Keywords
Computer Science and Digital Science
 A3.4.2. Unsupervised learning
 A3.4.7. Kernel methods
 A3.4.8. Deep learning
 A5.3. Image processing and analysis
 A5.3.2. Sparse modeling and image representation
 A5.3.3. Pattern recognition
 A5.3.5. Computational photography
 A5.7. Audio modeling and processing
 A5.7.3. Speech
 A5.7.4. Analysis
 A5.9. Signal processing
 A5.9.2. Estimation, modeling
 A5.9.3. Reconstruction, enhancement
 A5.9.5. Sparsityaware processing
Other Research Topics and Application Domains
 B2. Health
 B2.2. Physiology and diseases
 B2.2.1. Cardiovascular and respiratory diseases
 B2.2.6. Neurodegenerative diseases
 B3. Environment and planet
 B3.3. Geosciences
 B3.3.2. Water: sea & ocean, lake & river
 B3.3.4. Atmosphere
1 Team members, visitors, external collaborators
Research Scientists
 Hussein Yahia [Team leader, Inria, Researcher, HDR]
 Nicolas Brodu [Inria, Researcher, HDR]
 Khalid Daoudi [Inria, Researcher, HDR]
PostDoctoral Fellows
 Anis Fradi [Inria, from Nov 2021]
PhD Students
 Biswajit Das [Inria, until Sep 2021]
 Arash Rashidi [Groupe I2S]
Technical Staff
 Zhe Li [Inria, Engineer, from Oct 2021]
 Gabriel Augusto Zebadua Garcia [Inria, Engineer]
Administrative Assistant
 Sabrina Duthil [Inria]
2 Overall objectives
GEOSTAT is a research project which investigates the analysis of some classes of natural complex signals (physiological time series, turbulent universe and earth observation data sets) by determining, in acquired signals, the properties that are predicted by commonly admitted or new physical models best fitting the phenomenon. Consequently, when statistical properties discovered in the signals do not match closely enough those predicted by accepted physical models, we question the validity of existing models or propose, whenever possible, modifications or extensions of existing models. A new direction of research, based on the CONCAUST exploratory action and the newly accepted (in February 2021) associated team COMCAUSA proposed by N. Brodu with USA / UC Davis, Complexity Sciences Center, Physics Department is developped in the team.
An important aspect of the methodological approach is that we don't rely on a predetermined "universal" signal processing model to analyze natural complex signals. Instead, we take into consideration existing approaches in nonlinear signal processing (wavelets, multifractal analysis tools such as logcumulants or microcanonical multifractal formalism, time frequency analysis etc.) which are used to determine the micro structures or other micro features inside the acquired signals. Then, statistical analysis of these micro data are determined and compared to expected behaviour from theoretical physical models used to describe the phenomenon from which the data is acquired. From there different possibilities can be contemplated:
 The statistics match behaviour predicted by the model: complexity parameters predicted by the model are extracted from signals to analyze the dynamics of underlying phenomena. Examples: analysis of turbulent data sets in Oceanography and Astronomy.
 The signals displays statistics that cannot be attainable by the common lore of accepted models: how to extend or modify the models according to the behaviour of observed signals ? Example: audio speech signals.
GEOSTAT is a research project in nonlinear signal processing which develops on these considerations: it considers the signals as the realizations of complex extended dynamical systems. The driving approach is to describe the relations between complexity (or information content) and the geometric organization of information in a signal. For instance, for signals which are acquisitions of turbulent fluids, the organization of information may be related to the effective presence of a multiscale hierarchy of coherent structures, of multifractal nature, which is strongly related to intermittency and multiplicative cascade phenomena ; the determination of this geometric organization unlocks key nonlinear parameters and features associated to these signals; it helps understand their dynamical properties and their analysis. We use this approach to derive novel solution methods for superresolution and data fusion in Universe Sciences acquisitions 12. Specific advances are obtained in GEOSTAT in using this type of statistical/geometric approach to get validated dynamical information of signals acquired in Universe Sciences, e.g. Oceanography or Astronomy. The research in GEOSTAT encompasses nonlinear signal processing and the study of emergence in complex systems, with a strong emphasis on geometric approaches to complexity. Consequently, research in GEOSTAT is oriented towards the determination, in real signals, of quantities or phenomena, usually unattainable through linear methods, that are known to play an important role both in the evolution of dynamical systems whose acquisitions are the signals under study, and in the compact representations of the signals themselves.
Signals studied in GEOSTAT belong to two broad classes:
 Acquisitions in Astronomy and Earth Observation.
 Physiological time series.
3 Research program
3.1 General methodology
 Fully Developed Turbulence (FDT) Turbulence at very high Reynolds numbers; systems in FDT are beyond deterministic chaos, and symmetries are restored in a statistical sense only, and multiscale correlated structures are landmarks. Generalizing to more random uncorrelated multiscale structured turbulent fields.
 Compact Representation Reduced representation of a complex signal (dimensionality reduction) from which the whole signal can be reconstructed. The reduced representation can correspond to points randomly chosen, such as in Compressive Sensing, or to geometric localization related to statistical information content (framework of reconstructible systems).
 Sparse representation The representation of a signal as a linear combination of elements taken in a dictionary (frame or Hilbertian basis), with the aim of finding as less as possible nonzero coefficients for a large class of signals.
 Universality class In theoretical physics, the observation of the coincidence of the critical exponents (behaviour near a second order phase transition) in different phenomena and systems is called universality. Universality is explained by the theory of the renormalization group, allowing for the determination of the changes followed by structured fluctuations under rescaling, a physical system is the stage of. The notion is applicable with caution and some differences to generalized outofequilibrium or disordered systems. Nonuniversal exponents (without definite classes) exist in some universal slowing dynamical phenomena like the glass transition and kindred. As a consequence, different macroscopic phenomena displaying multiscale structures (and their acquisition in the form of complex signals) may be grouped into different sets of generalized classes.
Every signal conveys, as a measure experiment, information on the physical system whose signal is an acquisition of. As a consequence, it seems natural that signal analysis or compression should make use of physical modelling of phenomena: the goal is to find new methodologies in signal processing that goes beyond the simple problem of interpretation. Physics of disordered systems, and specifically physics of (spin) glasses is putting forward new algorithmic resolution methods in various domains such as optimization, compressive sensing etc. with significant success notably for NP hard problem heuristics. Similarly, physics of turbulence introduces phenomenological approaches involving multifractality. Energy cascades are indeed closely related to geometrical manifolds defined through random processes. At these structures’ scales, information in the process is lost by dissipation (close to the lower bound of inertial range). However, all the cascade is encoded in the geometric manifolds, through long or short distance correlations depending on cases. How do these geometrical manifold structures organize in space and time, in other words, how does the scale entropy cascades itself ? To unify these two notions, a description in term of free energy of a generic physical model is sometimes possible, such as an elastic interface model in a random nonlinear energy landscape : This is for instance the correspondence between compressible stochastic Burgers equation and directed polymers in a disordered medium. Thus, trying to unlock the fingerprints of cascadelike structures in acquired natural signals becomes a fundamental problem, from both theoretical and applicative viewpoints.
3.2 Turbulence in insterstellar clouds and Earth observation data
