The estimation of high resolution spectrometer Instrument Spectral Response Functions (ISRFs) is crucial because an imperfect knowledge of these functions can induce errors in the measurements. The state-of-the-art for this problem currently relies on the use of parametric models, which frequently lack flexibility to accurately model real-world ISRFs. To address this limitation, this paper proposes and investigates the use of sparse representations for modeling and estimating ISRFs, where the ISRFs are decomposed in a fixed dictionary of atoms. To estimate the sparse coefficient vector, a novel sparsity inducing regularization of the problem based on quadratic envelopes is studied and compared to the classical LASSO estimator and to a greedy method based on the Orthogonal Matching Pursuit (OMP) algorithm. Results for simulated ISRFs from the MicroCarb mission indicate that the proposed spectral representations yield excellent ISRF estimates, and that the use of quadratic envelopes can yield significantly better precision than competing methods.

Multifractal analysis (MFA) provides a framework for the global characterization of image textures by describing the spatial fluctuations of their local regularity based on the multifractal spectrum. Several works have shown the interest of using MFA for the description of homogeneous textures in images. Nevertheless, natural images can be composed of several textures and, in turn, multifractal properties associated with those textures. This paper introduces an unsupervised Bayesian multifractal segmentation method to model and segment multifractal textures by jointly estimating the multifractal parameters and labels on images, at the pixel-level. For this, a computationally and statistically efficient multifractal parameter estimation model for wavelet leaders is firstly developed, defining different multifractality parameters for different regions of an image. Then, a multiscale Potts Markov random field is introduced as a prior to model the inherent spatial and scale correlations (referred to as cross-scale correlations) between the labels of the wavelet leaders. A Gibbs sampling methodology is finally used to draw samples from the posterior distribution of the unknown model parameters. Numerical experiments are conducted on synthetic multifractal images to evaluate the performance of the proposed segmentation approach. The proposed method achieves superior performance compared to traditional unsupervised segmentation techniques as well as modern deep learning-based approaches, showing its effectiveness for multifractal image segmentation.

Understanding greenhouse gas fluxes at the Earth’s surface is becoming crucial in the context of climate change. The aim of the CNES/UKSA MicroCarb mission is therefore to map, on a planetary scale, the sources and sinks of carbon, the main greenhouse gas in the atmosphere. To do this, a spectrometer will be sent in space to acquire spectra in 4 narrow bands around wavelengths associated with O2and CO2. However, measurement errors can occur due to the instrument used, and induce errors in the resulting trace gas concentrations. It is therefore crucial to estimate the spectral response of the instrument as accurately as possible. This paper investigates a new estimation method for this spectral response that uses a sparse representation in a dictionary of appropriate basis functions. This sparse representation is performed using the LASSO and Orthogonal Matching Pursuit (OMP) algorithms. Simulations conducted on data mimicking observations resulting from the MicroCarb instrument allow the performance of this method to be appreciated.

This paper derives and studies Bayesian Cramér-Rao lower bounds for the mean squared error of covariance matrices that are structured as weighted sums of symmetric positive definite matrices associated with a circularly-symmetric Gaussian statistical model. This model naturally appears in a number of important applications, including multivariate multifractal analysis and vector-valued additive Gaussian processes. As an intermediary result, we derive a novel expression for the expectation of compositions of Wishart random matrices. We provide extensive numerical simulation results for analyzing the derived bounds and their properties, and illustrate their use for the multifractal analysis of bivariate time series.

Multivariate selfsimilarity has become a classical tool to analyze collections of time series recorded jointly on one same system. Often, it amounts to estimating as many scaling exponents as time series. However, this leaves open the important question how many such scaling exponents are actually different. Elaborating on earlier work aiming to test the hypothesis that all exponents are equal, we intend here to count the number of different scaling exponents from a single finite size multivariate time series. To this end, we devise an original clustering procedure that combines a wavelet domain block multivariate bootstrap scheme with a test strategy for a reduced set of multiple hypotheses on the pairwise equality of scaling exponents that are relevant to clustering. Monte Carlo simulations, making use of synthetic multivariate selfsimilar processes, assess the relevance and performance of the proposed procedure under different scenarios and demonstrate that the proposed method yields practically satisfactory cluster number and size estimations.

