis. These methods often rely on the minimization of some variational formulation on the image space for which the minimizers are assumed to be the solutions of the synthesis problem. In this paper we investigate, both theoretically and experimentally, another framework to deal with this problem using an alternate sampling/minimization scheme. First, we use results from information geometry to assess that our method yields a probability measure which has maximum entropy under some constraints in expectation. Then, we turn to the analysis of our method and we show, using recent results from the Markov chain literature, that its error can be explicitly bounded with constants which depend polynomially in the dimension even in the non-convex setting. This includes the case where the constraints are defined via a differentiable neural network. Finally, we present an extensive experimental study of the model, including a comparison with state-of-the-art methods and an extension to style transfer.

On quantitative Laplace-type convergence results for some exponential probability measures, with two applications

2021-01-01
Valentin De Bortoli , Agnès Desolneux
Laplace-type results characterize the limit of sequence of measures $(\pi_\varepsilon)_{\varepsilon >0}$ with density w.r.t the Lebesgue measure $(\mathrm{d} \pi_\varepsilon / \mathrm{d} \mathrm{Leb})(x) \propto \exp[-U(x)/\varepsilon]$ when the temperature $\varepsilon>0$ converges to … Laplace-type results characterize the limit of sequence of measures $(\pi_\varepsilon)_{\varepsilon >0}$ with density w.r.t the Lebesgue measure $(\mathrm{d} \pi_\varepsilon / \mathrm{d} \mathrm{Leb})(x) \propto \exp[-U(x)/\varepsilon]$ when the temperature $\varepsilon>0$ converges to $0$. If a limiting distribution $\pi_0$ exists, it concentrates on the minimizers of the potential $U$. Classical results require the invertibility of the Hessian of $U$ in order to establish such asymptotics. In this work, we study the particular case of norm-like potentials $U$ and establish quantitative bounds between $\pi_\varepsilon$ and $\pi_0$ w.r.t. the Wasserstein distance of order $1$ under an invertibility condition of a generalized Jacobian. One key element of our proof is the use of geometric measure theory tools such as the coarea formula. We apply our results to the study of maximum entropy models (microcanonical/macrocanonical distributions) and to the convergence of the iterates of the Stochastic Gradient Langevin Dynamics (SGLD) algorithm at low temperatures for non-convex minimization.
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A survey of exemplar-based texture synthesis

2017-07-22
Lara Raad , Axel Davy , Agnès Desolneux +1 more
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 … 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.

Macrocanonical Models for Texture Synthesis

2019-04-12
De Bortoli Valentin , Agnès Desolneux , Bruno Galerne +1 more
In this article we consider macrocanonical models for texture synthesis. In these models samples are generated given an input texture image and a set of features which should be matched … In this article we consider macrocanonical models for texture synthesis. In these models samples are generated given an input texture image and a set of features which should be matched in expectation. It is known that if the images are quantized, macrocanonical models are given by Gibbs measures, using the maximum entropy principle. We study conditions under which this result extends to real-valued images. If these conditions hold, finding a macrocanonical model amounts to minimizing a convex function and sampling from an associated Gibbs measure. We analyze an algorithm which alternates between sampling and minimizing. We present experiments with neural network features and study the drawbacks and advantages of using this sampling scheme.

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

2006-01-01
I. Abraham , Romain Abraham , Agnès Desolneux +1 more
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 … 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 $¶(\norm{\nabla I}\geq s(\eps) \bigm| ΔI = 0) \leq\eps$. 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.

Unidentified Floating Object detection in maritime environment using dictionary learning

2020-07-30
Darshan Venkatrayappa , Agnès Desolneux , Jean-Michel Hubert +1 more
Maritime domain is one of the most challenging scenarios for object detection due to the complexity of the observed scene. In this article, we present a new approach to detect … Maritime domain is one of the most challenging scenarios for object detection due to the complexity of the observed scene. In this article, we present a new approach to detect unidentified floating objects in the maritime environment. The proposed approach is capable of detecting floating objects without any prior knowledge of their visual appearance, shape or location. The input image from the video stream is denoised using a visual dictionary learned from a K-SVD algorithm. The denoised image is made of self-similar content. Later, we extract the residual image, which is the difference between the original image and the denoised (self-similar) image. Thus, the residual image contains noise and salient structures (objects). These salient structures can be extracted using an a contrario model. We demonstrate the capabilities of our algorithm by testing it on videos exhibiting varying maritime scenarios.

On quantitative Laplace-type convergence results for some exponential probability measures, with two applications

2021-10-26
Valentin De Bortoli , Agnès Desolneux
Laplace-type results characterize the limit of sequence of measures $(\pi_\varepsilon)_{\varepsilon >0}$ with density w.r.t the Lebesgue measure $(\mathrm{d} \pi_\varepsilon / \mathrm{d} \mathrm{Leb})(x) \propto \exp[-U(x)/\varepsilon]$ when the temperature $\varepsilon>0$ converges to … Laplace-type results characterize the limit of sequence of measures $(\pi_\varepsilon)_{\varepsilon >0}$ with density w.r.t the Lebesgue measure $(\mathrm{d} \pi_\varepsilon / \mathrm{d} \mathrm{Leb})(x) \propto \exp[-U(x)/\varepsilon]$ when the temperature $\varepsilon>0$ converges to $0$. If a limiting distribution $\pi_0$ exists, it concentrates on the minimizers of the potential $U$. Classical results require the invertibility of the Hessian of $U$ in order to establish such asymptotics. In this work, we study the particular case of norm-like potentials $U$ and establish quantitative bounds between $\pi_\varepsilon$ and $\pi_0$ w.r.t. the Wasserstein distance of order $1$ under an invertibility condition of a generalized Jacobian. One key element of our proof is the use of geometric measure theory tools such as the coarea formula. We apply our results to the study of maximum entropy models (microcanonical/macrocanonical distributions) and to the convergence of the iterates of the Stochastic Gradient Langevin Dynamics (SGLD) algorithm at low temperatures for non-convex minimization.
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Patch redundancy in images: a statistical testing framework and some applications

2019-01-01
De Bortoli Valentin , Agnès Desolneux , Bruno Galerne +1 more
In this work we introduce a statistical framework in order to analyze the spatial redundancy in natural images. This notion of spatial redundancy must be defined locally and thus we … In this work we introduce a statistical framework in order to analyze the spatial redundancy in natural images. This notion of spatial redundancy must be defined locally and thus we give some examples of functions (auto-similarity and template similarity) which, given one or two images, computes a similarity measurement between patches. Two patches are said to be similar if the similarity measurement is small enough. To derive a criterion for taking a decision on the similarity between two patches we present an a contrario model. Namely, two patches are said to be similar if the associated similarity measurement is unlikely to happen in a background model. Choosing Gaussian random fields as background models we derive non-asymptotic expressions for the probability distribution function of similarity measurements. We introduce a fast algorithm in order to assess redundancy in natural images and present applications in denoising, periodicity analysis and texture ranking.
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Crossings of smooth Shot Noise Processes