The research described in this section is a collaboration effort of GEOSTAT, CNRS LEGOS (Toulouse), CNRS LAM (Marseille Laboratory for Astrophysics), MERCATOR (Toulouse), IIT Roorkee, Moroccan Royal Center for Teledetection (CRST), Moroccan Center for Science CNRST, Rabat University, University of Heidelberg. Researchers involved:
 GEOSTAT: H. Yahia, N. Brodu, K. Daoudi, A. El Aouni, A. Tamim
 CNRS LAB: S. Bontemps, N. Schneider
 CNRS LEGOS: V. Garçon, I. HernandezCarrasco, J. Sudre, B. Dewitte
 CNRST, CRTS, Rabat University: D. Aboutajdine, A. Atillah, K. Minaoui
The analysis and modeling of natural phenomena, specially those observed in geophysical sciences and in astronomy, are influenced by statistical and multiscale phenomenological descriptions of turbulence; indeed these descriptions are able to explain the partition of energy within a certain range of scales. A particularly important aspect of the statistical theory of turbulence lies in the discovery that the support of the energy transfer is spatially highly non uniform, in other terms it is intermittent42. Because of the absence of localization of the Fourier transform, linear methods are not successful to unlock the multiscale structures and cascading properties of variables which are of primary importance as stated by the physics of the phenomena. This is the reason why new approaches, such as DFA (Detrented Fluctuation Analysis), Timefrequency analysis, variations on curvelets 40 etc. have appeared during the last decades. Recent advances in dimensionality reduction, and notably in Compressive Sensing, go beyond the Nyquist rate in sampling theory using nonlinear reconstruction, but data reduction occur at random places, independently of geometric localization of information content, which can be very useful for acquisition purposes, but of lower impact in signal analysis. We are successfully making use of a microcanonical formulation of the multifractal theory, based on predictability and reconstruction, to study the turbulent nature of interstellar molecular or atomic clouds. Another important result obtained in GEOSTAT is the effective use of multiresolution analysis associated to optimal inference along the scales of a complex system. The multiresolution analysis is performed on dimensionless quantities given by the singularity exponents which encode properly the geometrical structures associated to multiscale organization. This is applied successfully in the derivation of high resolution ocean dynamics, or the high resolution mapping of gaseous exchanges between the ocean and the atmosphere; the latter is of primary importance for a quantitative evaluation of global warming. Understanding the dynamics of complex systems is recognized as a new discipline, which makes use of theoretical and methodological foundations coming from nonlinear physics, the study of dynamical systems and many aspects of computer science. One of the challenges is related to the question of emergence in complex systems: largescale effects measurable macroscopically from a system made of huge numbers of interactive agents 21, 37. Some quantities related to nonlinearity, such as Lyapunov exponents, KolmogorovSinai entropy etc. can be computed at least in the phase space 22. Consequently, knowledge from acquisitions of complex systems (which include complex signals) could be obtained from information about the phase space. A result from F. Takens 41 about strange attractors in turbulence has motivated the theoretical determination of nonlinear characteristics associated to complex acquisitions. Emergence phenomena can also be traced inside complex signals themselves, by trying to localize information content geometrically. Fundamentally, in the nonlinear analysis of complex signals there are broadly two approaches: characterization by attractors (embedding and bifurcation) and timefrequency, multiscale/multiresolution approaches. In real situations, the phase space associated to the acquisition of a complex phenomenon is unknown. It is however possible to relate, inside the signal's domain, local predictability to local reconstruction 13 and to deduce relevant information associated to multiscale geophysical signals 14. A multiscale organization is a fundamental feature of a complex system, it can be for example related to the cascading properties in turbulent systems. We make use of this kind of description when analyzing turbulent signals: intermittency is observed within the inertial range and is related to the fact that, in the case of FDT (fully developed turbulence), symmetry is restored only in a statistical sense, a fact that has consequences on the quality of any nonlinear signal representation by frames or dictionaries.
The example of FDT as a standard "template" for developing general methods that apply to a vast class of complex systems and signals is of fundamental interest because, in FDT, the existence of a multiscale hierarchy ${\mathcal{F}}_{h}$ which is of multifractal nature and geometrically localized can be derived from physical considerations. This geometric hierarchy of sets is responsible for the shape of the computed singularity spectra, which in turn is related to the statistical organization of information content in a signal. It explains scale invariance, a characteristic feature of complex signals. The analogy from statistical physics comes from the fact that singularity exponents are direct generalizations of critical exponents which explain the macroscopic properties of a system around critical points, and the quantitative characterization of universality classes, which allow the definition of methods and algorithms that apply to general complex signals and systems, and not only turbulent signals: signals which belong to a same universality class share common statistical organization. During the past decades, canonical approaches permitted the development of a wellestablished analogy taken from thermodynamics in the analysis of complex signals: if $\mathcal{F}$ is the free energy, $\mathcal{T}$ the temperature measured in energy units, $\mathcal{U}$ the internal energy per volume unit $\mathcal{S}$ the entropy and $\widehat{\beta}=1/\mathcal{T}$, then the scaling exponents associated to moments of intensive variables $p\to {\tau}_{p}$ corresponds to $\widehat{\beta}\mathcal{F}$, $\mathcal{U}\left(\widehat{\beta}\right)$ corresponds to the singularity exponents values, and $\mathcal{S}\left(\mathcal{U}\right)$ to the singularity spectrum 20. The research goal is to be able to determine universality classes associated to acquired signals, independently of microscopic properties in the phase space of various complex systems, and beyond the particular case of turbulent data 35.
We show in figure 1 the result of the computation of singularity exponents on an Herschel astronomical observation map (the Musca galactic cloud) which has been edgeaware filtered using sparse ${L}^{1}$ filtering to eliminate the cosmic infrared background (or CIB), a type of noise that can modify the singularity spectrum of a signal.
3.3 Causal modeling
The team is working on a new class of models for modeling physical systems, starting from measured data and accounting for their dynamics 27. The idea is to statistically describe the evolution of a system in terms of causallyequivalent states; states that lead to the same predictions 23. Transitions between these states can be reconstructed from data, leading to a theoreticallyoptimal predictive model 39. In practice, however, no algorithm is currently able to reconstruct these models from data in a reasonable time and without substantial discrete approximations. Recent progress now allows a continuous formulation of predictive causal models. Within this framework, more efficient algorithms may be found. The broadened class of predictive models promises a new perspective on structural complexity in many applications.
3.4 Speech analysis
Phonetic and subphonetic analysis: We developed a novel algorithm for automatic detection of Glottal Closure Instants (GCI) from speech signals using the Microcanonical Multiscale Formalism (MMF). This state of the art algorithm is considered as a reference in this field. We made a Matlab code implementing it available to the community (link). Our approach is based on the Microcanonical Multiscale Formalism. We showed that in the case of clean speech, our algorithm performs almost as well as a recent stateoftheart method. In presence of different types of noises, we showed that our method is considerably more accurate (particularly for very low SNRs). Moreover, our method has lower computational times does not rely on an estimate of pitch period nor any critical choice of parameters. Using the same MMF, we also developed a method for phonetic segmentation of speech signal. We showed that this method outperforms state of the art ones in term of accuracy and efficiency.