Multifractal analysis provides a global description for the spatial fluctuations of the strengths of the pointwise regularity of image amplitudes. A global image characterization leads to robust estimation, but is blind to and corrupted by small regions in the image whose multifractality differs from that of the rest of the image. Prior detection of such zones with anomalous multifractality is thus crucial for relevant analysis, and their delineation of central interest in applications, yet has never been achieved so far. The goal of this work is to devise and study such a multifractal anomaly detection scheme. Our approach combines three original key ingredients: i) a recently proposed generic model for the statistics of the multiresolution coefficients used in multifractal estimation (wavelet leaders), ii) an original surrogate data generation procedure for simulating a hypothesized global multifractality and iii) a combination of multiple hypothesis tests to achieve pixel-wise detection. Numerical simulations using synthetic multifractal images show that our procedure is operational and leads to good multifractal anomaly detection results for a range of target sizes and parameter values of practical relevance.

This paper introduces a novel approach to learn visually grounded meaning representations of words as low-dimensional node embeddings on an underlying graph hierarchy.The lower level of the hierarchy models modality-specific word representations through dedicated but communicating graphs, while the higher level puts these representations together on a single graph to learn a representation jointly from both modalities.The topology of each graph models similarity relations among words, and is estimated jointly with the graph embedding.The assumption underlying this model is that words sharing similar meaning correspond to communities in an underlying similarity graph in a lowdimensional space.We named this model Hierarchical Multi-Modal Similarity Graph Embedding (HM-SGE).Experimental results validate the ability of HM-SGE to simulate human similarity judgements and concept categorization, outperforming the state of the art. 1

Recently, transfer subspace learning based approaches have shown to be a valid alternative to unsupervised subspace clustering and temporal data clustering for human motion segmentation (HMS). These approaches leverage prior knowledge from a source domain to improve clustering performance on a target domain, and currently they represent the state of the art in HMS. Bucking this trend, in this paper, we propose a novel unsupervised model that learns a representation of the data and digs clustering information from the data itself. Our model is reminiscent of temporal subspace clustering, but presents two critical differences. First, we learn an auxiliary data matrix that can deviate from the initial data, hence confers more degrees of freedom to the coding matrix. Second, we introduce a regularization term for this auxiliary data matrix that preserves the local geometrical structure present in the high-dimensional space. The proposed model is efficiently optimized by using an original Alternating Direction Method of Multipliers (ADMM) formulation allowing to learn jointly the auxiliary data representation, a nonnegative dictionary and a coding matrix. Experimental results on four benchmark datasets for HMS demonstrate that our approach achieves significantly better clustering performance then state-of-the-art methods, including both unsupervised and more recent semi-supervised transfer learning approaches1.

Recently, self-supervised learning has proved to be effective to learn representations of events suitable for temporal segmentation in image sequences, where events are understood as sets of temporally adjacent images that are semantically perceived as a whole. However, although this approach does not require expensive manual annotations, it is data hungry and suffers from domain adaptation problems. As an alternative, in this work, we propose a novel approach for learning event representations named Dynamic Graph Embedding (DGE). The assumption underlying our model is that a sequence of images can be represented by a graph that encodes both semantic and temporal similarity. The key novelty of DGE is to learn jointly the graph and its graph embedding. At its core, DGE works by iterating over two steps: 1) updating the graph representing the semantic and temporal similarity of the data based on the current data representation, and 2) updating the data representation to take into account the current data graph structure. The main advantage of DGE over state-of-the-art self-supervised approaches is that it does not require any training set, but instead learns iteratively from the data itself a low-dimensional embedding that reflects their temporal and semantic similarity. Experimental results on two benchmark datasets of real image sequences captured at regular time intervals demonstrate that the proposed DGE leads to event representations effective for temporal segmentation. In particular, it achieves robust temporal segmentation on the EDUBSeg and EDUBSeg-Desc benchmark datasets, outperforming the state of the art. Additional experiments on two Human Motion Segmentation benchmark datasets demonstrate the generalization capabilities of the proposed DGE.