2012-12-01
Hermine Biermé , Agnès Desolneux
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Maximum entropy methods for texture synthesis: theory and practice

2019-01-01
Valentin De Bortoli , Agnès Desolneux , Alain Durmus +2 more
Recent years have seen the rise of convolutional neural network techniques in exemplar-based image synthesis. These methods often rely on the minimization of some variational formulation on the image space … Recent years have seen the rise of convolutional neural network techniques in exemplar-based image synthesis. These methods often rely on the minimization of some variational formulation on the image space for which the minimizers are assumed to be the solutions of the synthesis problem. In this paper we investigate, both theoretically and experimentally, another framework to deal with this problem using an alternate sampling/minimization scheme. First, we use results from information geometry to assess that our method yields a probability measure which has maximum entropy under some constraints in expectation. Then, we turn to the analysis of our method and we show, using recent results from the Markov chain literature, that its error can be explicitly bounded with constants which depend polynomially in the dimension even in the non-convex setting. This includes the case where the constraints are defined via a differentiable neural network. Finally, we present an extensive experimental study of the model, including a comparison with state-of-the-art methods and an extension to style transfer.
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Crossings of smooth Shot Noise Processes

2012-12-01
Hermine Biermé , Agnès Desolneux
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Unidentified Floating Object detection in maritime environment using dictionary learning

2020-01-01
Darshan Venkatrayappa , Agnès Desolneux , Jean-Michel Hubert +1 more
Maritime domain is one of the most challenging scenarios for object detection due to the complexity of the observed scene. In this article, we present a new approach to detect … Maritime domain is one of the most challenging scenarios for object detection due to the complexity of the observed scene. In this article, we present a new approach to detect unidentified floating objects in the maritime environment. The proposed approach is capable of detecting floating objects without any prior knowledge of their visual appearance, shape or location. The input image from the video stream is denoised using a visual dictionary learned from a K-SVD algorithm. The denoised image is made of self-similar content. Later, we extract the residual image, which is the difference between the original image and the denoised (self-similar) image. Thus, the residual image contains noise and salient structures (objects). These salient structures can be extracted using an a contrario model. We demonstrate the capabilities of our algorithm by testing it on videos exhibiting varying maritime scenarios.
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Papillonnage et mathématiques des images

2011-01-28
Julie Delon , Agnès Desolneux
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Self-Organized Percolation Power Laws with and without Fractal Geometry in the Etching of Random Solids

2003-01-01
Agnès Desolneux , B. Sapoval , Andrea Baldassarri
Classically, percolation critical exponents are linked to the power laws that characterize percolation cluster fractal properties. It is found here that the gradient percolation power laws are conserved even for … Classically, percolation critical exponents are linked to the power laws that characterize percolation cluster fractal properties. It is found here that the gradient percolation power laws are conserved even for extreme gradient values for which the frontier of the infinite cluster is no more fractal. In particular the exponent 7/4 which was recently demonstrated to be the exact value for the dimension of the so-called "hull" or external perimeter of the incipient percolation cluster, controls the width and length of gradient percolation frontiers whatever the gradient magnitude. This behavior is extended to previous model studies of etching by a finite volume of etching solution in contact with a disordered solid. In such a model, the dynamics stop spontaneously on an equilibrium self-similar surface similar to the fractal frontier of gradient percolation. It is shown that the power laws describing the system geometry involves also the fractal dimension of the percolation hull, whatever the value of the dynamically generated gradient, i.e. even for a non-fractal frontier. The comparison between numerical results and the exact results that can be obtained analytically for extreme values of the gradient suggests that there exist a unique power law valid from the smallest possible scale up to infinity. These results suggest the possible existence of an underlying conservation law, relating the length and the statistical width of percolation gradient frontiers.

Macrocanonical Models for Texture Synthesis

2019-01-01
De Bortoli Valentin , Agnès Desolneux , Bruno Galerne +1 more
In this article we consider macrocanonical models for texture synthesis. In these models samples are generated given an input texture image and a set of features which should be matched … In this article we consider macrocanonical models for texture synthesis. In these models samples are generated given an input texture image and a set of features which should be matched in expectation. It is known that if the images are quantized, macrocanonical models are given by Gibbs measures, using the maximum entropy principle. We study conditions under which this result extends to real-valued images. If these conditions hold, finding a macrocanonical model amounts to minimizing a convex function and sampling from an associated Gibbs measure. We analyze an algorithm which alternates between sampling and minimizing. We present experiments with neural network features and study the drawbacks and advantages of using this sampling scheme.
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The anisotropy of 2D or 3D Gaussian random fields through their Lipschitz-Killing curvature densities

2024-10-16
Hermine Biermé , Agnès Desolneux
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Gromov-Wassertein-like Distances in the Gaussian Mixture Models Space

2023-01-01
Antoine Salmona , Julie Delon , Agnès Desolneux
The Gromov-Wasserstein (GW) distance is frequently used in machine learning to compare distributions across distinct metric spaces. Despite its utility, it remains computationally intensive, especially for large-scale problems. Recently, a … The Gromov-Wasserstein (GW) distance is frequently used in machine learning to compare distributions across distinct metric spaces. Despite its utility, it remains computationally intensive, especially for large-scale problems. Recently, a novel Wasserstein distance specifically tailored for Gaussian mixture models and known as MW (mixture Wasserstein) has been introduced by several authors. In scenarios where data exhibit clustering, this approach simplifies to a small-scale discrete optimal transport problem, which complexity depends solely on the number of Gaussian components in the GMMs. This paper aims to extend MW by introducing new Gromov-type distances. These distances are designed to be isometry-invariant in Euclidean spaces and are applicable for comparing GMMs across different dimensional spaces. Our first contribution is the Mixture Gromov Wasserstein distance (MGW), which can be viewed as a Gromovized version of MW. This new distance has a straightforward discrete formulation, making it highly efficient for estimating distances between GMMs in practical applications. To facilitate the derivation of a transport plan between GMMs, we present a second distance, the Embedded Wasserstein distance (EW). This distance turns out to be closely related to several recent alternatives to Gromov-Wasserstein. We show that EW can be adapted to derive a distance as well as optimal transportation plans between GMMs. We demonstrate the efficiency of these newly proposed distances on medium to large-scale problems, including shape matching and hyperspectral image color transfer.
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A Wasserstein-type distance in the space of Gaussian Mixture Models

2019-01-01
Julie Delon , Agnès Desolneux
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 … 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.

