Marianne Clausel

Self-Similar Anisotropic Texture Analysis: The Hyperbolic Wavelet Transform Contribution

Textures in images can often be well modeled using self-similar processes while they may simultaneously display anisotropy. The present contribution thus aims at studying jointly selfsimilarity and anisotropy by focusing on a specific classical class of Gaussian anisotropic selfsimilar processes. It will be first shown that accurate joint estimates of the anisotropy and selfsimilarity parameters are performed by replacing the standard 2D-discrete wavelet transform with the hyperbolic wavelet transform, which permits the use of different dilation factors along the horizontal and vertical axes. Defining anisotropy requires a reference direction that needs not a priori match the horizontal and vertical axes according to which the images are digitized; this discrepancy defines a rotation angle. Second, we show that this rotation angle can be jointly estimated. Third, a nonparametric bootstrap based procedure is described, which provides confidence intervals in addition to the estimates themselves and enables us to construct an isotropy test procedure, which can be applied to a single texture image. Fourth, the robustness and versatility of the proposed analysis are illustrated by being applied to a large variety of different isotropic and anisotropic self-similar fields. As an illustration, we show that a true anisotropy built-in self-similarity can be disentangled from an isotropic self-similarity to which an anisotropic trend has been superimposed.

Some prevalent results about strongly monoHölder functions

We study the typical behaviour of strongly monoH?lder functions from the prevalence point of view. To this end we first prove wavelet-based criteria for strongly monoH?lder functions. We then use the notion of prevalence to show that the functions of C?(Rd) are almost surely strongly monoH?lder with H?lder exponent ?. Finally, we prove that for any ? (0, 1) on a prevalent set of C?(Rd) the Hausdorff dimension of the graph is equal to d + 1 ? ?.

Streaming-LDA: A Copula-based Approach to Modeling Topic Dependencies in Document Streams

We propose in this paper two new models for modeling topic and word-topic dependencies between consecutive documents in document streams. The first model is a direct extension of Latent Dirichlet Allocation model (LDA) and makes use of a Dirichlet distribution to balance the influence of the LDA prior parameters wrt to topic and word-topic distribution of the previous document. The second extension makes use of copulas, which constitute a generic tools to model dependencies between random variables. We rely here on Archimedean copulas, and more precisely on Franck copulas, as they are symmetric and associative and are thus appropriate for exchangeable random variables. Our experiments, conducted on three standard collections that have been used in several studies on topic modeling, show that our proposals outperform previous ones (as dynamic topic models and temporal \LDA), both in terms of perplexity and for tracking similar topics in a document stream.

Wavelets Techniques for Pointwise Anti-Hölderian Irregularity

In this paper, we introduce a notion of weak pointwise Hölder regularity, starting from the definition of the pointwise anti-Hölder irregularity. Using this concept, a weak spectrum of singularities can be defined as for the usual pointwise Hölder regularity. We build a class of wavelet series satisfying the multifractal formalism and thus show the optimality of the upper bound. We also show that the weak spectrum of singularities is disconnected from the casual one (referred to here as strong spectrum of singularities) by exhibiting a multifractal function made of Davenport series whose weak spectrum differs from the strong one.

On a Topic Model for Sentences

Probabilistic topic models are generative models that describe the content of documents by discovering the latent topics underlying them. However, the structure of the textual input, and for instance the grouping of words in coherent text spans such as sentences, contains much information which is generally lost with these models. In this paper, we propose sentenceLDA, an extension of LDA whose goal is to overcome this limitation by incorporating the structure of the text in the generative and inference processes. We illustrate the advantages of sentenceLDA by comparing it with LDA using both intrinsic (perplexity) and extrinsic (text classification) evaluation tasks on different text collections.

Hyperbolic Wavelet-Fisz Denoising for a Model Arising in Ultrasound Imaging

We present an algorithm and its fully data-driven extension for noise reduction in ultrasound imaging. The proposed method computes the hyperbolic wavelet transform of the image, before applying a multiscale variance stabilization technique, via a Fisz transformation. This adapts the wavelet coefficients statistics to the wavelet thresholding paradigm. The use of hyperbolic wavelets makes it possible to recover the image while respecting the anisotropic nature of structural details. The data-driven extension obviates the need for any prior knowledge of the noise model parameters by estimating the noise variance using an isotonic Nadaraya–Watson estimator. Experiments on synthetic and real data demonstrate the potential of the proposed algorithm to recover ultrasound images while preserving tissue details. Furthermore, comparisons with other noise-reduction methods show that our technique is competitive with the state-of-the-art OBNLM filter. Finally, the variance estimation procedure is applied to real images emphasizing the noise model.

Asymptotic behavior of the quadratic variation of the sum of two Hermite processes of consecutive orders

Abstract Hermite processes are self-similar processes with stationary increments which appear as limits of normalized sums of random variables with long range dependence. The Hermite process of order 1 is fractional Brownian motion and the Hermite process of order 2 is the Rosenblatt process. We consider here the sum of two Hermite processes of orders q ≥ 1 and q + 1 and of different Hurst parameters. We then study its quadratic variations at different scales. This is akin to a wavelet decomposition. We study both the cases where the Hermite processes are dependent and where they are independent. In the dependent case, we show that the quadratic variation, suitably normalized, converges either to a normal or to a Rosenblatt distribution, whatever the order of the original Hermite processes.

Gaussian Fields Satisfying Simultaneous Operator Scaling Relations

In this chapter we define a special class of group of self-similar Gaussian fields. We present a harmonizable representation of m-parameter group self-similar Gaussian fields by utilizing the Haar measure of this group. These fields also have stationary rectangular increments according to special directions linked to coreduction of matrices of the considered m-parameter group.

Convex super-resolution detection of lines in images

In this paper, we present a new convex formulation for the problem of recovering lines in degraded images. Following the recent paradigm of super-resolution, we formulate a dedicated atomic norm penalty and we solve this optimization problem by means of a primal-dual algorithm. This parsimonious model enables the reconstruction of lines from lowpass measurements, even in presence of a large amount of noise or blur. Furthermore, a Prony method performed on rows and columns of the restored image, provides a spectral estimation of the line parameters, with subpixel accuracy.

A variational Shearlet-based model for aortic stent detection

In medical applications, stent segmentation in the abdominal aorta has to be carried out in challenging conditions, since one has to deal with noise, low contrast, objects having similar appearances and missing or blurred edges. Variational segmentation methods eases this task by carrying prior information on the target region or on the regularity of its boundaries. In this paper, we propose a new approach based on the global minimization of the Active Contour model using the L1-norm of the Shearlet Transform instead of Total Variation (TV -norm). One of the distinctive features of such a regularization is that it allows the detection of anisotropic structures in images like stents boundaries. The sparsity imposed by the minimization provides piecewise smooth solutions with C2-singularities. We also use the shearlet coefficients to construct an edge function for more faithful contour detection. Performances of our algorithm are evaluated on a stent segmentation from post-operative CT data. Results show that the proposed method drastically improves the detection of the stent placement compared to the TV based approach.