Nonnegative matrix factorization (NMF) is a popular method for audio spectral unmixing. While NMF is traditionally applied to off-the-shelf time-frequency representations based on the short-time Fourier or Cosine transforms, the ability to learn transforms from raw data attracts increasing attention. However, this adds an important computational overhead. When assumed orthogonal (like the Fourier or Cosine transforms), learning the transform yields a non-convex optimization problem on the orthogonal matrix manifold. In this paper, we derive a quasi-Newton method on the manifold using sparse approximations of the Hessian. Experiments on synthetic and real audio data show that the proposed algorithm outperforms state-of-the-art first-order and coordinate-descent methods by orders of magnitude in terms of speed. A Python package for fast TL-NMF is released online at https://github.com/pierreablin/tlnmf.

The self-similarity paradigm enables the analysis of scale-free temporal dynamics and has been widely used in a large set of real-world applications. However, in a multivariate setting, delays amongst components significantly impair the estimation of scale-free parameters. The first framework for the modeling, detection and estimation of delay parameters and for the joint estimation of scale-free parameters is proposed here. It is assumed that a single realization of a multivariate, self-similar time series is available. Use is made of ℂ-valued wavelets and, based on the imaginary part of the wavelet coherence, an original bootstrap-based delay estimation procedure based is constructed. Moreover, a consistent wavelet eigenanalysis-based semiparametric estimation for scale-free parameters that accounts for delay is defined. Monte Carlo experiments conducted over various instances of the model show that the proposed methodology enables the detection of delays with high probability and provides very satisfactory estimates of the delay and scale-free parameters.

Nonnegative matrix factorization (NMF) has become a method of choice for spectrogram decomposition. However, its inability to capture dependencies across columns of the input motivated the introduction of a variant, convolutive NMF. While algorithms for solving the convolutive NMF problem were previously proposed, they rely on the use of a heuristic that does not insure the convergence of the algorithm (in particular in terms of objective function values). The goal of this work is to propose rigorous update rules, based on a majorization-minimization (MM) approach, for convolutive NMF with the β-divergence (a standard family of measures of fit). Specifically, we derive and study two variants of a convolutive NMF algorithm that are guaranteed to decrease the objective function value at each iteration. The complexity of the algorithms is studied, and the performance in terms of execution time and objective function are evaluated and compared in several numerical experiments using real-world audio data. Experiments show that the proposed MM algorithms consistently provide lower values of the objective function than the heuristic, at similar computational cost.

Self-similarity has become a well-established modeling framework in several fields of application and its multivariate formulation is of ever-increasing importance in the Big Data era. Multivariate Hurst exponent estimation has thus received a great deal of attention recently, with wavelet eigenvalue-based regression becoming a focal point. The present work tackles the issue of the presence of significant finite-sample bias in wavelet eigenvalue regression stemming from the eigenvalue repulsion effect, whose origin and impact are analyzed and quantified. Furthermore, an original wavelet domain bias reduction technique is developed assuming a single multivariate time series is available. The protocol consists of a bootstrap resampling scheme that preserves the joint covariance structure of multivariate wavelet coefficients. Extensive numerical simulations show that this proposed method is effective in counteracting the bias at the price of a small increase in variance. This leads to wavelet eigenanalysis-based estimation of multivariate Hurst exponents with significantly improved finite-sample performance than earlier state-of-the-art formulations.

Temporal segmentation of untrimmed videos and photo-streams is currently an active area of research in computer vision and image processing. This paper proposes a new approach to improve the temporal segmentation of photo-streams. The method consists in enhancing image representations by encoding long-range temporal dependencies. Our key contribution is to take advantage of the temporal stationarity assumption of photostreams for modeling each frame by its nonlocal self-similarity function. The proposed approach is put to test on the EDUB-Seg dataset, a standard benchmark for egocentric photostream temporal segmentation. Starting from seven different (CNN based) image features, the method yields consistent improvements in event segmentation quality, leading to an average increase of F-measure of 3.71% with respect to the state of the art.