A zero-one law for first-order logic on random images

2006-01-01
David Coupier , Agnès Desolneux , Bernard Ycart
For an $n\times n$ random image with independent pixels, black with probability $p(n)$ and white with probability $1-p(n)$, the probability of satisfying any given first-order sentence tends to 0 or … For an $n\times n$ random image with independent pixels, black with probability $p(n)$ and white with probability $1-p(n)$, the probability of satisfying any given first-order sentence tends to 0 or 1, provided both $p(n)n^{\frac{2}{k}}$ and $(1-p(n))n^{\frac{2}{k}}$ tend to 0 or $+\infty$, for any integer $k$. The result is proved by computing the threshold function for basic local sentences, and applying Gaifman's theorem.

Image denoising by statistical area thresholding

2006-01-01
David Coupier , Agnès Desolneux , Bernard Ycart
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 … 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.

Constrained Approximate Optimal Transport Maps

2024-07-18
Eloi Tanguy , Agnès Desolneux , Julie Delon
We investigate finding a map $g$ within a function class $G$ that minimises an Optimal Transport (OT) cost between a target measure $\nu$ and the image by $g$ of a … We investigate finding a map $g$ within a function class $G$ that minimises an Optimal Transport (OT) cost between a target measure $\nu$ and the image by $g$ of a source measure $\mu$. This is relevant when an OT map from $\mu$ to $\nu$ does not exist or does not satisfy the desired constraints of $G$. We address existence and uniqueness for generic subclasses of $L$-Lipschitz functions, including gradients of (strongly) convex functions and typical Neural Networks. We explore a variant that approaches a transport plan, showing equivalence to a map problem in some cases. For the squared Euclidean cost, we propose alternating minimisation over a transport plan $\pi$ and map $g$, with the optimisation over $g$ being the $L^2$ projection on $G$ of the barycentric mapping $\overline{\pi}$. In dimension one, this global problem equates the $L^2$ projection of $\overline{\pi^*}$ onto $G$ for an OT plan $\pi^*$ between $\mu$ and $\nu$, but this does not extend to higher dimensions. We introduce a simple kernel method to find $g$ within a Reproducing Kernel Hilbert Space in the discrete case. Finally, we present numerical methods for $L$-Lipschitz gradients of $\ell$-strongly convex potentials.
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BN-SCAFFOLD: controlling the drift of Batch Normalization statistics in Federated Learning

2024-10-04
Gonzalo Iñaki Quintana , Laurence Vancamberg , Vincent Jugnon +2 more
Federated Learning (FL) is gaining traction as a learning paradigm for training Machine Learning (ML) models in a decentralized way. Batch Normalization (BN) is ubiquitous in Deep Neural Networks (DNN), … Federated Learning (FL) is gaining traction as a learning paradigm for training Machine Learning (ML) models in a decentralized way. Batch Normalization (BN) is ubiquitous in Deep Neural Networks (DNN), as it improves convergence and generalization. However, BN has been reported to hinder performance of DNNs in heterogeneous FL. Recently, the FedTAN algorithm has been proposed to mitigate the effect of heterogeneity on BN, by aggregating BN statistics and gradients from all the clients. However, it has a high communication cost, that increases linearly with the depth of the DNN. SCAFFOLD is a variance reduction algorithm, that estimates and corrects the client drift in a communication-efficient manner. Despite its promising results in heterogeneous FL settings, it has been reported to underperform for models with BN. In this work, we seek to revive SCAFFOLD, and more generally variance reduction, as an efficient way of training DNN with BN in heterogeneous FL. We introduce a unified theoretical framework for analyzing the convergence of variance reduction algorithms in the BN-DNN setting, inspired of by the work of Wang et al. 2023, and show that SCAFFOLD is unable to remove the bias introduced by BN. We thus propose the BN-SCAFFOLD algorithm, which extends the client drift correction of SCAFFOLD to BN statistics. We prove convergence using the aforementioned framework and validate the theoretical results with experiments on MNIST and CIFAR-10. BN-SCAFFOLD equals the performance of FedTAN, without its high communication cost, outperforming Federated Averaging (FedAvg), SCAFFOLD, and other FL algorithms designed to mitigate BN heterogeneity.
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Bridging Contrastive Learning and Domain Adaptation: Theoretical Perspective and Practical Application

2025-01-28
Gonzalo Iñaki Quintana , Laurence Vancamberg , Vincent Jugnon +2 more
This work studies the relationship between Contrastive Learning and Domain Adaptation from a theoretical perspective. The two standard contrastive losses, NT-Xent loss (Self-supervised) and Supervised Contrastive loss, are related to … This work studies the relationship between Contrastive Learning and Domain Adaptation from a theoretical perspective. The two standard contrastive losses, NT-Xent loss (Self-supervised) and Supervised Contrastive loss, are related to the Class-wise Mean Maximum Discrepancy (CMMD), a dissimilarity measure widely used for Domain Adaptation. Our work shows that minimizing the contrastive losses decreases the CMMD and simultaneously improves class-separability, laying the theoretical groundwork for the use of Contrastive Learning in the context of Domain Adaptation. Due to the relevance of Domain Adaptation in medical imaging, we focused the experiments on mammography images. Extensive experiments on three mammography datasets - synthetic patches, clinical (real) patches, and clinical (real) images - show improved Domain Adaptation, class-separability, and classification performance, when minimizing the Supervised Contrastive loss.
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Bridging Contrastive Learning and Domain Adaptation: Theoretical Perspective and Practical Application

2025-01-28
Gonzalo Iñaki Quintana , Laurence Vancamberg , Vincent Jugnon +2 more
This work studies the relationship between Contrastive Learning and Domain Adaptation from a theoretical perspective. The two standard contrastive losses, NT-Xent loss (Self-supervised) and Supervised Contrastive loss, are related to … This work studies the relationship between Contrastive Learning and Domain Adaptation from a theoretical perspective. The two standard contrastive losses, NT-Xent loss (Self-supervised) and Supervised Contrastive loss, are related to the Class-wise Mean Maximum Discrepancy (CMMD), a dissimilarity measure widely used for Domain Adaptation. Our work shows that minimizing the contrastive losses decreases the CMMD and simultaneously improves class-separability, laying the theoretical groundwork for the use of Contrastive Learning in the context of Domain Adaptation. Due to the relevance of Domain Adaptation in medical imaging, we focused the experiments on mammography images. Extensive experiments on three mammography datasets - synthetic patches, clinical (real) patches, and clinical (real) images - show improved Domain Adaptation, class-separability, and classification performance, when minimizing the Supervised Contrastive loss.
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The anisotropy of 2D or 3D Gaussian random fields through their Lipschitz-Killing curvature densities

2024-10-16
Hermine Biermé , Agnès Desolneux
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BN-SCAFFOLD: controlling the drift of Batch Normalization statistics in Federated Learning