Pathological speech analysis and classification: we made a critical analysis of some widely used methodologies in pathological speech classification. We then introduced some novel methods for extracting some common features used in pathological speech analysis and proposed more robust techniques for classification.
Speech analysis of patients with Parkinsonism: with our collaborators from the Czech Republic, we started preliminary studies of some machine learning issues in the field essentially due the small amount of training data.
3.5 Databased identification of characteristic scales and automated modeling
Data are often acquired at the highest possible resolution, but that scale is not necessarily the best for modeling and understanding the system from which data was measured. The intrinsic properties of natural processes do not depend on the arbitrary scale at which data is acquired; yet, usual analysis techniques operate at the acquisition resolution. When several processes interact at different scales, the identification of their characteristic scales from empirical data becomes a necessary condition for properly modeling the system. A classical method for identifying characteristic scales is to look at the work done by the physical processes, the energy they dissipate over time. The assumption is that this work matches the most important action of each process on the studied natural system, which is usually a reasonable assumption. In the framework of timefrequency analysis 31, the power of the signal can be easily computed in each frequency band, itself matching a temporal scale.
However, in open and dissipating systems, energy dissipation is a prerequisite and thus not necessarily the most useful metric to investigate. In fact, most natural, physical and industrial systems we deal with fall in this category, while balanced quasistatic assumptions are practical approximation only for scales well below the characteristic scale of the involved processes. Open and dissipative systems are not locally constrained by the inevitable rise in entropy, thus allowing the maintaining through time of mesoscopic ordered structures. And, according to information theory 33, more order and less entropy means that these structures have a higher information content than the rest of the system, which usually gives them a high functional role.
We propose to identify characteristic scales not only with energy dissipation, as usual in signal processing analysis, but most importantly with information content. Information theory can be extended to look at which scales are most informative (e.g. multiscale entropy 26, $\epsilon $entropy 25). Complexity measures quantify the presence of structures in the signal (e.g. statistical complexity 28, MPR 36 and others 30). With these notions, it is already possible to discriminate between random fluctuations and hidden order, such as in chaotic systems 27, 36. The theory of how information and structures can be defined through scales is not complete yet, but the state of art is promising 29. Current research in the team focuses on how informative scales can be found using collections of random paths, assumed to capture local structures as they reach out 24.
Building on these notions, it should also possible to fully automate the modeling of a natural system. Once characteristic scales are found, causal relationships can be established empirically. They are then clustered together in internal states of a special kind of Markov models called $\u03f5$machines 28. These are known to be the optimal predictors of a system, with the drawback that it is currently quite complicated to build them properly, except for small system 38. Recent extensions with advanced clustering techniques 23, 32, coupled with the physics of the studied system (e.g. fluid dynamics), have proved that $\u03f5$machines are applicable to large systems, such as global wind patterns in the atmosphere 34. Current research in the team focuses on the use of reproducing kernels, coupled possibly with sparse operators, in order to design better algorithms for $\u03f5$machines reconstruction. In order to help with this longterm project, a collaboration is ongoing with J. Crutchfield lab at UC Davis.
4 Application domains
4.1 Sparse signals & optimization
This research topic involves Geostat team and is used to set up an InnovationLab with I2S company
Sparsity can be used in many ways and there exist various sparse models in the literature; for instance minimizing the ${l}_{0}$ quasinorm is known to be an NPhard problem as one needs to try all the possible combinations of the signal's elements. The ${l}_{1}$ norm, which is the convex relation of the ${l}_{0}$ quasinorm results in a tractable optimization problem. The ${l}_{p}$ pseudonorms with $0<p<1$ are particularly interesting as they give closer approximation of ${l}_{0}$ but result in a nonconvex minimization problem. Thus, finding a global minimum for this kind of problem is not guaranteed. However, using a nonconvex penalty instead of the ${l}_{1}$ norm has been shown to improve significantly various sparsitybased applications. Nonconvexity has a lot of statistical implications in signal and image processing. Indeed, natural images tend to have a heavytailed (kurtotic) distribution in certain domains such as wavelets and gradients. Using the ${l}_{1}$ norm comes to consider a Laplacian distribution. More generally, the hyperLaplacian distribution is related to the ${l}_{p}$ pseudonorm ($0<p<1$) where the value of $p$ controls how the distribution is heavytailed. As the hyperLaplacian distribution for $0<p<1$ represents better the empirical distribution of the transformed images, it makes sense to use the ${l}_{p}$ pseudonorms instead of ${l}_{1}$. Other functions that better reflect heavytailed distributions of images have been used as well such as Studentt or Gaussian Scale Mixtures. The internal properties of natural images have helped researchers to push the sparsity principle further and develop highly efficient algorithms for restoration, representation and coding. Group sparsity is an extension of the sparsity principle where data is clustered into groups and each group is sparsified differently. More specifically, in many cases, it makes sense to follow a certain structure when sparsifying by forcing similar sets of points to be zeros or nonzeros simultaneously. This is typically true for natural images that represent coherent structures. The concept of group sparsity has been first used for simultaneously shrinking groups of wavelet coefficients because of the relations between wavelet basis elements. Lastly, there is a strong relationship between sparsity, nonpredictability and scale invariance.
We have shown that the two powerful concepts of sparsity and scale invariance can be exploited to design fast and efficient imaging algorithms. A general framework has been set up for using nonconvex sparsity by applying a firstorder approximation. When using a proximal solver to estimate a solution of a sparsitybased optimization problem, sparse terms are always separated in subproblems that take the form of a proximal operator. Estimating the proximal operator associated to a nonconvex term is thus the key component to use efficient solvers for nonconvex sparse optimization. Using this strategy, only the shrinkage operator changes and thus the solver has the same complexity for both the convex and nonconvex cases. While few previous works have also proposed to use nonconvex sparsity, their choice of the sparse penalty is rather limited to functions like the ${l}_{p}$ pseudonorm for certain values of $p\ge 0.5$ or the Minimax Concave (MC) penalty because they admit an analytical solution. Using a firstorder approximation only requires calculating the (super)gradient of the function, which makes it possible to use a wide range of penalties for sparse regularization. This is important in various applications where we need a flexible shrinkage function such as in edgeaware processing. Apart from nonconvexity, using a firstorder approximation makes it easier to verify the optimality condition of proximal operatorbased solvers via fixedpoint interpretation. Another problem that arises in various imaging applications but has attracted less works is the problem of multisparsity, when the minimization problem includes various sparse terms that can be nonconvex. This is typically the case when looking for a sparse solution in a certain domain while rejecting outliers in the datafitting term. By using one intermediate variable per sparse term, we show that proximalbased solvers can be efficient. We give a detailed study of the Alternating Direction Method of Multipliers (ADMM) solver for multisparsity and study its properties. The following subjects are addressed and receive new solutions:

Edge aware smoothing: given an input image $g$, one seeks a smooth image $u$ "close" to $g$ by minimizing:
$$\underset{u}{\text{argmin}}\phantom{\rule{3.33333pt}{0ex}}{\displaystyle \frac{\lambda}{2}{\parallel ug\parallel}_{2}^{2}+\psi \left(\nabla u\right)}$$where $\psi $ is a sparcityinducing nonconvex function and $\lambda $ a positive parameter. Splitting and alternate minimization lead to the subproblems:
$$\begin{array}{ccc}\text{(sp1)}\hfill & :& {\displaystyle {v}^{(k+1)}\leftarrow \underset{v}{\text{argmin}}\phantom{\rule{3.33333pt}{0ex}}\psi \left(v\right)+\frac{\beta}{2}{\parallel \nabla {u}^{\left(k\right)}v\parallel}_{2}^{2}}\hfill \\ \text{(sp2)}\hfill & :& {\displaystyle {u}^{(k+1)}\leftarrow \underset{u}{\text{argmin}}{\phantom{\rule{3.33333pt}{0ex}}\lambda \parallel ug\parallel}_{2}^{2}+\beta {\parallel \nabla u{v}^{(k+1)}\parallel}_{2}^{2}.}\hfill \end{array}$$We solve subproblem $\text{(sp2)}$ through deconvolution and efficient estimation via separable filters and warmstart initialization for fast GPU implementation, and subproblem $\text{(sp1)}$ through nonconvex proximal form.
 Structuretexture separation: design of an efficient algorithm using nonconvex terms on both the datafitting and the prior. The resulting problem is solved via a combination of HalfQuadratic (HQ) and MaximizationMinimization (MM) methods. We extract challenging texture layers outperforming existing techniques while maintaining a low computational cost. Using spectral sparsity in the framework of lowrank estimation, we propose to use robust Principal Component Analysis (RPCA) to perform robust separation on multichannel images such as glare and artifacts removal of flash/noflash photographs. As in this case, the matrix to decompose has much less columns than lines, we propose to use a QR decomposition trick instead of a direct singular value decomposition (SVD) which makes the decomposition faster.
 Robust integration: in many applications, we need to reconstruct an image from corrupted gradient fields. The corruption can take the form of outliers only when the vector field is the result of transformed gradient fields (lowlevel vision), or mixed outliers and noise when the field is estimated from corrupted measurements (surface reconstruction, gradient camera, Magnetic Resonance Imaging (MRI) compressed sensing, etc.). We use nonconvexity and multisparsity to build efficient integrability enforcement algorithms. We present two algorithms : 1) a local algorithm that uses sparsity in the gradient field as a prior together with a sparse datafitting term, 2) a nonlocal algorithm that uses sparsity in the spectral domain of nonlocal patches as a prior together with a sparse datafitting term. Both methods make use of a multisparse version of the HalfQuadratic solver. The proposed methods were the first in the literature to propose a sparse regularization to improve integration. Results produced with these methods significantly outperform previous works that use no regularization or simple ${l}_{1}$ minimization. Exact or nearexact recovery of surfaces is possible with the proposed methods from highly corrupted gradient fields with outliers.
 Learning image denoising: deep convolutional networks that consist in extracting features by repeated convolutions with highpass filters and pooling/downsampling operators have shown to give nearhuman recognition rates. Training the filters of a multilayer network is costly and requires powerful machines. However, visualizing the first layers of the filters shows that they resemble wavelet filters, leading to sparse representations in each layer. We propose to use the concept of scale invariance of multifractals to extract invariant features on each sparse representation. We build a biLipschitz invariant descriptor based on the distribution of the singularities of the sparsified images in each layer. Combining the descriptors of each layer in one feature vector leads to a compact representation of a texture image that is invariant to various transformations. Using this descriptor that is efficient to calculate with learning techniques such as classifiers combination and artificially adding training data, we build a powerful texture recognition system that outperforms previous works on 3 challenging datasets. In fact, this system leads to quite close recognition rates compared to latest advanced deep nets while not requiring any filters training.
5 Social and environmental responsibility
5.1 Participation in the Covid19 Inria mission
GeoStat is participating in the Covid19 Inria mission: : Vocal biomarkers of respiratory diseases.
6 Highlights of the year
6.1 Awards
 The article Description of turbulent dynamics in the interstellar medium: multifractal/microcanonical analysis I. Application to Herschel observations of the Musca filament has been selected as Highlights by Astronomy and Astrophysics: link to the Highlights section of Astronomy and Astrophysics. The corresponding research topic has also been the subject of an INRIA article on inria.fr: link to to the INRIA article.
 Researchbusiness partnership in image processing: report between I2S and Geostat. Article published on inria.fr: link to to the INRIA article.
6.2 HDRs
 Nicolas Brodu defended his HDR entitled Systèmes complexes : inférence, dynamique et applications on June 9, 2021.
 K. Daoudi defended his HDR entitled Nouveaux paradigmes en traitement de la parole et de ses troubles on March 17, 2021.
7 New software and platforms
7.1 New software
7.1.1 Fluex