Traditional NMF-based signal decomposition relies on the factorization of spectral data, which is typically computed by means of short-time frequency transform. In this paper we propose to relax the choice of a pre-fixed transform and learn a short-time orthogonal transform together with the factorization. To this end, we formulate a regularized optimization problem reminiscent of conventional NMF, yet with the transform as additional unknown parameters, and design a novel block-descent algorithm enabling to find stationary points of this objective function. The proposed joint transform learning and factorization approach is tested for two audio signal processing experiments' illustrating its conceptual and practical benefits.

Multifractal analysis, notably with its recent wavelet-leader based formulation, has nowadays become a reference tool to characterize scale-free temporal dynamics in time series. It proved successful in numerous applications very diverse in nature. However, such successes remained restricted to univariate analysis while many recent applications call for the joint analysis of several components. Surprisingly, multivariate multifractal analysis remained mostly overlooked. The present contribution aims at defining a wavelet-leader based framework for multivariate multifractal analysis and at studying its properties and estimation performance. To better understand what properties of multivariate data are actually captured in multivariate multifractal analysis, a multivariate multifractal model is used as representative paradigm and permits to show that multivariate multifractal analysis puts in evidence transient and local dependencies that are not well quantified or even evidenced by the classical Pearson correlation coefficient.

As part of the Historic Photographic Paper Classification Challenge, a multitude of approaches to quantifying paper texture similarity have been developed. These approaches have yielded encouraging results when applied to very controlled datasets containing photomicrographs of familiar specimens. In this paper, we report on the k-nearest neighbors classification performance of two multiscale analysis-based texture similarity approaches when applied to a much larger reference collection of silver gelatin photographic papers. The clusters for this data set were derived from a visual sorting experiment conducted by art conservators and paper experts later extended through crowd-sourcing. The results show that these texture similarity approaches, when combined with a simple k-nearest neighbors classification algorithm, yield workable performances with accuracy of up to 69%. We discuss this outcome in the context of available data and the cross-validation procedure used, then provide suggestions for improvement.

Texture characterization of natural images using the mathematical framework of multifractal analysis (MFA) enables the study of the fluctuations in the regularity of image intensity. Although successfully applied in various contexts, the use of MFA has so far been limited to the independent analysis of a single image, while the data available in applications are increasingly multivariate. This paper addresses this limitation and proposes a joint Bayesian model and associated estimation procedure for multifractal parameters of multivariate images. It builds on a recently introduced generic statistical model that enabled the Bayesian estimation of multifractal parameters for a single image and relies on the following original key contributions: First, we develop a novel Fourier domain statistical model for a single image that permits the use of a likelihood that is separable in the multifractal parameters via data augmentation. Second, a joint Bayesian model for multivariate images is formulated in which prior models based on gamma Markov random fields encode the assumption of the smooth evolution of multifractal parameters between the image components. The design of the likelihood and of conjugate prior models is such that exploitation of the conjugacy between the likelihood and prior models enables an efficient estimation procedure that can handle a large number of data components. Numerical simulations conducted using sequences of multifractal images demonstrate that the proposed procedure significantly outperforms previous univariate benchmark formulations at a competitive computational cost.

Scale-free dynamics commonly appear in individual components of multivariate data. Yet, while the behavior of cross-components is crucial in modeling real-world multivariate data, their examination often suggests departures from exact multivariate self-similarity (also termed fractal connectivity). The present paper introduces a multivariate Gaussian stochastic process with Hadamard (i.e., entry-wise) self-similar scale-free dynamics, controlled by a matrix Hurst parameter H, that allows departures from fractal connectivity. The properties of its wavelet coefficients and wavelet spectrum are studied, enabling the estimation of H and of the fractal connectivity parameter. Furthermore, it permits the computation of closed-form confidence intervals for the estimates based on approximate (wavelet) covariances. Finally, these developments enable us to devise a test for fractal connectivity. Monte Carlo simulations are used to assess the accuracy of the proposed approximate confidence intervals and the performance of the fractal connectivity test.