2024-10-04
Gonzalo Iñaki Quintana , Laurence Vancamberg , Vincent Jugnon +2 more
Federated Learning (FL) is gaining traction as a learning paradigm for training Machine Learning (ML) models in a decentralized way. Batch Normalization (BN) is ubiquitous in Deep Neural Networks (DNN), … Federated Learning (FL) is gaining traction as a learning paradigm for training Machine Learning (ML) models in a decentralized way. Batch Normalization (BN) is ubiquitous in Deep Neural Networks (DNN), as it improves convergence and generalization. However, BN has been reported to hinder performance of DNNs in heterogeneous FL. Recently, the FedTAN algorithm has been proposed to mitigate the effect of heterogeneity on BN, by aggregating BN statistics and gradients from all the clients. However, it has a high communication cost, that increases linearly with the depth of the DNN. SCAFFOLD is a variance reduction algorithm, that estimates and corrects the client drift in a communication-efficient manner. Despite its promising results in heterogeneous FL settings, it has been reported to underperform for models with BN. In this work, we seek to revive SCAFFOLD, and more generally variance reduction, as an efficient way of training DNN with BN in heterogeneous FL. We introduce a unified theoretical framework for analyzing the convergence of variance reduction algorithms in the BN-DNN setting, inspired of by the work of Wang et al. 2023, and show that SCAFFOLD is unable to remove the bias introduced by BN. We thus propose the BN-SCAFFOLD algorithm, which extends the client drift correction of SCAFFOLD to BN statistics. We prove convergence using the aforementioned framework and validate the theoretical results with experiments on MNIST and CIFAR-10. BN-SCAFFOLD equals the performance of FedTAN, without its high communication cost, outperforming Federated Averaging (FedAvg), SCAFFOLD, and other FL algorithms designed to mitigate BN heterogeneity.
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Constrained Approximate Optimal Transport Maps

2024-07-18
Eloi Tanguy , Agnès Desolneux , Julie Delon
We investigate finding a map $g$ within a function class $G$ that minimises an Optimal Transport (OT) cost between a target measure $\nu$ and the image by $g$ of a … We investigate finding a map $g$ within a function class $G$ that minimises an Optimal Transport (OT) cost between a target measure $\nu$ and the image by $g$ of a source measure $\mu$. This is relevant when an OT map from $\mu$ to $\nu$ does not exist or does not satisfy the desired constraints of $G$. We address existence and uniqueness for generic subclasses of $L$-Lipschitz functions, including gradients of (strongly) convex functions and typical Neural Networks. We explore a variant that approaches a transport plan, showing equivalence to a map problem in some cases. For the squared Euclidean cost, we propose alternating minimisation over a transport plan $\pi$ and map $g$, with the optimisation over $g$ being the $L^2$ projection on $G$ of the barycentric mapping $\overline{\pi}$. In dimension one, this global problem equates the $L^2$ projection of $\overline{\pi^*}$ onto $G$ for an OT plan $\pi^*$ between $\mu$ and $\nu$, but this does not extend to higher dimensions. We introduce a simple kernel method to find $g$ within a Reproducing Kernel Hilbert Space in the discrete case. Finally, we present numerical methods for $L$-Lipschitz gradients of $\ell$-strongly convex potentials.
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Gromov-Wassertein-like Distances in the Gaussian Mixture Models Space

2023-01-01
Antoine Salmona , Julie Delon , Agnès Desolneux
The Gromov-Wasserstein (GW) distance is frequently used in machine learning to compare distributions across distinct metric spaces. Despite its utility, it remains computationally intensive, especially for large-scale problems. Recently, a … The Gromov-Wasserstein (GW) distance is frequently used in machine learning to compare distributions across distinct metric spaces. Despite its utility, it remains computationally intensive, especially for large-scale problems. Recently, a novel Wasserstein distance specifically tailored for Gaussian mixture models and known as MW (mixture Wasserstein) has been introduced by several authors. In scenarios where data exhibit clustering, this approach simplifies to a small-scale discrete optimal transport problem, which complexity depends solely on the number of Gaussian components in the GMMs. This paper aims to extend MW by introducing new Gromov-type distances. These distances are designed to be isometry-invariant in Euclidean spaces and are applicable for comparing GMMs across different dimensional spaces. Our first contribution is the Mixture Gromov Wasserstein distance (MGW), which can be viewed as a Gromovized version of MW. This new distance has a straightforward discrete formulation, making it highly efficient for estimating distances between GMMs in practical applications. To facilitate the derivation of a transport plan between GMMs, we present a second distance, the Embedded Wasserstein distance (EW). This distance turns out to be closely related to several recent alternatives to Gromov-Wasserstein. We show that EW can be adapted to derive a distance as well as optimal transportation plans between GMMs. We demonstrate the efficiency of these newly proposed distances on medium to large-scale problems, including shape matching and hyperspectral image color transfer.
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Gromov–Wasserstein distances between Gaussian distributions

2022-08-18
Julie Delon , Agnès Desolneux , Antoine Salmona
Abstract Gromov–Wasserstein distances were proposed a few years ago to compare distributions which do not lie in the same space. In particular, they offer an interesting alternative to the Wasserstein … Abstract Gromov–Wasserstein distances were proposed a few years ago to compare distributions which do not lie in the same space. In particular, they offer an interesting alternative to the Wasserstein distances for comparing probability measures living on Euclidean spaces of different dimensions. We focus on the Gromov–Wasserstein distance with a ground cost defined as the squared Euclidean distance, and we study the form of the optimal plan between Gaussian distributions. We show that when the optimal plan is restricted to Gaussian distributions, the problem has a very simple linear solution, which is also a solution of the linear Gromov–Monge problem. We also study the problem without restriction on the optimal plan, and provide lower and upper bounds for the value of the Gromov–Wasserstein distance between Gaussian distributions.
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Can Push-forward Generative Models Fit Multimodal Distributions?

2022-01-01
Antoine Salmona , Valentin De Bortoli , Agnès Desolneux +1 more
Many generative models synthesize data by transforming a standard Gaussian random variable using a deterministic neural network. Among these models are the Variational Autoencoders and the Generative Adversarial Networks. In … Many generative models synthesize data by transforming a standard Gaussian random variable using a deterministic neural network. Among these models are the Variational Autoencoders and the Generative Adversarial Networks. In this work, we call them "push-forward" models and study their expressivity. We show that the Lipschitz constant of these generative networks has to be large in order to fit multimodal distributions. More precisely, we show that the total variation distance and the Kullback-Leibler divergence between the generated and the data distribution are bounded from below by a constant depending on the mode separation and the Lipschitz constant. Since constraining the Lipschitz constants of neural networks is a common way to stabilize generative models, there is a provable trade-off between the ability of push-forward models to approximate multimodal distributions and the stability of their training. We validate our findings on one-dimensional and image datasets and empirically show that generative models consisting of stacked networks with stochastic input at each step, such as diffusion models do not suffer of such limitations.
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On quantitative Laplace-type convergence results for some exponential probability measures, with two applications

