M2 Internship

Multifractal analysis of complex boundariesSpring 2019

[•] Context. Multifractal analysis is a mathematical modeling and analysis framework that describes afunction f : Rd 7→ R based on the global geometry of the fluctuations of their pointwise regularity [1, 2]. Inpractice, after averaging, this assumes the absence of characteristic scales and intermittent, complex transientdependence. Essentially, multifractal analysis relies on the identification of the power law exponents ζ(q)that relate the qth moments of multiresolution coefficients Tf (a,x), such as discrete wavelet coefficients, atdifferent scales a, ∑

k

|TX(a,x = k)|q ' cq|a|ζ(q). (1)

Multifractal analysis is an active area of research and is becoming a standard data analysis tool that nat-urally encompasses rich nonlinear and multiscale data properties. It has been successfully applied for theanalysis of real-world signals and images in a broad range of contexts, from Physics (e.g. material science[3]) over man-made signals (Internet traffic [4], perception and Art [5]) to biomedicine [6] to quote but afew examples (see also [2] and references therein).

[•] Limitations. Despite such massive successes in applications, today’s state-of-the-art for practi-cal multifractal analysis of images remains essentially restricted to the study of a single homogeneous

MUTATIONDéfi B.11 Grands enjeux transverses - Axe 1 AAPG ANR 2018I Context. A complex boundary – i.e., the interface separating two distinct regions in an image, suchas a coastline – may be the most emblematic example of a fractal object [51,52]. It may therefore appear

(a) (b)Synthetic multifractal texture (a) andcomplex boundary of its level sets (b).

surprising that the state-of-the-art tool for studying collections of objectsthat convey information not only in their different textures, but also in theheterogeneity and irregularity of the boundaries separating them, currentlyremains limited to calculating its fractal dimension [52,61,72], which can-not fully account for the rich irregularity properties of the interfaces foundin multidimensional data. This constitutes a significant deadlock in viewof the paramount importance of characterizing boundary irregularity inapplications from so distinct disciplines as material sciences [33], quantitative urbanism [29], perceptionalpsychology and Art [32], medical and tumor imaging [4, 12, 46], etc.

III.2.1 WP 1.1: Boundary multifractal analysisI Issues. Multifractal analysis for collections of objects with complex shapes does not exist.I Goals. Develop and assess a boundary irregularity multifractal formalism and matching practical tools.Boundary multifractal formalism. To go beyond the sole fractal dimension df for an indicator func-tion 1�, we propose to build on preliminary intuitions that theoretically defined regularity exponentsh�(y) for the boundary of � [36], yet remained unnoticed in the literature except for the study of scholarlymathematical objects. To construct a multifractal formalism, a key challenge will consist in defining mul-tiresolution quantities that reproduce h�(y) in the limit of fine scales. Exploratory analyses [36] lead toconjecture that the variation with p of the decay rate of p-leaders (4) is linked to the degree of irregularityof � at position y, yet with the major difficulty of extremely fast decay rates (similar to oscillations asdefined in [54]). We will investigate how the thermodynamical arguments leading to (3) can be modifiedto establish a theoretical link between these variations and the boundary multifractal spectrum D�(h),leading to stable formulas for the estimation of D�(h), which will be assessed numerically.From texture to boundary regularity. The fact that decay rates of `(p)

X quantify regularity exponentsh(y) for texture (when varying q in (2)) and h�(y) for indicator functions (when varying p) points toan entirely new, never explored theoretical perspective for the multifractal analysis of textured objects:jointly tuning p and q could permit to assess a theoretical continuum of singularity properties, fromtextures to boundaries. We aim to first identify the formal mechanisms that balance between the textureand boundary singularities’ control over the multiscale statistics of `(p)

X . We expect first answers for thisdifficult issue from studying the evolution of the norms of `(p)

X between pure texture and texture with added,gradually stronger step singularity with irregular geometry. Theoretical studies will be guided by empiricalconjectures from simulations on multifractal textures with boundaries, to be defined. This will also yieldgeneral intuitions on the natures of captured regularity information. The challenge of how to formalizea mixed multifractal formalism for texture and boundary regularity will be investigated (e.g. building ongrand-canonical formalisms [37]). Further, we will study the relative impacts of discretization in space andamplitude, different in nature, whose understanding is critical for robust estimation of D(h) and D�(h).

III.2.2 WP 1.2: Synthetic models for multifractal boundariesI Issues. Except from scholar examples, no realistic synthetic models exist for boundary irregularity.I Goals. Develop theoretical models and practical synthesis procedures that are crucially needed to assessanalysis tools and to model and understand real-world mechanisms generating complex boundaries.Deterministic multifractal boundary models. We will study D�(h) for the difficult case when bound-aries cannot be unfolded by projection (since they would be multivalued functions). We will also exploregeneralizing classical objects (e.g. Von Koch curve with different parameters for left and right children)to yield boundaries with same df , but different D�(h). To test the ability of the boundary multifractalformalism to assess different types of heterogeneity, D�(h) for objects with zero mass (e.g. Sierpinsky car-pets) and with mass but fractal boundary will be compared. Further, multifractal texture on deterministicfractal sets � will enable probing p-exponent regularity jointly for texture and boundary.Stochastic model processes. Founding models for multifractal boundaries will be inspired by studyingstochastic indicator functions 1X>µ defined by the level-sets of multifractal textures X. To model theevolution between and co-occurrence of textures & boundaries, non-linear or non-smooth transformations,e.g. � (X�µ)

max(X) +(1��)sign(X�µ), will be used to pick a level-set µ and continuously tune between its