This contribution focuses, within the ℓ1-Potts model, on the automated estimation of the regularization parameter balancing the ℓ1data fidelity term and the TVℓ0penalization. Variational approaches based on total variation gained considerable interest to solve piecewise constant denoising problems thanks to their deterministic setting and low computational cost. However, the quality of the achieved solution strongly depends on the tuning of the regularization parameter. While recent works have tailored various hierarchical Bayesian procedures to additionally estimate the regularization parameter for Gaussian noise, less attention has been granted to Laplacian noise, of interested in numerous applications. This contribution promotes a fast and parameter-free denoising procedure for piecewise constant signals corrupted by Laplacian noise, that includes automated selection of the regularization parameter. It relies on the minimization of a Bayesian-driven criterion whose similarities with the ℓ1-Potts model permit to derive a computationally efficient algorithm.

Among many research efforts devoted to automated art investigations, the problem of quantification of literary style remains current. Meanwhile, linguists and computer scientists have tried to sort out texts according to their types or authors. We use the recently-introduced p-leader multifractal formalism to analyze a corpus of novels written for adults and young adults, with the goal of assessing if a difference in style can be found. Our results agree with the interpretation that novels written for young adults largely follow conventions of the genre, whereas novels written for adults are less homogeneous.

In the mid 1990s, it was shown that the statistics of aggregated time series from Internet traffic departed from those of traditional short range-dependent models, and were instead characterized by asymptotic self-similarity. Following this seminal contribution, over the years, many studies have investigated the existence and form of scaling in Internet traffic. This contribution first aims at presenting a methodology, combining multiscale analysis (wavelet and wavelet leaders) and random projections (or sketches), permitting a precise, efficient and robust characterization of scaling, which is capable of seeing through non-stationary anomalies. Second, we apply the methodology to a data set spanning an unusually long period: 14 years, from the MAWI traffic archive, thereby allowing an in-depth longitudinal analysis of the form, nature, and evolutions of scaling in Internet traffic, as well as network mechanisms producing them. We also study a separate three-day long trace to obtain complementary insight into intra-day behavior. We find that a biscaling (two ranges of independent scaling phenomena) regime is systematically observed: long-range dependence over the large scales, and multifractallike scaling over the fine scales. We quantify the actual scaling ranges precisely, verify to high accuracy the expected relationship between the long range dependent parameter and the heavy tail parameter of the flow size distribution, and relate fine scale multifractal scaling to typical IP packet inter-arrival and to round-trip time distributions.

Texture analysis is an image processing task that can be conducted using the mathematical framework of multifractal analysis to study the regularity fluctuations of image intensity and the practical tools for their assessment, such as (wavelet) leaders. A recently introduced statistical model for leaders enables the Bayesian estimation of multifractal parameters. It significantly improves performance over standard (linear regression based) estimation. However, the computational cost induced by the associated nonstandard posterior distributions limits its application. The present work proposes an alternative Bayesian model for multifractal analysis that leads to more efficient algorithms. It relies on three original contributions: A novel generative model for the Fourier coefficients of log-leaders; an appropriate reparametrization for handling its inherent constraints; a data-augmented Bayesian model yielding standard conditional posterior distributions that can be sampled exactly. Numerical simulations using synthetic multifractal images demonstrate the excellent performance of the proposed algorithm, both in terms of estimation quality and computational cost.

Texture analysis can be embedded in the mathematical framework of multifractal (MF) analysis, enabling the study of the fluctuations in regularity of image intensity and providing practical tools for their assessment, wavelet leaders. A statistical model for leaders was proposed permitting Bayesian estimation of MF parameters for images yielding improved estimation quality over linear regression based estimation. This present work proposes an extension of this Bayesian model for patch-wise MF analysis of images. Classical MF analysis assumes space homogeneity of the MF properties whereas here we assume MF properties may change between texture elements and we do not know where the changes are located. This paper proposes a joint Bayesian model for patches formulated using spatially smoothing gamma Markov Random Field priors to counterbalance the increased statistical variability of estimates caused by small patch sizes. Numerical simulations based on synthetic multi-fractal images demonstrate that the proposed algorithm outperforms previous formulations and standard estimators.

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.