2021-10-26
Valentin De Bortoli , Agnès Desolneux
Laplace-type results characterize the limit of sequence of measures $(\pi_\varepsilon)_{\varepsilon >0}$ with density w.r.t the Lebesgue measure $(\mathrm{d} \pi_\varepsilon / \mathrm{d} \mathrm{Leb})(x) \propto \exp[-U(x)/\varepsilon]$ when the temperature $\varepsilon>0$ converges to … Laplace-type results characterize the limit of sequence of measures $(\pi_\varepsilon)_{\varepsilon >0}$ with density w.r.t the Lebesgue measure $(\mathrm{d} \pi_\varepsilon / \mathrm{d} \mathrm{Leb})(x) \propto \exp[-U(x)/\varepsilon]$ when the temperature $\varepsilon>0$ converges to $0$. If a limiting distribution $\pi_0$ exists, it concentrates on the minimizers of the potential $U$. Classical results require the invertibility of the Hessian of $U$ in order to establish such asymptotics. In this work, we study the particular case of norm-like potentials $U$ and establish quantitative bounds between $\pi_\varepsilon$ and $\pi_0$ w.r.t. the Wasserstein distance of order $1$ under an invertibility condition of a generalized Jacobian. One key element of our proof is the use of geometric measure theory tools such as the coarea formula. We apply our results to the study of maximum entropy models (microcanonical/macrocanonical distributions) and to the convergence of the iterates of the Stochastic Gradient Langevin Dynamics (SGLD) algorithm at low temperatures for non-convex minimization.
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The effect of discretization on the mean geometry of a 2D random field

2021-09-22
Hermine Biermé , Agnès Desolneux
The study of the geometry of excursion sets of 2D random fields is a question of interest from both the theoretical and the applied viewpoints. In this paper we are … The study of the geometry of excursion sets of 2D random fields is a question of interest from both the theoretical and the applied viewpoints. In this paper we are interested in the relationship between the perimeter (resp. the total curvature, related to the Euler characteristic by Gauss–Bonnet Theorem) of the excursion sets of a function and the ones of its discretization. Our approach is a weak framework in which we consider the functions that map the level of the excursion set to the perimeter (resp. the total curvature) of the excursion set. We will be also interested in a stochastic framework in which the sets are the excursion sets of 2D random fields. We show in particular that, under some stationarity and isotropy conditions on the random field, in expectation, the perimeter is always biased (with a 4/π factor), whereas the total curvature is not. We illustrate all our results on different examples of random fields.
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Determinantal Point Processes for Image Processing

2021-01-01
Claire Launay , Agnès Desolneux , Bruno Galerne
Determinantal point processes (DPPs) are probabilistic models of configurations that favor diversity or repulsion. They have recently gained influence in the machine learning community, mainly because of their ability to … Determinantal point processes (DPPs) are probabilistic models of configurations that favor diversity or repulsion. They have recently gained influence in the machine learning community, mainly because of their ability to elegantly and efficiently subsample large sets of data. In this paper, we consider DPPs from an image processing perspective, meaning that the data we want to subsample are pixels or patches of a given image. To this end, our framework is discrete and finite. First, we adapt their basic definition and properties to DPPs defined on the pixels of an image, that we call determinantal pixel processes (DPixPs). We are mainly interested in the repulsion properties of such a process and we apply DPixPs to texture synthesis using shot noise models. Finally, we study DPPs on the set of patches of an image. Because of their repulsive property, DPPs provide a strong tool to subsample discrete distributions such as that of image patches.
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Maximum Entropy Methods for Texture Synthesis: Theory and Practice

2021-01-01
Valentin De Bortoli , Agnès Desolneux , Alain Durmus +2 more
Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 18 December 2019Accepted: 20 October 2020Published online: 26 January 2021Keywordstexture synthesis, information geometry, … Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 18 December 2019Accepted: 20 October 2020Published online: 26 January 2021Keywordstexture synthesis, information geometry, maximum entropy, Markov chains, Langevin algorithm, convolutional neural networkAMS Subject Headings94A17, 94A20, 62B10, 62L20, 62P99, 60J05, 60J22, 65C05, 65C40, 68U10Publication DataISSN (online): 2577-0187Publisher: Society for Industrial and Applied MathematicsCODEN: sjmdaq
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Gromov-Wasserstein Distances between Gaussian Distributions

2021-01-01
Antoine Salmona , Julie Delon , Agnès Desolneux
The Gromov-Wasserstein distances were proposed a few years ago to compare distributions which do not lie in the same space. In particular, they offer an interesting alternative to the Wasserstein … The Gromov-Wasserstein distances were proposed a few years ago to compare distributions which do not lie in the same space. In particular, they offer an interesting alternative to the Wasserstein distances for comparing probability measures living on Euclidean spaces of different dimensions. In this paper, we focus on the Gromov-Wasserstein distance with a ground cost defined as the squared Euclidean distance and we study the form of the optimal plan between Gaussian distributions. We show that when the optimal plan is restricted to Gaussian distributions, the problem has a very simple linear solution, which is also solution of the linear Gromov-Monge problem. We also study the problem without restriction on the optimal plan, and provide lower and upper bounds for the value of the Gromov-Wasserstein distance between Gaussian distributions.
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On quantitative Laplace-type convergence results for some exponential probability measures, with two applications

2021-01-01
Valentin De Bortoli , Agnès Desolneux
Laplace-type results characterize the limit of sequence of measures $(\pi_\varepsilon)_{\varepsilon >0}$ with density w.r.t the Lebesgue measure $(\mathrm{d} \pi_\varepsilon / \mathrm{d} \mathrm{Leb})(x) \propto \exp[-U(x)/\varepsilon]$ when the temperature $\varepsilon>0$ converges to … Laplace-type results characterize the limit of sequence of measures $(\pi_\varepsilon)_{\varepsilon >0}$ with density w.r.t the Lebesgue measure $(\mathrm{d} \pi_\varepsilon / \mathrm{d} \mathrm{Leb})(x) \propto \exp[-U(x)/\varepsilon]$ when the temperature $\varepsilon>0$ converges to $0$. If a limiting distribution $\pi_0$ exists, it concentrates on the minimizers of the potential $U$. Classical results require the invertibility of the Hessian of $U$ in order to establish such asymptotics. In this work, we study the particular case of norm-like potentials $U$ and establish quantitative bounds between $\pi_\varepsilon$ and $\pi_0$ w.r.t. the Wasserstein distance of order $1$ under an invertibility condition of a generalized Jacobian. One key element of our proof is the use of geometric measure theory tools such as the coarea formula. We apply our results to the study of maximum entropy models (microcanonical/macrocanonical distributions) and to the convergence of the iterates of the Stochastic Gradient Langevin Dynamics (SGLD) algorithm at low temperatures for non-convex minimization.
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Exact sampling of determinantal point processes without eigendecomposition

