arXiv subjects

Béatrice Vedel

Publications and source records attributed to Béatrice Vedel.

14 recordsLinked to original sources

arXiv2026

The p-spectrum of Random Wavelet Series

The goal of multifractal analysis is to characterize variations in local regularity of functions by computing the Hausdorff dimension of sets of points sharing the same regularity. While classical approaches rely on Hölder exponents and are restricted to locally bounded functions, $p$-exponents extend this framework to functions locally in $L^p$ and allow one to describe negative regularities. We establish a wavelet-based upper bound for the $p$-spectrum in terms of the asymptotic distribution of wavelet coefficients, extending the classical Hölder case. We then compute the exact $p$-spectrum of \textit{Random Wavelet Series} and show that, for non-negative regularities, this bound is sharp and is also attained by a prevalent set of functions with a prescribed wavelet statistic. Finally, we show that a different phenomenon occurs for negative regularities: contrary to the classical Hölder case, Random Wavelet Series do not in general realize the maximal $p$-spectrum compatible with a prescribed distribution of wavelet coefficients.

math.CA↗

arXiv2026

Material Decomposition in Photon-Counting Computed Tomography with Diffusion Models: Comparative Study and Hybridization with Variational Regularizers

Photon-counting computed tomography (PCCT) has emerged as a promising imaging technique, enabling spectral imaging and material decomposition (MD). However, images typically suffer from a low signal-to-noise ratio (SNR) due to constraints such as low photon counts and sparse-view settings which provoke artifacts. To prevent this, variational methods minimize a data-fit function coupled with handcrafted regularizers that mimic a prior by enforcing image properties such as gradient sparsity. In the last few years, diffusion models (DMs) have become predominant in the field of generative models and have been used as a learned prior for image reconstruction. This work investigates the use of DMs as regularizers for MD tasks in PCCT, specifically using diffusion posterior sampling (DPS) guidance. Three DPS-based approaches -- image-domain two-step DPS (im-TDPS), projection-domain two-step DPS (proj-TDPS), and one-step DPS (ODPS) -- are evaluated. The first two methods achieve MD in two steps by performing reconstruction and MD separately. The last method, ODPS, samples the material images directly from the measurement data. The results indicate that ODPS achieves superior performance compared to im-TDPS and proj-TDPS, providing sharper, noise-free and crosstalk-free images. Furthermore, we introduce a novel hybrid method for scenarios involving materials absent from the training dataset which combines DM priors with standard variational handcrafted regularizers for the materials unknown to the DM. This hybrid method demonstrates improved MD quality compared to a standard variational method and does not require additional training of the DM neural network (NN).

physics.med-ph↗

arXiv2025

Weighted tensorized fractional Brownian textures

This paper presents a new model of textures, obtained as realizations of a new class of fractional Brownian fields. These fields, called weighted tensorized fractional Brownian fields, are obtained by a relaxation of the tensor-product structure that appears in the definition of fractional Brownian sheets. Statistical properties such as self-similarity, stationarity of rectangular increments and regularity properties are obtained. An operator scaling extension is defined and we provide simulations of the fields using their spectral representation.

math.PR↗

arXiv2025

Constructing self-similar subsets within the fractal support of Lacunary Wavelet Series for their multifractal analysis

Given a fractal $\mathcal{I}$ whose Hausdorff dimension matches with the upper-box dimension, we propose a new method which consists in selecting inside $\mathcal{I}$ some subsets (called quasi-Cantor sets) of almost same dimension and with controled properties of self-similarties at prescribed scales. It allows us to estimate below the Hausdorff dimension $\mathcal{I}$ intersected to limsup sets of contracted balls selected according a Bernoulli law, in contexts where classical Mass Transference Principles cannot be applied. We apply this result to the computation of the increasing multifractal spectrum of lacunary wavelet series supported on $\mathcal{I}$.

math.CA↗

arXiv2024

Regularity of Weighted Tensorized Fractional Brownian Fields and associated function spaces

We investigate a new class of self-similar fractional Brownian fields, called Weighted Tensorized Fractional Brownian Fields (WTFBS). These fields, introduced in the companion paper \cite{ELLV}, generalize the well-known fractional Brownian sheet (FBs) by relaxing its tensor-product structure, resulting in new self-similar Gaussian fields with stationary rectangular increments that differ from the FBs. We analyze the local regularity properties of these fields and introduce a new concept of regularity through the definition of Weighted Tensorized Besov Spaces. These spaces combine aspects of mixed dominating smoothness spaces and hyperbolic Besov spaces, which are similar in structure to classical Besov spaces. We provide a detailed characterization of these spaces using Littlewood-Paley theory and hyperbolic wavelet analysis.

math.PR↗

arXiv2024

Spectral CT Two-step and One-step Material Decomposition using Diffusion Posterior Sampling

This paper proposes a novel approach to spectral computed tomography (CT) material decomposition that uses the recent advances in generative diffusion models (DMs) for inverse problems. Spectral CT and more particularly photon-counting CT (PCCT) can perform transmission measurements at different energy levels which can be used for material decomposition. It is an ill-posed inverse problem and therefore requires regularization. DMs are a class of generative model that can be used to solve inverse problems via diffusion posterior sampling (DPS). In this paper we adapt DPS for material decomposition in a PCCT setting. We propose two approaches, namely Two-step Diffusion Posterior Sampling (TDPS) and One-step Diffusion Posterior Sampling (ODPS). Early results from an experiment with simulated low-dose PCCT suggest that DPSs have the potential to outperform state-of-the-art model-based iterative reconstruction (MBIR). Moreover, our results indicate that TDPS produces material images with better peak signal-to-noise ratio (PSNR) than images produced with ODPS with similar structural similarity (SSIM).