Keywords:
Signal, Signal processing

Scientific Description:
Fluex is a package consisting of the Microcanonical Multiscale Formalism for 1D, 2D 3D and 3D+t general signals.

Functional Description:
Fluex is a C++ library developed under Gforge. Fluex is a library in nonlinear signal processing. Fluex is able to analyze turbulent and natural complex signals, Fluex is able to determine low level features in these signals that cannot be determined using standard linear techniques.

Contact:
Hussein Yahia

Participants:
Hussein Yahia, Rémi Paties
7.1.2 FluidExponents

Keywords:
Signal processing, Wavelets, Fractal, Spectral method, Complexity

Functional Description:
FluidExponents is a signal processing software dedicated to the analysis of complex signals displaying multiscale properties. It analyzes complex natural signals by use of nonlinear methods. It implements the multifractal formalism and allows various kinds of signal decomposition and reconstruction. One key aspect of the software lies in its ability to evaluate key concepts such as the degree of impredictability around a point in a signal, and provides different kinds of applications. The software can be used for times series or multidimensional signals.

Contact:
Hussein Yahia

Participants:
Antonio Turiel, Hussein Yahia
7.1.3 ProximalDenoising

Name:
ProximalDenoising

Keywords:
2D, Image filter, Filtering, Minimizing overall energy, Noise, Signal processing, Image reconstruction, Image processing

Scientific Description:
Image filtering is contemplated in the form of a sparse minimization problem in a nonconvex setting. Given an input image I, one seeks to compute a denoised output image u such that u is close to I in the L2 norm. To do so, a minimization term is added which favors sparse gradients for output image u. Imposing sparse gradients lead to a nonconvex minimization term: for instance a pseudonorm Lp with 0 < p < 1 or a Cauchy or Welsh function. Halfquadratic algorithm is used by adding a new variable in the minimization functionnal which leads to two subproblems, the first subproblem is nonconvex and solved by use of proximal operators. The second subproblem can be written in variational form, and is best solved in Fourier space: it takes the form of a deconvolution operator whose kernel can be approximated by a finite sum of separable filters. This solution method produces excellent computation times even on big images.

Functional Description:
Use of proximal and non quadratic minimization. GPU implementation. If f is an input image, one seeks an output g such that the following functional is minimized:
l/2*(norme2(fg) + psi(grad(g))) with : l positive constant, norme2 = L2 norm, psi is a Cauchy function used for parcimony.
This functional is also applied for debayerization.

Release Contributions:
This software implements H. Badri PhD thesis results.
 URL:

Authors:
Marie Martin, Chiheb Sakka, Hussein Yahia, Nicolas Brodu, Gabriel Zebadua Garcia, Khalid Daoudi

Contact:
Hussein Yahia

Partner:
Innovative Imaging Solutions I2S
7.1.4 Manzana

Name:
Manzana

Keywords:
2D, Image processing, Filtering

Scientific Description:
Software library developed in the framework of I2SGEOSTAT innovationlab and made of highlevel image processing functionalities based on sparsity and nonconvex optimization.

Functional Description:
Library of software in image processing: filtering, hdr, inpainting etc.