2020-11-23
Claire Launay , Bruno Galerne , Agnès Desolneux
Abstract Determinantal point processes (DPPs) enable the modeling of repulsion: they provide diverse sets of points. The repulsion is encoded in a kernel K that can be seen, in a … Abstract Determinantal point processes (DPPs) enable the modeling of repulsion: they provide diverse sets of points. The repulsion is encoded in a kernel K that can be seen, in a discrete setting, as a matrix storing the similarity between points. The main exact algorithm to sample DPPs uses the spectral decomposition of K , a computation that becomes costly when dealing with a high number of points. Here we present an alternative exact algorithm to sample in discrete spaces that avoids the eigenvalues and the eigenvectors computation. The method used here is innovative, and numerical experiments show competitive results with respect to the initial algorithm.
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Unidentified Floating Object detection in maritime environment using dictionary learning

2020-07-30
Darshan Venkatrayappa , Agnès Desolneux , Jean-Michel Hubert +1 more
Maritime domain is one of the most challenging scenarios for object detection due to the complexity of the observed scene. In this article, we present a new approach to detect … Maritime domain is one of the most challenging scenarios for object detection due to the complexity of the observed scene. In this article, we present a new approach to detect unidentified floating objects in the maritime environment. The proposed approach is capable of detecting floating objects without any prior knowledge of their visual appearance, shape or location. The input image from the video stream is denoised using a visual dictionary learned from a K-SVD algorithm. The denoised image is made of self-similar content. Later, we extract the residual image, which is the difference between the original image and the denoised (self-similar) image. Thus, the residual image contains noise and salient structures (objects). These salient structures can be extracted using an a contrario model. We demonstrate the capabilities of our algorithm by testing it on videos exhibiting varying maritime scenarios.

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

2020-04-01
Hermine Biermé , Agnès Desolneux
We introduce the level perimeter integral and the total curvature integral associated with a real-valued function $f$ defined on the plane $\mathbb{R}^{2}$, as integrals allowing to compute the perimeter of … We introduce the level perimeter integral and the total curvature integral associated with a real-valued function $f$ defined on the plane $\mathbb{R}^{2}$, as integrals allowing to compute the perimeter of the excursion set of $f$ above level $t$ and the total (signed) curvature of its boundary for almost every level $t$. Thanks to the Gauss–Bonnet theorem, the total curvature is directly related to the Euler characteristic of the excursion set. We show that the level perimeter and the total curvature integrals can be 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: for piecewise constant shot noise random fields, we give some examples of completely explicit formulas, and for smooth shot noise random fields the provided examples are only partly explicit, since the formulas are given under the form of integrals of some special functions.
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Redundancy in Gaussian random fields

2020-01-01
Valentin De Bortoli , Agnès Desolneux , Bruno Galerne +1 more
In this paper, we introduce a notion of spatial redundancy in Gaussian random fields. This study is motivated by applications of the a contrario method in image processing. We define … In this paper, we introduce a notion of spatial redundancy in Gaussian random fields. This study is motivated by applications of the a contrario method in image processing. We define similarity functions on local windows in random fields over discrete or continuous domains. We derive explicit Gaussian asymptotics for the distribution of similarity functions when computed on Gaussian random fields. Moreover, for the special case of the squared L 2 norm, we give non-asymptotic expressions in both discrete and continuous periodic settings. Finally, we present fast and accurate approximations of these non-asymptotic expressions using moment methods and matrix projections.
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A Wasserstein-Type Distance in the Space of Gaussian Mixture Models

2020-01-01
Julie Delon , Agnès Desolneux
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 … 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.
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Unidentified Floating Object detection in maritime environment using dictionary learning

2020-01-01
Darshan Venkatrayappa , Agnès Desolneux , Jean-Michel Hubert +1 more
Maritime domain is one of the most challenging scenarios for object detection due to the complexity of the observed scene. In this article, we present a new approach to detect … Maritime domain is one of the most challenging scenarios for object detection due to the complexity of the observed scene. In this article, we present a new approach to detect unidentified floating objects in the maritime environment. The proposed approach is capable of detecting floating objects without any prior knowledge of their visual appearance, shape or location. The input image from the video stream is denoised using a visual dictionary learned from a K-SVD algorithm. The denoised image is made of self-similar content. Later, we extract the residual image, which is the difference between the original image and the denoised (self-similar) image. Thus, the residual image contains noise and salient structures (objects). These salient structures can be extracted using an a contrario model. We demonstrate the capabilities of our algorithm by testing it on videos exhibiting varying maritime scenarios.
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Maximum entropy methods for texture synthesis: theory and practice

2019-12-03
Valentin De Bortoli , Agnès Desolneux , Alain Durmus +2 more
Recent years have seen the rise of convolutional neural network techniques in exemplar-based image synthesis. These methods often rely on the minimization of some variational formulation on the image space … Recent years have seen the rise of convolutional neural network techniques in exemplar-based image synthesis. These methods often rely on the minimization of some variational formulation on the image space for which the minimizers are assumed to be the solutions of the synthesis problem. In this paper we investigate, both theoretically and experimentally, another framework to deal with this problem using an alternate sampling/minimization scheme. First, we use results from information geometry to assess that our method yields a probability measure which has maximum entropy under some constraints in expectation. Then, we turn to the analysis of our method and we show, using recent results from the Markov chain literature, that its error can be explicitly bounded with constants which depend polynomially in the dimension even in the non-convex setting. This includes the case where the constraints are defined via a differentiable neural network. Finally, we present an extensive experimental study of the model, including a comparison with state-of-the-art methods and an extension to style transfer.

Detection of Small Anomalies on Moving Background

2019-08-26
Axel Davy , Agnès Desolneux , Jean‐Michel Morel
We consider the problem of detecting small targets in videos where the textured background is also possibly moving. The proposed method is based on a two steps statistical framework. In … We consider the problem of detecting small targets in videos where the textured background is also possibly moving. The proposed method is based on a two steps statistical framework. In a first step, the optical flow is computed using a pyramidal scheme incorporating statistical tests for a result with reliability guarantees. In the second step, the detection of targets is changed into a problem of anomaly detection in noise, and statistical testing ensures a control of the number of false detections.

Macrocanonical Models for Texture Synthesis

2019-04-12
De Bortoli Valentin , Agnès Desolneux , Bruno Galerne +1 more
In this article we consider macrocanonical models for texture synthesis. In these models samples are generated given an input texture image and a set of features which should be matched … In this article we consider macrocanonical models for texture synthesis. In these models samples are generated given an input texture image and a set of features which should be matched in expectation. It is known that if the images are quantized, macrocanonical models are given by Gibbs measures, using the maximum entropy principle. We study conditions under which this result extends to real-valued images. If these conditions hold, finding a macrocanonical model amounts to minimizing a convex function and sampling from an associated Gibbs measure. We analyze an algorithm which alternates between sampling and minimizing. We present experiments with neural network features and study the drawbacks and advantages of using this sampling scheme.