The SPOT 6-7 satellite ground segment includes a systematic and automatic cloud detection step in order to feed a catalogue with a binary cloud mask and an appropriate confidence measure. In order to significantly improve the SPOT cloud detection and get rid of frequent manual re-labelings, we study a new automatic cloud detection technique that is adapted to large datasets. The proposed method is based on a modified distributed boosting algorithm. Experiments conducted using the framework Apache Spark on a SPOT 6 image database with various landscapes and cloud coverage show promising results.

Recent advances in wearable and/or biomedical sensing technologies have made it possible to record very long-term, continuous biomedical signals, referred to as biomedical intensive longitudinal data (ILD). To link ILD to clinical applications, such as personalized healthcare and disease prevention, the development of robust and reliable data analysis techniques is considered important. In this review, we introduce multiscale analysis methods for and the applications to two types of intensive longitudinal biomedical signals, heart rate variability (HRV) and spontaneous physical activity (SPA) time series. It has been shown that these ILD have robust characteristics unique to various multiscale complex systems, and some parameters characterizing the multiscale complexity are in fact altered in pathological states, showing potential usability as a new type of ambient diagnostic and/or prognostic tools. For example, parameters characterizing increased intermittency of HRV are found to be potentially useful in detecting abnormality in the state of the autonomic nervous system, in particular the sympathetic hyperactivity, and intermittency parameters of SPA might also be useful in evaluating symptoms of psychiatric patients with depressive as well as manic episodes, all in the daily settings. Therefore, multiscale analysis might be a useful tool to extract information on clinical events occurring at multiple time scales during daily life and the underlying physiological control mechanisms from biomedical ILD.

This paper presents a new strategy to correct the Earth data corrupted by spurious samples that are randomly included in the multiplexed data stream provided by the MADRAS instrument. The proposed strategy relies on the construction of a trellis associated with each scan of the multichannel image, modeling the possible occurrences of these erroneous data. A specific weight that promotes the smooth behavior of the signals recorded in each channel is assigned to each transition between trellis states. The joint detection and correction of the erroneous data are conducted using a dynamic programming algorithm for minimizing the overall cost function throughout the trellis. Simulation results obtained on synthetic and real MADRAS data demonstrate the effectiveness of the proposed solution.

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We present a new anomaly detector for data traffic, ‘SMS’, based on combining random projections (sketches) with multiscale analysis, which has low computational complexity. The sketches allow ‘normal’ traffic to be automatically and robustly extracted, and anomalies detected, without the need for training data. The multiscale analysis extracts statistical descriptors, using wavelet leader tools developed recently for multifractal analysis, without any need for timescales to be selected a priori. The proposed detector is illustrated using a large recent dataset of Internet backbone traffic from the MAWI archive, and compared against existing detectors.

This work aims to segment a texture into different regions, each characterized by a priori unknown multifractal properties. The multifractal properties are quantified using the multiscale function C1, jthat quantifies the evolution along analysis scales 2jof the empirical mean of the log of the wavelet leaders. The segmentation procedure is applied to local estimate of C1, j. It involves a multivariate Mumford-Shah relaxation formulated as a convex optimization problem involving a structure tensor penalization and an efficient algorithmic solution based on primal-dual proximal algorithm. The performances are evaluated on synthetic textures.

We develop a representation of reverse-time migration (RTM) in terms of Fourier integral operators, the canonical relations of which are graphs. Through the dyadic parabolic decomposition of phase space, we obtain the solution of the wave equation with a boundary source and homogeneous initial conditions using wave packets. On this basis, we develop a numerical procedure for the reverse-time continuation from the boundary of scattering data and for RTM. The algorithms are derived from those we recently developed for the discrete approximate evaluation of the action of Fourier integral operators and inherit their conceptual and numerical properties.