Contact:
Hussein Yahia

Partner:
Innovative Imaging Solutions I2S
7.2 New platforms
Participants:
8 New results
8.1 Turbulent dynamics in the interstellar medium
Participants: H. Yahia, N. Schneider, S. Bontemps, L. Bonne, et al.
Observations of the interstellar medium (ISM) show a complex density and velocity structure, which is in part attributed to turbulence. Consequently, the multifractal formalism should be applied to observation maps of the ISM in order to characterize its turbulent and multiplicative cascade properties. However, the multifractal formalism, even in its more advanced and recent canonical versions, requires a large number of realizations of the system, which usually cannot be obtained in astronomy. We present a selfcontained introduction to the multifractal formalism in a "microcanonical" version, which allows us, for the first time, to compute precise turbulence characteristic parameters from a single observational map without the need for averages in a grand ensemble of statistical observables (e.g., a temporal sequence of images). We compute the singularity exponents and the singularity spectrum for both observations and magnetohydrodynamic simulations, which include key parameters to describe turbulence in the ISM. For the observations we focus on the 250 µm Herschel map of the Musca filament. Scaling properties are investigated using spatial 2D structure functions, and we apply a twopoint logcorrelation magnitude analysis over various lines of the spatial observation, which is known to be directly related to the existence of a multiplicative cascade under precise conditions. It reveals a clear signature of a multiplicative cascade in Musca with an inertial range from 0.05 to 0.65 pc. We show that the proposed microcanonical approach provides singularity spectra that are truly scale invariant, as required to validate any method used to analyze multifractality. The obtained singularity spectrum of Musca, which is sufficiently precise for the first time, is clearly not as symmetric as usually observed in lognormal behavior. We claim that the singularity spectrum of the ISM toward Musca features a more logPoisson shape. Since logPoisson behavior is claimed to exist when dissipation is stronger for rare events in turbulent flows, in contrast to more homogeneous (in volume and time) dissipation events, we suggest that this deviation from lognormality could trace enhanced dissipation in rare events at small scales, which may explain, or is at least consistent with, the dominant filamentary structure in Musca. Moreover, we find that subregions in Musca tend to show different multifractal properties: While a few regions can be described by a lognormal model, other regions have singularity spectra better fitted by a logPoisson model. This strongly suggests that different types of dynamics exist inside the Musca cloud. We note that this deviation from lognormality and these differences between subregions appear only after reducing noise features, using a sparse edgeaware algorithm, which have the tendency to "lognormalize" an observational map. Implications for the star formation process are discussed. Our study establishes fundamental tools that will be applied to other galactic clouds and simulations in forthcoming studies.Publication: Astronomy & Astrophysics, HAL. This publication has been selected in the "Highlights" section of Astronomy & Astrophysics.
8.2 CONCAUST Exploratory Action
Participants: N. Brodu, J. P. Crutchfield, L. Bourel, P. Rau, A. Rupe, Y. Li.
Concaust Exploratory Action Web page.The article detailing the exploratory action core method “Discovering Causal Structure with ReproducingKernel Hilbert Space $\epsilon $Machines”, has been accepted for publication in the Chaos journal. The remaining budget was reallocated into 2 years of postdoctorate positions:  1 year for working on the core methodology  1 year for researching the applicability of the method on real data, in particular the study of El Niño phenomena. Due to covid constraints, the postdoctorate candidate (Alexandra Jurgens) that was selected for the core methodology could not come to France in fall 2021. She may come again in fall 2022, if the situation allows, so the corresponding postdoc is reserved for her. The second postdoctorate offer on El Niño is currently opened for applications.
8.3 COMCAUSA Associated team
Participants: N. Brodu, J. P. Crutchfield, et al. (see Website).
Web page The associate team Comcausa was created as part of the Inria@SiliconValley international lab, between Inria Geostat and the Complexity Sciences Center at University of California, Davis. This team is managed by Nicolas Brodu (Inria) and Jim Crutchfield (UC Davis) and the full list of collaborators is given on the web site. We organized a series of 10 online seminars “Inference for Dynamical Systems A seminar series” in which we invited team members and external researchers to present their results. This online seminars series was a federative moment during the covid lockdowns and fostered new collaborations. Additional funding was obtained (coPIs Nicolas Brodu, Jim Crutchfield, Sarah Marzen) from the Templeton Foundation in the form of 2×1 years postdoctorate, to work on bioacoustic signatures in whale communication signals. We recruited Alexandra Jurgens for one year, renewable, on an exploratory topic of research: seeking new methods for inferring how much information is being transferred at every scale in a signal. Nicolas Brodu is actively cosupervising her on this program, which she may pursue at Inria in fall 2022 on the Concaust exploratory action postdoc budget. More preliminary results from this associate team were obtained on CO and water flux in₂ ecosystems (collaboration between Nicolas Brodu, Yao Liu and Adam Rupe). Nicolas Brodu presented these at the yearly meeting of the ICOS network on monitoring stations, run mostly by INRAE. This in turn lead to the writing of a proposal for the joint InriaINRAE « Agroécologie et numérique » PEPR, which passed the preselection phase in December 2021: this project is being proposed, jointly with a partner at INRAE, as one of the 10 flagship projects retained for the round 1 of this PEPR. The final decision for whether this PEPR will be funded or not will be made in 2022. Similarly, preliminary results from the El Niño data (collaboration between Nicolas Brodu and Luc Bourrel), have lead to the submission of an ANR proposal. This ANR funding would allow us to extend the work of the postdoctorate researcher which we will cosupervise on the Concaust budget. The Associated Team budget of 2021 could only be partially used as travels were restricted for most of the year. An extensive lab tour was still possible (Nov.Dec. 2021), where Nicolas Brodu has met with most US associate team members. This tour was scientifically fruitful and we are currently preparing articles detailing our new results8.4 Vocal biomarkers of Parkinsonian disorders
Participants: K. Daoudi, B. Das, S. Milhé de Saint Victor, A. FoubertSamier, A. PavyLe Traon, O. Rascol, W. G. Meissner, V. Woisard.
In the early stage of disease, the symptoms of Parkinson's disease (PD) are similar to atypical Parkinsonian disorders (APD) such as Progressive Supranuclear Palsy (PSP) and Multiple System Atrophy (MSA). The early differential diagnosis between PD and APD and between APD groups is thus a very challenging task. It turns out that speech disorder is an early and common symptom to PD and APD. The goal of our research is to develop a digital biomarker based on speech analysis in order to assist the neurologists in their diagnosis. We identified distinctive speech features for discrimination between PD and MSAP, the variant of MSA where Parkinsonism dominates. These features were inferred from French speaking patients and consist in the detection of distortions in the production of particular voiced consonants (plosives and fricatives). We also continued of work on differential diagnosis between PSP and MSA Czech speaking patients. We designed two composite speech indices based on two categorizations of speech features, production subsystems and dysarthria type. These new indices led to high accuracy discrimination not only between PSP and MSA but also between MSA/PSP and PD. These results made us confident to step on to the second clinical phase of the Voice4PDMSA project, the validation phase, thought we didn’t reach the expected number of patient inclusions in the first phase (because of the pandemic). The clinical protocol of this second phase will be submitted to CPP in January 2022.Publications: PhD thesis (B. Das HAL), Proc. Interspeech’ 2021 (K. Daoudi et al.HAL).
8.5 Vocal biomarkers of respiratory diseases
Participants: K. Daoudi, T. Similowski.
GeoStat continues its contribution to the Covid19 mission of Inria via the VocaPnée project in partnership with APHP and codirected by K. Daoudi and Thomas Similowski, responsible for the pulmonology and resuscitation service at La PitiéSalpêtrière hospital and UMRS 1158. The objective of the VocaPnée project is to bring together skills available at Inria to develop and validate a vocal biomarker for the remote monitoring of patients at home suffering from acute respiratory diseases (such as Covid19) or chronic one (such as asthma). This biomarker will then be integrated into a telemedicine platform, ORTIF or COVIDOM for Covid, to assist the doctors in assessing the patient's respiratory status. VocaPnée is divided into 2 longitudinal clinical studies, a hospital study, as a proof of concept, followed by another in telemedicine. The former received the clearance of CPP and we are preparing the file for the CNIL clearance. In this context, the ADT project VocaPy started in November for a 2 years duration. The goal of VocaPy is to develop a Python library dedicated to pathological speech analysis and vocal biomarkers conception. VocaPy works in synergy with the ADT VocaPnéeInfra dedicated to the development of an infrastructure to interact with protected health data.8.6 Bayesian approach in a learningbased hyperspectral image denoising framework
Participants: H. Aetesam, S. Kumar Maji, H. Yahia.
Hyperspectral images are corrupted by a combination of Gaussianimpulse noise. On one hand, the traditional approach of handling the denoising problem using maximum a posteriori criterion is often restricted by the timeconsuming iterative optimization process and design of handcrafted priors to obtain an optimal result. On the other hand, the discriminative learningbased approaches offer fast inference speed over a trained model; but are highly sensitive to the noise level used for training. A discriminative model trained with a loss function which does not accord with the Bayesian degradation process often leads to suboptimal results. In this paper, we design the training paradigm emphasizing the role of loss functions; similar to as observed in modelbased optimization methods. As a result; loss functions derived in Bayesian setting and employed in neural network training boosts the denoising performance. Extensive analysis and experimental results on synthetically corrupted and real hyperspectral dataset suggest the potential applicability of the proposed technique under a wide range of homogeneous and heterogeneous noisy settings.Publication: IEEE Access , HAL
8.7 InnovationLab with I2S
Participants: A. Zebadua, A. Rashidi, H. Yahia, A. Cherif [I2S], J. L. Vallancogne [I2S], A. Cailly [I2S].
The InnovationLab with I2S is extended one year starting 1st February 2021.8.8 InnovationLab with I2S:Acceleration of optimizationbased methods using convolutional neural networks (Emulation)
Participants: A. Zebadua, H. Yahia, A. Cherif [I2S], J. L. Vallancogne [I2S], A. Cailly [I2S].
During the Inriai2s Innovationlab partnership (20172020), several iterative optimizationbased image processing algorithms were developed. Such methods are able to provide significant image quality improvements with respect to the previous i2s image processing methods. However, such algorithms seek the minimum of a mathematical function iteratively. In some cases, their convergence can be significantly slow and demand an execution time prohibitive for production goals.This motivated a deeplearning approach in order to reduce computational time. We called this approach Emulation. A convolutional neural network (CNN) is designed and trained it to reproduce the results obtained by the optimization approach. This makes it possible to take advantage of the quality level of an expensive iterative optimization algorithm at a lower computational cost and shorter execution time.
Iterative and deep learning methods make both use of numerical optimization to solve a mathematical problem. In the case of deep learning, however, the optimization takes places during training, which is performed offline. Once trained, a CNN can be used online to provide a fast execution time to process an image (which is commonly referred to as inference). This approach was shown to be effective and allowed comparable image quality levels along with a reduction in execution time 5 times shorter.
8.9 InnovationLab with I2S: Fast image deconvolution for enhancement of the resolution in the video rate terahertz imaging
Participants: A. Rashidi, A. Cailly [I2S], A. Cherif [I2S], H. Yahia.
A. Rashidi has pursued the implementation of deconvolution and denoising algorithms within the algorithms of I2S. A fast image deconvolution algorithm is used to demonstrate the resolution enhancement of video rate camera acquired Terahertz images. Our algorithm is based on variable splitting technique with the use of a family of sparsity inducing regularizers for the first time in an image deconvolution application, it is also suitable for practical applications in industry with computationally constrained conditions. The results of the proposed process provide substantial enhancement on the quality and resolution of THz images.A first work concerned the classical cameras of I2S which is called "Eagle".
Industrial Quality assessment of the images and results showed effective and encouraging results. The images showed better image quality contrast.
Following on, the superresolution algorithm was integrated into the new cameras of I2S. The new camera is called "Xtra". we are currently in the stage of verifying the effectiveness of the results and their quality with standard measurement methods.
Publication: 46th International Conference on Infrared, Millimeter, and Terahertz Waves, HAL
9 Bilateral contracts and grants with industry
9.1 Bilateral contracts with industry
InnovationLab with I2S company, starting scheduled after 1st 2019 COPIL in January 2019. This InnovationLab is extended one year starting February 2021.
10 Partnerships and cooperations
10.1 International initiatives
10.1.1 Associate Teams in the framework of an Inria International Lab or in the framework of an Inria International Program
COMCAUSA