Patch Redundancy in Images: A Statistical Testing Framework and Some Applications

2019-01-01
Valentin De Bortoli , Agnès Desolneux , Bruno Galerne +1 more
In this work we introduce a statistical framework in order to analyze the spatial redundancy in natural images. This notion of spatial redundancy must be defined locally. To do so, … In this work we introduce a statistical framework in order to analyze the spatial redundancy in natural images. This notion of spatial redundancy must be defined locally. To do so, we define an auto-similarity function which, given one image, computes a dissimilarity measurement between patches. To derive a criterion for taking a decision on the similarity between two patches, we present an a contrario model. Namely, two patches are said to be similar if the associated dissimilarity measurement is unlikely to happen in a background model. Choosing Gaussian random fields as background models, we derive nonasymptotic expressions for the probability distribution function of similarity measurements. We present an algorithm in order to assess redundancy in natural images and discuss applications in denoising, periodicity analysis, and texture ranking.
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Macrocanonical Models for Texture Synthesis

2019-01-01
Valentin De Bortoli , Agnès Desolneux , Bruno Galerne +1 more
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Patch redundancy in images: a statistical testing framework and some applications

2019-01-01
De Bortoli Valentin , Agnès Desolneux , Bruno Galerne +1 more
In this work we introduce a statistical framework in order to analyze the spatial redundancy in natural images. This notion of spatial redundancy must be defined locally and thus we … In this work we introduce a statistical framework in order to analyze the spatial redundancy in natural images. This notion of spatial redundancy must be defined locally and thus we give some examples of functions (auto-similarity and template similarity) which, given one or two images, computes a similarity measurement between patches. Two patches are said to be similar if the similarity measurement is small enough. To derive a criterion for taking a decision on the similarity between two patches we present an a contrario model. Namely, two patches are said to be similar if the associated similarity measurement is unlikely to happen in a background model. Choosing Gaussian random fields as background models we derive non-asymptotic expressions for the probability distribution function of similarity measurements. We introduce a fast algorithm in order to assess redundancy in natural images and present applications in denoising, periodicity analysis and texture ranking.
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Maximum entropy methods for texture synthesis: theory and practice

2019-01-01
Valentin De Bortoli , Agnès Desolneux , Alain Durmus +2 more
Recent years have seen the rise of convolutional neural network techniques in exemplar-based image synthesis. These methods often rely on the minimization of some variational formulation on the image space … Recent years have seen the rise of convolutional neural network techniques in exemplar-based image synthesis. These methods often rely on the minimization of some variational formulation on the image space for which the minimizers are assumed to be the solutions of the synthesis problem. In this paper we investigate, both theoretically and experimentally, another framework to deal with this problem using an alternate sampling/minimization scheme. First, we use results from information geometry to assess that our method yields a probability measure which has maximum entropy under some constraints in expectation. Then, we turn to the analysis of our method and we show, using recent results from the Markov chain literature, that its error can be explicitly bounded with constants which depend polynomially in the dimension even in the non-convex setting. This includes the case where the constraints are defined via a differentiable neural network. Finally, we present an extensive experimental study of the model, including a comparison with state-of-the-art methods and an extension to style transfer.
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Macrocanonical Models for Texture Synthesis

2019-01-01
De Bortoli Valentin , Agnès Desolneux , Bruno Galerne +1 more
In this article we consider macrocanonical models for texture synthesis. In these models samples are generated given an input texture image and a set of features which should be matched … In this article we consider macrocanonical models for texture synthesis. In these models samples are generated given an input texture image and a set of features which should be matched in expectation. It is known that if the images are quantized, macrocanonical models are given by Gibbs measures, using the maximum entropy principle. We study conditions under which this result extends to real-valued images. If these conditions hold, finding a macrocanonical model amounts to minimizing a convex function and sampling from an associated Gibbs measure. We analyze an algorithm which alternates between sampling and minimizing. We present experiments with neural network features and study the drawbacks and advantages of using this sampling scheme.
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A Wasserstein-type distance in the space of Gaussian Mixture Models

2019-01-01
Julie Delon , Agnès Desolneux
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 … 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.

A survey of exemplar-based texture synthesis

2018-01-01
Lara Raad , Axel Davy , Agnès Desolneux +1 more
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 … 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 combines ideas from both approaches to avoid their hurdles. Current methods, including the recent CNN approaches, are able to produce impressive synthesis on various kinds of textures. Nevertheless, 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.
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A survey of exemplar-based texture synthesis

2017-07-22
Lara Raad , Axel Davy , Agnès Desolneux +1 more
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 … 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.

A survey of exemplar-based texture synthesis

2017-01-01
Lara Raad , Axel Davy , Agnès Desolneux +1 more
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 … 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.
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On the perimeter of excursion sets of shot noise random fields

2016-01-01
Hermine Biermé , Agnès Desolneux
In this paper, we use the framework of functions of bounded variation and the coarea formula to give an explicit computation for the expectation of the perimeter of excursion sets … In this paper, we use the framework of functions of bounded variation and the coarea formula to give an explicit computation for the expectation of the perimeter of excursion sets of shot noise random fields in dimension $n\geq1$. This will then allow us to derive the asymptotic behavior of these mean perimeters as the intensity of the underlying homogeneous Poisson point process goes to infinity. In particular, we show that two cases occur: we have a Gaussian asymptotic behavior when the kernel function of the shot noise has no jump part, whereas the asymptotic is non-Gaussian when there are jumps.
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Crossings of smooth Shot Noise Processes

2012-12-01
Hermine Biermé , Agnès Desolneux
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Crossings of smooth Shot Noise Processes

2012-12-01
Hermine Biermé , Agnès Desolneux
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Crossings of smooth shot noise processes

2012-11-23
Hermine Biermé , Agnès Desolneux
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 … 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.
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A Fourier Approach for the Level Crossings of Shot Noise Processes with Jumps

2012-03-01
Hermine Biermé , Agnès Desolneux
We use a change-of-variable formula in the framework of functions of bounded variation to derive an explicit formula for the Fourier transform of the level crossing function of shot noise … We use a change-of-variable formula in the framework of functions of bounded variation to derive an explicit formula for the Fourier transform of the level crossing function of shot noise processes with jumps. We illustrate the result in some examples and give some applications. In particular, it allows us to study the asymptotic behavior of the mean number of level crossings as the intensity of the Poisson point process of the shot noise process goes to infinity.
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A compact representation of random phase and Gaussian textures

