arXiv subjects

D. Van De Ville

Publications and source records attributed to D. Van De Ville.

5 recordsLinked to original sources

arXiv2026

Graph Diffusion-Advection Operator for Directed Graph Signal Processing

Graph signal processing (GSP) provides a framework for analyzing data on irregular domains, with applications in neuroscience, finance, chemistry, and social sciences. Classical GSP primarily models symmetric relationships using undirected graphs, yet many real-world systems exhibit asymmetric interactions, motivating extensions to directed graphs. Central to directed GSP is the graph shift operator, typically defined via the directed graph Laplacian. Building on the well-known link between the undirected graph Laplacian and the diffusion operator, we establish a correspondence between the directed graph Laplacian and the diffusion-advection operator. This perspective opens new avenues for addressing crucial points such as frequency ordering, smoothness definition, and the design of spectral and graph filters. Specifically, we introduce two new orderings of frequencies based on the modulus and argument of the eigenvalues, naturally leading to new definitions of smoothness. Then we present two kernels reflecting diffusive and advective processes, namely the heat and transport kernels, respectively. Finally, we propose novel graph filters obtained by composing diffusive and advective parts, which approximate ideal spectral filters accurately and characterize the evolution of graph signals in richer ways. All aforementioned developments are illustrated on both synthetic and real graphs, including an application to temperature graph and signals.

eess.SP↗

arXiv2013

Sparse image reconstruction on the sphere: implications of a new sampling theorem

We study the impact of sampling theorems on the fidelity of sparse image reconstruction on the sphere. We discuss how a reduction in the number of samples required to represent all information content of a band-limited signal acts to improve the fidelity of sparse image reconstruction, through both the dimensionality and sparsity of signals. To demonstrate this result we consider a simple inpainting problem on the sphere and consider images sparse in the magnitude of their gradient. We develop a framework for total variation (TV) inpainting on the sphere, including fast methods to render the inpainting problem computationally feasible at high-resolution. Recently a new sampling theorem on the sphere was developed, reducing the required number of samples by a factor of two for equiangular sampling schemes. Through numerical simulations we verify the enhanced fidelity of sparse image reconstruction due to the more efficient sampling of the sphere provided by the new sampling theorem.

cs.IT↗

arXiv2011

Implications for compressed sensing of a new sampling theorem on the sphere

A sampling theorem on the sphere has been developed recently, requiring half as many samples as alternative equiangular sampling theorems on the sphere. A reduction by a factor of two in the number of samples required to represent a band-limited signal on the sphere exactly has important implications for compressed sensing, both in terms of the dimensionality and sparsity of signals. We illustrate the impact of this property with an inpainting problem on the sphere, where we show the superior reconstruction performance when adopting the new sampling theorem compared to the alternative.

cs.IT↗

arXiv2011

Sampling theorems and compressive sensing on the sphere

We discuss a novel sampling theorem on the sphere developed by McEwen & Wiaux recently through an association between the sphere and the torus. To represent a band-limited signal exactly, this new sampling theorem requires less than half the number of samples of other equiangular sampling theorems on the sphere, such as the canonical Driscoll & Healy sampling theorem. A reduction in the number of samples required to represent a band-limited signal on the sphere has important implications for compressive sensing, both in terms of the dimensionality and sparsity of signals. We illustrate the impact of this property with an inpainting problem on the sphere, where we show superior reconstruction performance when adopting the new sampling theorem.

cs.IT↗

arXiv2011

Harmonic analysis of spherical sampling in diffusion MRI

In the last decade diffusion MRI has become a powerful tool to non-invasively study white-matter integrity in the brain. Recently many research groups have focused their attention on multi-shell spherical acquisitions with the aim of effectively mapping the diffusion signal with a lower number of q-space samples, hence enabling a crucial reduction of acquisition time. One of the quantities commonly studied in this context is the so-called orientation distribution function (ODF). In this setting, the spherical harmonic (SH) transform has gained a great deal of popularity thanks to its ability to perform convolution operations efficiently and accurately, such as the Funk-Radon transform notably required for ODF computation from q-space data. However, if the q-space signal is described with an unsuitable angular resolution at any b-value probed, aliasing (or interpolation) artifacts are unavoidably created. So far this aspect has been tackled empirically and, to our knowledge, no study has addressed this problem in a quantitative approach. The aim of the present work is to study more theoretically the efficiency of multi-shell spherical sampling in diffusion MRI, in order to gain understanding in HYDI-like approaches, possibly paving the way to further optimization strategies.

q-bio.QM↗