Title:
Computation of causal structures

Duration:
2021 >

Coordinators:
J. P. Crutchfield (chaos@cse.ucdavis.edu), N. Brodu

Partners:
 University of California Davis

Inria contact:
N. Brodu
 Summary: The associate team Comcausa was created as part of the Inria@SiliconValley international lab, between Inria Geostat and the Complexity Sciences Center at University of California, Davis. This team is managed by N. Brodu (Inria) and J. P. Crutchfield (UC Davis).
10.2 European initiatives
10.2.1 FP7 & H2020 projects
GENESIS Project (Geostat, Laboratoire d'Astrophysique de Bordeaux, Physics Inst. (Köln University) 5year contract, 20172022 (initially 3 years, extented).
Web site. GENeration and Evolution of Structures in the ISm. The objective of this research project is to better understand the structure and evolution of molecular clouds in the interstellar medium (ISM) and to link cloud structure with starformation. For that, farinfrared observations of dust (Herschel) and cooling lines (SOFIA) are combined with groundbased submillimetre observations of molecular lines. Dedicated analysis tools will be used and developed to analyse the maps and compared to simulations in order to disentangle the underlying physical processes such as gravity, turbulence, magnetic fields, and radiation.
10.3 National initiatives
 CONCAUST Exploratory Action The exploratory action « TRACME » was renamed « CONCAUST » and is going on with good progress. Collaboration with J. P. Crutchfield and its laboratory has lead to a first article, “Discovering Causal Structure with ReproducingKernel Hilbert Space $\epsilon $Machines”, available available here . That article poses the main theoretical fundations for building a new class of models, able to reconstruct a measured process « causal states » from data.