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texture. Yet, in many applications, the characterization of theboundaries separating different textures in an image is of cen-tral interest. To give a concrete example, the complex and het-erogeneous interaction of tumours with their neighboring micro-environment is believed to play an important role in the de-velopment of resistance to treatment therapy, and the quanti-tative analysis of such boundaries could reveal crucial informa-tion.

[•] Objectives. Building on preliminary works, the goal of the internship is to explore models and toolsfor the multifractal analysis of complex boundaries:

1. A formalism for studying multifractal boundaries has been theoretically proposed in [7, 8, 9]. Itrelies on specific multiresolution coefficients, the p-leaders, that have already been studied for themultifractal analysis of image texture [10, 11]. Making use of existing MATLAB implementations,the first objective of this internship will be to study the use of these tools for the analysis of simpleboundary models.

2. The second objective consists in studying synthetic models for multifractal boundaries. Both determin-istic models (such as generalizations of the famous fractal Von Koch curve) or stochastic models (forinstance, the level sets of complex textures) will be explored theoretically and validated numerically.

[•] Perspectives. This internship is part of larger national research programme aiming at the multifrac-tal modeling and analysis of biomedical images. The developed models, tools and concepts could thereforealso be put to test on real-world biomedical images. Depending on the outcome of the internship, it ispossible that the topic will be pursued by a PhD project (funding secured).

[•] Requirements. Candidates should have a solid background in mathematics and statistics, andshould be operational with MATLAB. An interest in the analysis and modeling of (real-world) images is ofadvantage but not indispensable.

[•] Contact. This internship will be co-advised by:- Herwig Wendt, Researcher, CNRS, Institut de Recherche en Informatique de Toulouse.- Clothilde Melot, Maitre de Conference, Institut de Mathematiques de Marseille, Universite Aix Marseille.

[•] Application.All applications must be sent electronically to the advisors (minimum: motivation letter, CV).

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References

[1] H. Wendt, P. Abry, and S. Jaffard, “Bootstrap for empirical multifractal analysis,” IEEE Signal Process.Mag., vol. 24, no. 4, pp. 38–48, 2007.

[2] S. Jaffard, P. Abry, and H. Wendt, Irregularities and Scaling in Signal and Image Processing: Multi-fractal Analysis, pp. 31–116. Singapore: World scientific publishing, 2015.

[3] A. Arnedo, N. Decoster, and S. Roux, “A wavelet-based method for multifractal image analysis. i.methodology and test applications on isotropic and anisotropic random rough surfaces,” Eur. Phys. J.B, vol. 15, pp. 567–600, June 2000.

[4] P. Abry, P. Flandrin, M. Taqqu, and D. Veitch, “Wavelets for the analysis, estimation and synthesis ofscaling data,” in Self-similar Network Traffic and Performance Evaluation, Wiley, spring 2000.

[5] P. Abry, S. G. Roux, H. Wendt, P. Messier, A. G. Klein, N. Tremblay, P. Borgnat, S. Jaffard, B. Vedel,J. Coddington, and L. Daffner, “Multiscale anisotropic texture analysis and classification of photo-graphic prints: Art scholarship meets image processing algorithms,” IEEE Signal Proces. Mag., vol. 32,pp. 18–27, July 2015.

[6] R. Lopes and N. Betrouni, “Fractal and multifractal analysis: a review,” Medical image analysis, vol. 13,no. 4, pp. 634–649, 2009.

[7] S. Jaffard and C. Melot, “Wavelet analysis of fractal boundaries. Part 1: local exponents.,” Comm.Math. Phys., vol. 258, no. 3, pp. 513–539, 2005.

[8] S. Jaffard and C. Melot, “Wavelet analysis of fractal boundaries. Part 2: Multifractal analysis.,” Comm.Math. Phys., vol. 258, no. 3, pp. 541–565, 2005.

[9] M. Ben Slimane and C. Melot, “Analysis of a fractal boundary: the graph of the Knopp function,” inAbstract and Applied Analysis, vol. 2015, Hindawi, 2015.

[10] S. Jaffard, C. Melot, R. Leonarduzzi, H. Wendt, P. Abry, S. G. Roux, and M. E. Torres, “p-exponentand p-leaders, part i: Negative pointwise regularity.,” Physica A, vol. 448, pp. 300–318, 2016.

[11] R. Leonarduzzi, H. Wendt, P. Abry, S. Jaffard, C. Melot, S. G. Roux, and M. E. Torres, “p-exponentand p-leaders, part ii: Multifractal analysis. relations to detrended fluctuation analysis.,” Physica A,vol. 448, pp. 319–339, 2016.

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