Texture characterization is a central element in many image processing applications. Multifractal analysis is a useful signal and image processing tool, yet, the accurate estimation of multifractal parameters for image texture remains a challenge. This is due in the main to the fact that current estimation procedures consist of performing linear regressions across frequency scales of the 2D dyadic wavelet transform, for which only a few such scales are computable for images. The strongly non-Gaussian nature of multifractal processes, combined with their complicated dependence structure, makes it difficult to develop suitable models for parameter estimation. Here, we propose a Bayesian procedure that addresses the difficulties in the estimation of the multifractality parameter. The originality of the procedure is threefold. The construction of a generic semiparametric statistical model for the logarithm of wavelet leaders; the formulation of Bayesian estimators that are associated with this model and the set of parameter values admitted by multifractal theory; the exploitation of a suitable Whittle approximation within the Bayesian model which enables the otherwise infeasible evaluation of the posterior distribution associated with the model. Performance is assessed numerically for several 2D multifractal processes, for several image sizes and a large range of process parameters. The procedure yields significant benefits over current benchmark estimators in terms of estimation performance and ability to discriminate between the two most commonly used classes of multifractal process models. The gains in performance are particularly pronounced for small image sizes, notably enabling for the first time the analysis of image patches as small as 64 × 64 pixels.

Scale invariance is a widely used concept to analyze real-world data from many different applications and multifractal analysis has become the standard corresponding signal processing tool. It characterizes data by describing globally and geometrically the fluctuations of local regularity, usually measured by means of the Ho¨lder exponent. A major limitation of the current procedure is that it applies only to locally bounded functions or signals, i.e., to signals with positive regularity. The present contribution proposes to characterize local regularity with a new quantity, the p-exponent, that permits negative regularity in data, a widely observed property in real-world data. Relations to Ho¨lder exponents are detailed and a corresponding p-leader multifractal formalism is devised and shown at work on synthetic multifractal processes, representative of a class of models often used in applications. We formulate a conjecture regarding the equivalence between Ho¨lder and p-exponents for a subclass of processes. Even when Ho¨lder and p-exponents coincide, the p-leader formalism is shown to achieve better estimation performance.

The identification of texture changes is a challenging problem that can be addressed by considering local regularity fluctuations in an image. This work develops a procedure for local regularity estimation that combines a convex optimization strategy with wavelet leaders, specific wavelet coefficients recently introduced in the context of multifractal analysis. The proposed procedure is formulated as an inverse problem that combines the joint estimation of both local regularity exponent and of the optimal weights underlying regularity measurement. Numerical experiments using synthetic texture indicate that the performance of the proposed approach compares favorably against other wavelet based local regularity estimation formulations. The method is also illustrated with an example involving real-world texture.

Texture segmentation constitutes a classical yet crucial task in image processing. In many applications of very different natures (biomedical, geophysics,...) textures are naturally defined in terms of their local regularity fluctuations, which can be quantified as the variations of local Hölder exponents. Furthermore, such images are often naturally embedded in the class of piece-wise constant local regularity functions. The present contribution aims at proposing and assessing a segmentation procedure for this class of images. Its originality is twofold: First, local regularity is estimated using wavelet leaders, a novel multiresolution quantity recently introduced for multifractal analysis but barely used in local regularity measurement, comparisons against wavelet coefficient based estimation are conducted; Second, the challenging minimal partition problem underlying segmentation is convexified and conducted within a customized proximal framework. The estimation of the number of regions and their target regularity is obtained from a total-variation estimate that enables the actual use of proximal minimization for texture segmentation. Performance is assessed and illustrated on synthetic textures.

Multifractal analysis has matured into a widely used signal and image processing tool. Due to the statistical nature of multifractal processes (strongly non-Gaussian and intricate dependence) the accurate estimation of multifractal parameters is very challenging in situations where the sample size is small (notably including a range of biomedical applications) and currently available estimators need to be improved. To overcome such limitations, the present contribution proposes a Bayesian estimation procedure for the multifractality (or intermittence) parameter. Its originality is threefold: First, the use of wavelet leaders, a recently introduced multiresolution quantity that has been shown to yield significant benefits for multifractal analysis; Second, the construction of a simple yet generic semi-parametric model for the marginals and covariance structure of wavelet leaders for the large class of multiplicative cascade based multifractal processes; Third, the construction of original Bayesian estimators associated with the model and the constraints imposed by multifractal theory. Performance are numerically assessed and illustrated for synthetic multifractal processes for a range of multifractal parameter values. The proposed procedure yields significantly improved estimation performance for small sample sizes.