2012-03-01
Agnès Desolneux , Lionel Moisan , Samuel Ronsin
In this paper, we are interested in the mathematical analysis of the micro-textures that have the property to be perceptually invariant under the randomization of the phases of their Fourier … In this paper, we are interested in the mathematical analysis of the micro-textures that have the property to be perceptually invariant under the randomization of the phases of their Fourier Transform. We propose a compact representation of these textures by considering a special instance of them: the one that has identically null phases, and we call it "texton". We show that this texton has many interesting properties, and in particular it is concentrated around the spatial origin. It appears to be a simple and useful tool for texture analysis and texture synthesis, and its definition can be extended to the case of color micro-textures.
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Regularity and Crossings of Shot Noise Processes

2011-03-31
Hermine Biermé , Agnès Desolneux
In this paper, we consider shot noise processes and their expected number of level crossings. When the kernel response function is sufficiently smooth, the crossings mean number function is obtained … In this paper, we consider shot noise processes and their expected number of level crossings. When the kernel response function is sufficiently smooth, the crossings mean number 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 is studied in detail and two different regimes are exhibited.
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Papillonnage et mathématiques des images

2011-01-28
Julie Delon , Agnès Desolneux
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Significant edges in the case of non-stationary Gaussian noise

2007-03-13
I. Abraham , Romain Abraham , Agnès Desolneux +1 more
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Significant edges in the case of a non-stationary Gaussian noise

2006-01-01
I. Abraham , Romain Abraham , Agnès Desolneux +1 more
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 … 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 $¶(\norm{\nabla I}\geq s(\eps) \bigm| ΔI = 0) \leq\eps$. 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.

A zero-one law for first-order logic on random images

2006-01-01
David Coupier , Agnès Desolneux , Bernard Ycart
For an $n\times n$ random image with independent pixels, black with probability $p(n)$ and white with probability $1-p(n)$, the probability of satisfying any given first-order sentence tends to 0 or … For an $n\times n$ random image with independent pixels, black with probability $p(n)$ and white with probability $1-p(n)$, the probability of satisfying any given first-order sentence tends to 0 or 1, provided both $p(n)n^{\frac{2}{k}}$ and $(1-p(n))n^{\frac{2}{k}}$ tend to 0 or $+\infty$, for any integer $k$. The result is proved by computing the threshold function for basic local sentences, and applying Gaifman's theorem.

Image denoising by statistical area thresholding

2006-01-01
David Coupier , Agnès Desolneux , Bernard Ycart
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 … 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.

Image Denoising by Statistical Area Thresholding

2005-04-14
David Coupier , Agnès Desolneux , Bernard Ycart
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A Zero-One Law for First-Order Logic on Random Images

2004-01-01
David Coupier , Agnès Desolneux , Bernard Ycart
For an $n\times n$ random image with independent pixels, black with probability $p(n)$ and white with probability $1-p(n)$, the probability of satisfying any given first-order sentence tends to 0 or … For an $n\times n$ random image with independent pixels, black with probability $p(n)$ and white with probability $1-p(n)$, the probability of satisfying any given first-order sentence tends to 0 or 1, provided both $p(n)n^{\frac{2}{k}}$ and $(1-p(n))n^{\frac{2}{k}}$ tend to 0 or $+\infty$, for any integer $k$. The result is proved by computing the threshold function for basic local sentences, and applying Gaifman's theorem.
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Self-organized percolation power laws with and without fractal geometry in the etching of random solids

2004-01-01
Agnès Desolneux , B. Sapoval , Andrea Baldassarri
Classically, percolation critical exponents are linked to the power laws that characterize percolation cluster fractal properties. It is found here that the gradient percolation power laws are conserved even for … Classically, percolation critical exponents are linked to the power laws that characterize percolation cluster fractal properties. It is found here that the gradient percolation power laws are conserved even for extreme gra- dient values for which the frontier of the infinite cluster is no more fractal. In particular the exponent 7/4 which was recently demonstrated to be the exact value for the dimension of the so-called hull or external perimeter of the incipient percolation cluster, controls the width and length of gradient percolation frontiers whatever the gradient magnitude. This behavior is extended to previous model studies of etching by a finite volume of etching solution in contact with a disordered solid. In such a model, the dynamics stop spontaneously on an equilibrium self-similar surface similar to the fractal frontier of gradient percolation. It is shown that the power laws describing the system geometry involves also the fractal dimension of the percolation hull, whatever the value of the dynamically generated gradient, i.e. even for a non-fractal frontier. The comparison between numerical results and the exact results that can be obtained analytically for extreme values of the gradient suggests that there exist a unique power law valid from the smallest possible scale up to infinity. These results suggest the possible existence of an underlying conservation law, relating the length and the statistical width of percolation gradient frontiers.
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Self-Organized Percolation Power Laws with and without Fractal Geometry in the Etching of Random Solids

2003-01-01
Agnès Desolneux , B. Sapoval , Andrea Baldassarri
Classically, percolation critical exponents are linked to the power laws that characterize percolation cluster fractal properties. It is found here that the gradient percolation power laws are conserved even for … Classically, percolation critical exponents are linked to the power laws that characterize percolation cluster fractal properties. It is found here that the gradient percolation power laws are conserved even for extreme gradient values for which the frontier of the infinite cluster is no more fractal. In particular the exponent 7/4 which was recently demonstrated to be the exact value for the dimension of the so-called "hull" or external perimeter of the incipient percolation cluster, controls the width and length of gradient percolation frontiers whatever the gradient magnitude. This behavior is extended to previous model studies of etching by a finite volume of etching solution in contact with a disordered solid. In such a model, the dynamics stop spontaneously on an equilibrium self-similar surface similar to the fractal frontier of gradient percolation. It is shown that the power laws describing the system geometry involves also the fractal dimension of the percolation hull, whatever the value of the dynamically generated gradient, i.e. even for a non-fractal frontier. The comparison between numerical results and the exact results that can be obtained analytically for extreme values of the gradient suggests that there exist a unique power law valid from the smallest possible scale up to infinity. These results suggest the possible existence of an underlying conservation law, relating the length and the statistical width of percolation gradient frontiers.
Coauthor Papers Together
Bruno Galerne 11
Valentin De Bortoli 9
Hermine Biermé 9
Julie Delon 8
Arthur Leclaire 8
Axel Davy 4
Bernard Ycart 4
Jean‐Michel Morel 4
David Coupier 4
Antoine Salmona 4
Alain Durmus 3
De Bortoli Valentin 3
Lara Raad 3
Romain Abraham 2
Sébastien Li‐Thiao‐Té 2
Laurence Vancamberg 2
Josselin Manceau 2
Mathilde Mougeot 2
Vincent Jugnon 2
Gonzalo Iñaki Quintana 2
B. Sapoval 2
I. Abraham 2
Darshan Venkatrayappa 2
Claire Launay 2
Andrea Baldassarri 2
Jean-Michel Hubert 2
L. Schultz Arthur 1
Eloi Tanguy 1
Samuel Ronsin 1
Lionel Moisan 1