Clothilde Mélot

Some proximal methods for CBCT and PET tomography

The reconstruction of the images obtained via the Cone Beam Computerized Tomography (CBCT) and Positron Emission Tomography (PET) Scanners are ill-posed inverse problems. One needs to adress carefully the problem of inversion of the mathematical operators involved. Recent advances in optimization have yielded efficient algorithms to solve very general classes of inverse problems via the minimization of non-differentiable convex functions. We show that such models are well suited to solve the CBCT and PET reconstruction problems. On the one hand, they can incorporate directly the physics of new acquisition devices, free of dark noise; on the other hand, they can take into account the specificity of the pure Poisson noise. We propose various fast numerical schemes to recover the original data and compare them to state-of-the-art algorithms on simulated data. We study more specifically how different contrasts and resolutions may be resolved as the level of noise and/or the number of projections of the acquired sinograms decrease. We conclude that the proposed algorithms compare favorably with respect to well-established methods in tomography.

p-exponent and p-leaders, Part II: Multifractal Analysis. Relations to Detrended Fluctuation Analysis

Multifractal analysis studies signals, functions, images or fields via the fluctuations of their local regularity along time or space, which capture crucial features of their temporal/spatial dynamics. It has become a standard signal and image processing tool and is commonly used in numerous applications of different natures. In its common formulation, it relies on the Holder exponent as a measure of local regularity, which is by nature restricted to positive values and can hence be used for locally bounded functions only. In this contribution, it is proposed to replace the Holder exponent with a collection of novel exponents for measuring local regularity, the p-exponents. One of the major virtues of p-exponents is that they can potentially take negative values. The corresponding wavelet-based multiscale quantities, the p-leaders, are constructed and shown to permit the definition of a new multifractal formalism, yielding an accurate practical estimation of the multifractal properties of real-world data. Moreover, theoretical and practical connections to and comparisons against another multifractal formalism, referred to as multifractal detrended fluctuation analysis, are achieved. The performance of the proposed p-leader multifractal formalism is studied and compared to previous formalisms using synthetic multifractal signals and images, illustrating its theoretical and practical benefits. The present contribution is complemented by a companion article studying in depth the theoretical properties of p-exponents and the rich classification of local singularities it permits.

Multifractal Analysis Based on p-Exponents and Lacunarity Exponents

Many examples of signals and images cannot be modeled by locally bounded functions, so that the standard multifractal analysis, based on the Holder exponent, is not feasible. We present a multifractal analysis based on another quantity, the p-exponent, which can take arbitrarily large negative values. We investigate some mathematical properties of this exponent, and show how it allows us to model the idea of “lacunarity” of a singularity at a point. We finally adapt the wavelet based multifractal analysis in this setting, and we give applications to a simple mathematical model of multifractal processes: Lacunary wavelet series.

p-exponent and p-leaders, Part I: Negative pointwise regularity

Multifractal analysis aims to characterize signals, functions, images or fields, via the fluctuations of their local regularity along time or space, hence capturing crucial features of their temporal/spatial dynamics. Multifractal analysis is becoming a standard tool in signal and image processing, and is nowadays widely used in numerous applications of different natures. Its common formulation relies on the measure of local regularity via the Holder exponent, by nature restricted to positive values, and thus to locally bounded functions or signals. It is here proposed to base the quantification of local regularity on p-exponents, a novel local regularity measure potentially taking negative values. First, the theoretical properties of p-exponents are studied in detail. Second, wavelet-based multiscale quantities, the p-leaders, are constructed and shown to permit accurate practical estimation of p-exponents. Exploiting the potential dependence with p, it is also shown how the collection of p-exponents enriches the classification of locally singular behaviors in functions, signals or images. The present contribution is complemented by a companion article developing the p-leader based multifractal formalism associated to p-exponents.

On the efficiency of proximal methods in CBCT and PET

Cone Beam Computerized Tomography (CBCT) and Positron Emission Tomography (PET) Scans are medical imaging devices that require solving ill-posed inverse problems. The models considered come directly from the physics of the acquisition devices, and take into account the specificity of the (Poisson) noise. We propose various fast numerical schemes to compute the solution. In particular, we show that a new algorithm recently introduced by A. Chambolle and T. Pock is well suited in the PET case when considering non differentiable regularizations such as total variation or wavelet ℓ1-regularization. Numerical experiments indicate that the proposed algorithms compare favorably with respect to well-established methods in tomography.

New Exponents for Pointwise Singularity Classification

We introduce new tools for pointwise singularity classification: We investigate the properties of the two-variable function which is defined at every point as the p-exponent of a fractional integral of order t; new exponents are derived which are not of regularity type but give a more precise description of the behavior of the function near a singularity. We revisit several classical examples (deterministic and random) of multifractal functions for which the additional information supplied by this classification is derived. Finally, a new example of multifractal function is studied, where these exponents prove pertinent.

Random Forests and Networks Analysis

AbstractWilson (Proceedings of the twenty-eight annual acm symposium on the theory of computing, pp. 296–303, 1996) in the 1990s described a simple and efficient algorithm based on loop-erased random walks to sample uniform spanning trees and more generally weighted trees or forests spanning a given graph. This algorithm provides a powerful tool in analyzing structures on networks and along this line of thinking, in recent works (Avena and Gaudillière in A proof of the transfer-current theorem in absence of reversibility, in Stat. Probab. Lett. 142, 17–22 (2018); Avena and Gaudillière in J Theor Probab, 2017. https://doi.org/10.1007/s10959-017-0771-3; Avena et al. in Approximate and exact solutions of intertwining equations though random spanning forests, 2017. arXiv:1702.05992v1; Avena et al. in Intertwining wavelets or multiresolution analysis on graphs through random forests, 2017. arXiv:1707.04616, to appear in ACHA (2018)) we focused on applications of spanning rooted forests on finite graphs. The resulting main conclusions are reviewed in this paper by collecting related theorems, algorithms, heuristics and numerical experiments. A first foundational part on determinantal structures and efficient sampling procedures is followed by four main applications: (1) a random-walk-based notion of well-distributed points in a graph, (2) a framework to describe metastable-like dynamics in finite settings by means of Markov intertwining dualities, (3) coarse graining schemes for networks and associated processes, (4) wavelets-like pyramidal algorithms for graph signals.