Recently, a growing interest in the exploration of the potential of signal or image processing tools for the purposes of art analysis has emerged. The wavelet leader based multifractal analysis consists of a mathematical tool recently introduced in image processing for the characterization of homogeneous textures based on their regularity properties. Here, this novel tool is applied to a set of digitized versions of drawings, made available by the NY Metropolitan Museum of Art, consisting of authentic Bruegel drawings and several imitations. Multifractal attributes are estimated from several patches of each of these drawings, and their ability to discriminate authentic drawings from impostors is investigated by means of subspace projections and quadratic discriminant analysis. Besides showing very satisfactory performance, the achieved discrimination provides interesting insights into the differences between the regularity of the textures of authentic Bruegel drawings versus imitations, potentially relating the fractal properties of the drawings to the artist's drawing style.

Within the framework of long memory multivariate processes, fractal connectivity is a particular model, in which the low frequencies (coarse scales) of the interspectrum of each pair of process components are determined by the autospectra of the components. The underlying intuition is that long memories in each components are likely to arise from a same and single mechanism. The present contribution aims at defining and characterizing a statistical procedure for testing actual fractal connectivity amongst data. The test is based on Fisher's Z transform and Pearson correlation coefficient, and anchored in a wavelet framework. Its performance are analyzed theoretically and validated on synthetic data. Its usefulness is illustrated on the analysis of Internet traffic Packet and Byte count time series.

Image classification often relies on texture characterization. Yet texture characterization has so far rarely been based on a true 2D multifractal analysis. Recently, a 2D wavelet Leader based multifractal formalism has been proposed. It allows to perform an accurate, complete and low computational and memory costs multifractal characterization of textures in images. This contribution describes the first application of such a formalism to a real large size (publicly available) image database, consisting of 25 classes of non traditional textures, with 40 high resolution images in each class. Multifractal attributes are estimated from each image and used as classification features within a standard k nearest neighbor classification procedure. The results reported here show that this Leader based multifractal analysis enables the effective discrimination of different textures, as performances in both classification scores and computational costs compare favorably against those of procedures previously proposed in the literature on the same database.

On open and controversial issue in empirical data analysis is to decide whether scaling and multifractal properties observed in empirical data actually exist, or whether they are induced by intricate non stationarities. To contribute to answering this question, we propose a procedure aiming at testing the constancy along time of multifractal attributes estimated over adjacent non overlapping time windows. The procedure is based on non parametric bootstrap resampling and on wavelet Leader estimations for the multifractal parameters.lt is shown, by means of numerical simulations on synthetic multifractal processes, that the proposed procedure is reliable and powerful for discriminating true scaling behavior against non stationarities. We end up with a practical procedure that can be applied to a single finite length observation of data with unknown statistical properties.

Multifractal analysis is nowadays commonly used in real-life data analyses and involved in standard signal processing tasks such as detection, identification or classification. In a number of situations, mostly in image processing, the data are available for the analyses only in (possibly severely) quantized versions. The present contribution aims at analyzing the robustness of standard multifractal estimation procedures against quantization. To this end, we analyze the behaviors and statistical performance of these procedures when applied to a large number of realizations of known synthetic multifractal processes subject to various quantization levels. Our study shows that immunity against quantization can be obtained by restricting the range of scales involved in multifractal parameter estimation to the largest ones. Comparing multifractal analyses based on different multiresolution quantities, increments, wavelet coefficients and leaders, we show that wavelets, thanks to their good frequency localization, bring robustness against quantization when increments do not. This study provides the practitioner with a clear guide line to perform multifractal analysis over quantized data.

Multifractal analysis, which mainly consists in estimating scaling exponents, has become a popular tool for empirical data analysis. Although widely used in different applications, the statistical performance and the reliability of the estimation procedures are still poorly known. Notably, little is known about confidence intervals, though they are of first importance in applications. The present work investigates the potential uses of bootstrap for multifractal estimation. Can bootstrap improve current estimation procedures or be used to obtain reliable confidence intervals? Comparing the statistical performance of different estimators, our major result is to show that bootstrap based procedures provide us both with accurate estimates and reliable confidence intervals. We believe that this brings substantial improvements to practical empirical multifractal analyses