CovidVoice project: Inria Coind19 mission. The CovidVoice project evolved into the VocaPnée project in partnership with APHP and codirected by K. Daoudi and Thomas Similowski, responsible for the pulmonology and resuscitation service at La PitiéSalpêtrière hospital and UMRS 1158. The objective of the VocaPnée project is to bring together all the skills available at Inria to develop and validate a vocal biomarker for the remote monitoring of patients at home suffering from an acute respiratory disease (such as Covid) or chronic (such as asthma) . This biomarker will then be integrated into a telemedicine platform, ORTIF or COVIDOM for Covid, to assist the doctors in assessing the patient's respiratory status. VocaPnée is divided into 2 longitudinal pilot clinical studies, a hospital study and another in telemedicine.
In this context, a voice data collection platform was developed by Inria's SED. This platform is used to collect data from healthy controls. It will then be migrated to the APHP servers to collect patient data.
 ANR project Voice4PDMSA, led by K. Daoudi, which targets the differential diagnosis between Parkinson's disease and Multiple System Atrophy. The total amount of the grant is 468555 euros, from which GeoStat has 203078 euros. Initial duration of the project was 42 months, and has been extended until January 2023. Partners: CHU Bordeaux (Bordeaux), CHU Toulouse, IRIT, IMT (Toulouse).
 GEOSTAT is a member of ISIS (Information, Image & Vision), AMF (Multifractal Analysis) GDRs.
 GEOSTAT is participating in the CNRS IMECO project Intermittence multiéchelles de champs océaniques : analyse comparative d’images satellitaires et de sorties de modèles numériques. CNRS call AO INSU 2018. PI: F. Schmitt, DR CNRS, UMR LOG 8187. Duration: 2 years.
10.4 Regional initiatives
The ADT (IA Plan) project VocaPy, led by K. Daoudi, started in November 2021 for a 2 years duration. The goal of VocaPy is to develop a Python library dedicated to pathological speech analysis and vocal biomarkers conception. Ms Zhe Li was recruited as an engineer on this project.
11 Dissemination
11.1 Promoting scientific activities
11.1.1 Journal
Member of the editorial boards
H. Yahia is a member of the editorial board of "Frontiers in Fractal Physiology" journal.
11.1.2 Invited talks
 N. Brodu: The ICOS days, Université de Reims, France, 1214 October 2021
11.1.3 Scientific expertise
 K. Daoudi was invited by ARS RhôneAlpes (on December 3th 2021) to give a talk on ethical risks of vocal biomarkers: link
 N. Brodu. Santa Fe  Nov 29/Dec 7: Meeting with Jim Crutchfield (Santa Fe Institute, UC Davis) and Adam Rupe (Los Alamos National Lab).
 N. Brodu. Los Alamos  Dec 7/8: Meeting with Balasubramanya Nadiga (Los Alamos National Lab).
 N. Brodu. Santa Barbara  Dec 8/9: Meeting with Nina Miolane (UC Santa Barbara).
 N. Brodu. Pasadena  9/12: Meeting with Stuart Bartlett (Caltech).
 N. Brodu. Claremont  12/15: Meeting with Sarah Marzen (Claremont College), Alexandra Jurgens (UC Davis), Alec Boyd (Caltech).
 N. Brodu: European Conference on Complex Systems, Lyon, 2529 October 2021
11.1.4 Juries
H. Yahia was a member of the PhD thesis jury of H. MAHAMAT. Thesis defended on March 30, 2021, 10.30 am, Université de Bourgogne, UFR Sciences et Techniques  BP 47870 21078 Dijon cedex.
11.2 Popularization
11.2.1 Articles and contents
12 Scientific production
12.1 Major publications
 1 articleMultifractal Desynchronization of the Cardiac Excitable Cell Network During Atrial Fibrillation. II. Modeling.Frontiers in Physiology10April 2019, 480 (118)
 2 articleMultifractal desynchronization of the cardiac excitable cell network during atrial fibrillation. I. Multifractal analysis of clinical data.Frontiers in Physiology8March 2018, 130
 3 articleA NonLocal LowRank Approach to Enforce Integrability.IEEE Transactions on Image ProcessingJune 2016, 10
 4 inproceedingsFast and Accurate Texture Recognition with Multilayer Convolution and Multifractal Analysis.European Conference on Computer VisionECCV 2014Zürich, SwitzerlandSeptember 2014
 5 articleIncreasing the Resolution of Ocean pCO₂ Maps in the South Eastern Atlantic Ocean Merging Multifractal SatelliteDerived Ocean Variables.IEEE Transactions on Geoscience and Remote SensingJune 2018, 1  15
 6 articleReconstruction of superresolution ocean pCO 2 and airsea fluxes of CO 2 from satellite imagery in the Southeastern Atlantic.BiogeosciencesSeptember 2015, 20
 7 articleDetection of Glottal Closure Instants based on the Microcanonical Multiscale Formalism.IEEE Transactions on Audio, Speech and Language ProcessingDecember 2014
 8 articleEfficient and robust detection of Glottal Closure Instants using Most Singular Manifold of speech signals.IEEE Transactions on Acoustics Speech and Signal Processingforthcoming2014
 9 articleEdges, Transitions and Criticality.Pattern RecognitionJanuary 2014, URL: http://hal.inria.fr/hal00924137
 10 articleA Multifractalbased Wavefront Phase Estimation Technique for Groundbased Astronomical Observations.IEEE Transactions on Geoscience and Remote SensingNovember 2015, 11
 11 articleSingularity analysis in digital signals through the evaluation of their Unpredictable Point Manifold.International Journal of Computer Mathematics2012, URL: http://hal.inria.fr/hal00688715
 12 articleOcean Turbulent Dynamics at Superresolution From Optimal Multiresolution Analysis and Multiplicative Cascade.IEEE Transactions on Geoscience and Remote Sensing5311June 2015, 12
 13 articleMicrocanonical multifractal formalism: a geometrical approach to multifractal systems. Part I: singularity analysis.Journal of Physics A: Math. Theor412008, URL: http://dx.doi.org/10.1088/17518113/41/1/015501
 14 articleMotion analysis in oceanographic satellite images using multiscale methods and the energy cascade.Pattern Recognition43102010, 35913604URL: http://dx.doi.org/10.1016/j.patcog.2010.04.011
12.2 Publications of the year
International journals
 15 articleBayesian Approach in a LearningBased Hyperspectral Image Denoising Framework.IEEE AccessDecember 2021, 11
 16 articleDescription of turbulent dynamics in the interstellar medium: Multifractalmicrocanonical analysis: I. Application to Herschel observations of the Musca filament.Astronomy and Astrophysics  A&AMarch 2021
Conferences without proceedings
 17 inproceedingsDistortion of voiced obstruents for differential diagnosis between parkinson's disease and multiple system atrophy.INTERSPEECH 2021Brno, Czech RepublicAugust 2021
 18 inproceedingsFast image deconvolution for enhancement of the resolution in the video rate terahertz imaging.IRMMWTHz46th International Conference on Infrared, Millimeter, and Terahertz WavesIRMMWTHzBeijing, ChinaSeptember 2021
Doctoral dissertations and habilitation theses
 19 thesisAutomated objective speech markers for differential diagnosis between parkinson's disease and atypical parkinsonian disorders.Université de BordeauxSeptember 2021
12.3 Cited publications
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 23 articleReconstruction of epsilonmachines in predictive frameworks and decisional states.Advances in Complex Systems14052011, 761794URL: https://doi.org/10.1142/S0219525911003347
 24 articleStochastic Texture Difference for ScaleDependent Data Analysis.arXiv preprint arXiv:1503.032782015
 25 bookChaos and coarse graining in statistical mechanics.Cambridge University Press Cambridge2008
 26 articleMultiscale entropy analysis of complex physiologic time series.Physical review letters8962002, 068102
 27 articleBetween order and chaos.Nature Physics812012, 1724
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 32 inproceedingsMixed LICORS: A nonparametric algorithm for predictive state reconstruction.Artificial Intelligence and Statistics2013, 289297
 33 bookEntropy and information theory.Springer Science & Business Media2011
 34 articleMultifield visualization using local statistical complexity.IEEE Transactions on Visualization and Computer Graphics1362007, 13841391
 35 book Statistical physics, statics, dynamics & renormalization.World Scie,tific2000
 36 articleGeneralized statistical complexity measures: Geometrical and analytical properties.Physica A: Statistical Mechanics and its Applications36922006, 439462
 37 bookFrom Statistical Physics to Statistical Inference and Back.New York Heidelberg BerlinSpringer1994, URL: http://www.springer.com/physics/complexity/book/9780792327752
 38 inproceedingsBlind construction of optimal nonlinear recursive predictors for discrete sequences.Proceedings of the 20th conference on Uncertainty in artificial intelligenceAUAI Press2004, 504511
 39 articleQuantifying SelfOrganization with Optimal Predictors.Phys. Rev. Lett.9311Sep 2004, 118701URL: https://link.aps.org/doi/10.1103/PhysRevLett.93.118701
 40 bookSparse Image and Signal Processing: Wavelets, Curvelets, Morphological Diversity.ISBN:9780521119139Cambridge University Press2010
 41 articleDetecting Strange Attractors in Turbulence.Non Linear Optimization8981981, 366381URL: http://www.springerlink.com/content/b254x77553874745/
 42 articleRevisting multifractality of high resolution temporal rainfall using a waveletbased formalism.Water Resources Research422006