Béatrice Vedel

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.

Multiscale Anisotropic Texture Analysis and Classification of Photographic Prints: Art scholarship meets image processing algorithms

Texture characterization of photographic prints can provide scholars with valuable information regarding photographers? aesthetic intentions and working practices. Currently, texture assessment is strictly based on the visual acuity of a range of scholars associated with collecting institutions, such as museum curators and conservators. Natural interindividual discrepancies, intraindividual variability, and the large size of collections present a pressing need for computerized and automated solutions for the texture characterization and classification of photographic prints. In the this article, this challenging image processing task is addressed using an anisotropic multiscale representation of texture, the hyperbolic wavelet transform (HWT), from which robust multiscale features are constructed. Cepstral distances aimed at ensuring balanced multiscale contributions are computed between pairs of images. The resulting large-size affinity matrix is then clustered using spectral clustering, followed by a Ward linkage procedure. For proof of concept, these procedures are first applied to a reference data set of historic photographic papers that combine several levels of similarity and second to a large data set of culturally valuable photographic prints held by the Museum of Modern Art in New York. The characterization and clustering results are interpreted in collaboration with art scholars with an aim toward developing new modes of art historical research and humanities-based collaboration.

On the impact of the number of vanishing moments on the dependence structures of compound Poisson motion and fractional Brownian motion in multifractal time

From a theoretical perspective, scale invariance, or simply scaling, can fruitfully be modeled with classes of multifractal stochastic processes, designed from positive multiplicative martingales (or cascades). From a practical perspective, scaling in real-world data is often analyzed by means of multiresolution quantities. The present contribution focuses on three different types of such multiresolution quantities, namely increment, wavelet and Leader coefficients, as well as on a specific multifractal processes, referred to as Infinitely Divisible Motions and fractional Brownian motion in multifractal time. It aims at studying, both analytically and by numerical simulations, the impact of varying the number of vanishing moments of the mother wavelet and the order of the increments on the decay rate of the (higher order) covariance functions of the (q-th power of the absolute values of these) multiresolution coefficients. The key result obtained here consist of the fact that, though it fastens the decay of the covariance functions, as is the case for fractional Brownian motions, increasing the number of vanishing moments of the mother wavelet or the order of the increments does not induce any faster decay for the (higher order) covariance functions

Wavelet decomposition of measures: Application to multifractal analysis of images

We show the relevance of multifractal analysis for some problems in image processing. We relate it to the standard question of the determination of correct function space settings. We show why a scale-invariant analysis, such as the one provided by wavelets, is pertinent for this purpose. Since a good setting for images is provided by spaces of measures, we give some insight into the problem of multifractal analysis of measures using wavelet techniques.

Hyperbolic wavelet transform for historic photographic paper classification challenge

Photographic paper texture characterization constitutes a challenging image processing task and an important stake both for manufacturers and art museums. The present contribution shows how the Hyperbolic Wavelet Transform, thanks to its joint multiscale and anisotropie nature, permits to achieve an accurate photographic paper texture analysis and characterization. A cepstral-type distance, constructed on the coefficients of the Hyperbolic Wavelet Transform, is then used to measure similarity between pairs of paper textures. Spectral clustering followed by Ascendant Hierarchical Clustering applied to the similarity matrix enables an unsupervised classification of photographic paper sheets. This methodology is applied to a test dataset made available in the framework of the Historic Photographic Paper Classification Challenge. The relevance of the proposed texture characterization and classification procedure is assessed by comparisons against the database documentation provided by experts.

Hyperbolic wavelet leaders for anisotropic multifractal texture analysis

Scale invariance has proven a crucial concept in texture modeling and analysis. Isotropic and self-similar fractional Brownian fields (2D-fBf) are often used as the natural reference process to model scale free textures. Its analysis is standardly conducted using the 2D discrete wavelet transform. Generalizations of 2D-fBf were considered independently in two respects: Anisotropy in the texture is allowed while preserving exact self-similarity, analysis then needs to be conducted using the 2D-Hyperbolic wavelet transform; Multifractality enables more versatile scale free models but requires isotropy, analysis is then achieved using wavelet leaders. The present paper proposes a first unifying extension, which is enabled through the following two key contributions: The definition of 2D process that incorporates jointly anisotropy and multi-fractality : The definition of the corresponding analysis tool, the hyperbolic wavelet leaders. Their relevance are studied by numerical simulations using synthetic scale free textures.