physics.med-ph↗

arXiv2024

Lacunary wavelet series on Cantor sets

We provide a multifractal analysis of lacunary wavelet series on Cantor sets. Byintroducing a desynchronization between the scales of the wavelets and the scales of the steps of the construction of the Cantor set, we obtain random processes that donot satisfy multifractal formalisms based on the Legendre transform and on the large deviation of wavelet leaders. Subsequently, we show how the computation of the leader large deviation spectra of "lacunarized" versions of such wavelet series can detect the failure in the multifractal formalism using only numerical quantities.

math.PR↗

arXiv2024

Diffusion Posterior Sampling for Synergistic Reconstruction in Spectral Computed Tomography

Using recent advances in generative artificial intelligence (AI) brought by diffusion models, this paper introduces a new synergistic method for spectral computed tomography (CT) reconstruction. Diffusion models define a neural network to approximate the gradient of the log-density of the training data, which is then used to generate new images similar to the training ones. Following the inverse problem paradigm, we propose to adapt this generative process to synergistically reconstruct multiple images at different energy bins from multiple measurements. The experiments suggest that using multiple energy bins simultaneously improves the reconstruction by inverse diffusion and outperforms state-of-the-art synergistic reconstruction techniques.

physics.med-ph↗

arXiv2023

Uconnect: Synergistic Spectral CT Reconstruction with U-Nets Connecting the Energy bins

Spectral computed tomography (CT) offers the possibility to reconstruct attenuation images at different energy levels, which can be then used for material decomposition. However, traditional methods reconstruct each energy bin individually and are vulnerable to noise. In this paper, we propose a novel synergistic method for spectral CT reconstruction, namely Uconnect. It utilizes trained convolutional neural networks (CNNs) to connect the energy bins to a latent image so that the full binned data is used synergistically. We experiment on two types of low-dose data: simulated and real patient data. Qualitative and quantitative analysis show that our proposed Uconnect outperforms state-of-art model-based iterative reconstruction (MBIR) techniques as well as CNN-based denoising.

physics.med-ph↗

arXiv2023

Regularity properties of random wavelet series

We study the regularity properties of random wavelet series constructed by multiplying the coefficients of a deterministic wavelet series with unbounded I.I.D. random variables. In particular, we show that, at the opposite to what happens for Fourier series, the randomization of almost every continuous function gives an almost surely nowhere locally bounded function.

math.PR↗

arXiv2013

Non-Local means est un algorithme de débruitage local (Non-Local means is a local image denoising algorithm)

The Non-Local Means (NLM) image denoising algorithm pushed the limits of denoising. But it introduced a new paradigm, according to which one could capture the similarity of images with the NLM weights. We show that, contrary to the prevailing opinion, the NLM weights do not allow to get a reliable measure of the similarity in a noisy image, unless one add a locality constraint. As an image denoising method, the Non-Local Means prove to be local. Some works had already pointed out that to get the best denoising performances with the NLM algorithm, one should run it locally. But no general conclusion has been yet proposed and the only explanation that was proposed to justify the experimental results is not sufficient. Our study based on experimental evidence proves that, on average on natural images, the bias of the NLM estimator is an increasing function of the radius of the similarity searching zone. The reason for this phenomenon is that noise disrupts the order of similarity between patches. Hence the mean squared error between the original image and the NLM estimation, which is the sum of the bias, the variance and the covariance of the estimator, has an absolute minimum for a disk of radius 3 to 4 pixels.

math.ST↗

arXiv2013

Un algorithme de débruitage Non-Local means espace-fréquence (A space-frequency Non-Local means image denoising algorithm)

The efficiency of the Non-Local means (NLM) image denoising algorithm relies on the identification of similar original pixels from noisy similar patches. Hence fine details and low-contrasted structures are badly recovered after the application of NLM. But as these structures tend to correspond to redundant ones in the Fourier domain, NLM filtering in this domain allows one to better denoise them. A mixed space-frequency approach improves the denoising performances of NLM because it ensures that the information is redundant enough, in the spatial domain or in the frequency domain. Our approach is simple : it consists in running two times the NLM algorithm (firstly in the frequency domain and secondly in the spatial domain). For fine textures and isolated points we get a better visual reconstruction than with the original NLM. In terms of PSNR, the improvement can be over 1 dB. Our approach gives intermediate results between the original NLM and state-of-the-art methods while at the same time having moderate complexity and leading to few visual artifacts.

math.FA↗

arXiv2012

Hyperbolic wavelet transform: an efficient tool for multifractal analysis of anisotropic textures

Global and local regularities of functions are analyzed in anisotropic function spaces, under a common framework, that of hyperbolic wavelet bases. Local and directional regularity features are characterized by means of global quantities constructed upon the coefficients of hyperbolic wavelet decompositions. A multifractal analysis is introduced, that jointly accounts for scale invariance and anisotropy. Its properties are studied in depth.

math.FA↗

arXiv2011

Explicit constructions of operator scaling Gaussian fields

We propose an explicit way to generate a large class of Operator scaling Gaussian random fields (OSGRF). Such fields are anisotropic generalizations of selfsimilar fields. More specifically, we are able to construct any Gaussian field belonging to this class with given Hurst index and exponent. Our construction provides - for simulations of texture as well as for detection of anisotropies in an image - a large class of models with controlled anisotropic geometries and structures.

math.PR↗