Pierre Vandergheynst

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Compression efficiency is mainly driven by redundancy of the overcomplete set of functions chosen for the signal decomposition. In Matching Pursuit algorithms for example, the redundancy of the dictionary influences the convergence of the residual energy. The set of functions or dictionary plays a crucial role into the non-orthogonal transform properties, and more particularly in the ability of this transform to compact the signal energy. Redundancy provides an important criteria in the design of dictionari

We present a purely group-theoretical derivation of the continuous wavelet transform (CWT) on the 2-sphere S2, based on the construction of general coherent states associated to square integrable group representations. The parameter space X of our CWT is the product of SO(3) for motions and +* for dilations on S2, which are embedded into the Lorentz group SO0(3, 1) via the Iwasawa decomposition, so that X SO0(3, 1) M Y S O L N, where N . We select an appropriate unitary representation of SO0(3, 1) acting in

Compressed Sensing (CS) and Total Variation (TV)- based iterative image reconstruction algorithms have received increased attention recently. This is due to the ability of such methods to reconstruct from limited and noisy data. Local TV methods fail to preserve texture details and fine structures, which are tedious for the method to distinguish from noise. In many cases local methods also create additional artifacts due to over smoothing. Non-Local Total Variation (NLTV) has been increasingly used for medi

In many applications - such as compression, de-noising and source separation - a good and efficient signal representation is characterized by sparsity. This means that many coefficients are close to zero, while only few ones have a non-negligible amplitude. On the other hand, real-world signals - such as audio or natural images - clearly present peculiar structures. In this paper we introduce a global optimization framework that aims at respecting the sparsity criterion while decomposing a signal over an ov

The problem of finding the missing values of a matrix given a few of its entries, called matrix completion, has gathered a lot of attention in the recent years. Although the problem is NP-hard, Candes and Recht showed that it can be exactly relaxed if the matrix is low-rank and the number of observed entries is sufficiently large. In this work, we introduce a novel matrix completion model that makes use of proximity information about rows and columns by assuming they form communities. This assumption makes

This paper presents an unequal error protection scheme for atomic image bitstreams. An atomic stream is the encoded version of a digital image, which is represented as a sum of bi-dimensional functions, as typically generated by Matching Pursuit encoders. The atomic structure of the compressed image presents an enormous advantage in terms of flexibility, since any atom of the stream can receive a different treatment, like a finely adapted protection against error. We take benefit from this property to propo

There is a growing interest on adapted signal expansions for efficient sparse approximations. For this purpose, signal expansions on over-complete bases are of high interest. Several strategies exist in order to get sparse approximations of a signal as a superposition of functions from a redundant dictionary. One of these strategies is the well known Matching Pursuit (MP). MP is an algorithm where complexity depends on the accuracy of the desired approximation. This is due to its greedy iterative nature. Fo

This paper presents a scalable video coding scheme (MP3D), based on the use of a redundant 3-D spatio-temporal dictionary of functions. The spatial component of the dictionary consists of directional and anisotropically scaled functions, which form a rich collection of visual primitives. The temporal component is tuned to capture most of the energy along motion trajectories in the video sequences. The MP3D video coding first finds motion trajectories. It then applies a spatio-temporal decomposition using an

The dynamics of neuron populations during diverse tasks often evolve on low-dimensional manifolds. However, it remains challenging to discern the contributions of geometry and dynamics for encoding relevant behavioural variables. Here, we introduce an unsupervised geometric deep learning framework for representing non-linear dynamical systems based on statistical distributions of local phase portrait features. Our method provides robust geometry-aware or geometry-agnostic representations for the unbiased co

This paper proposes a rate-distortion optimal a posteriori quantization scheme for Matching Pursuit coefficients. The a posteriori quantization applies to a Matching Pursuit expansion that has been generated off-line, and cannot benefit of any feedback loop to the encoder in order to compensate for the quantization noise. The redundancy of the Matching Pursuit dictionary provides an indicator of the relative importance of coefficients and atom indices, and subsequently on the quantization error. It is used

This paper addresses the reconstruction of high resolution omnidirectional images from a low resolution video acquired by an omnidirectional camera moving in a static scene. In order to exploit the additional information provided by the side images in the ...

We review the known construction of the continuous wavelet transform (CWT) on the two-sphere. Next we describe the construction of a CWT on the upper sheet of a two- sheeted hyperboloid, emphasizing the similarities between the two cases. Finally we give s ...

The real-time development of multi-camera systems is a great challenge. Synchronization and large data rates of the cameras adds to the complexity of these systems as well. The complexity of such system also increases as the number of their incorporating cameras increases. The customary approach to implementation of such system is a central type, where all the raw stream from the camera are first stored then processed for their target application. An alternative approach is to embed smart cameras to these s

In this paper, we introduce a framework that merges classical ideas borrowed from scale-space and multi-resolution segmentation with non-linear partial differential equations. A non-linear scale-space stack is constructed by means of an appropriate diffusion equation. This stack is analyzed and a tree of coherent segments is constructed based on relationships between different scale layers. Pruning this tree proves to be a very efficient tool for unsupervised segmentation of different classes of images (e.g

The decomposition of a signal on the sphere with the steerable wavelet constructed from the second Gaussian derivative gives access to the orientation, signed-intensity and elongation of the signal’s local features. In the present work, the non-Gaussianity ...

Graph Neural Networks (GNNs) often struggle to propagate information across long distances due to oversmoothing and oversquashing. Existing remedies such as graph transformers or rewiring typically incur high computational cost or require altering the graph structure. We introduce a Bakry-Émery graph Laplacian that integrates diffusion and advection through a learnable node-wise potential, inducing task-dependent propagation dynamics without modifying topology. This operator has a well-behaved spectral deco

Noisy 3D point clouds arise in many applications. They may be due to errors when creating a 3D model from images or simply to imprecise depth sensors. Point clouds can be given geometrical structure using graphs created from the similarity information betw ...

In this study, we address the problem of computing efficiently a dense optical flow between two images under a total variation (TV) regularization and an

L_1

norm data fidelity constraint using a variational method. We build upon Nesterov's framework for convex minimization. By keeping in memory the solution estimated at the previous iteration, this framework yields convergence rates of one order of magnitude faster than existing algorithms, hence is computationally more efficient. We show how to adapt th

Simulating the long-timescale dynamics of biomolecules is a central challenge in computational science. While enhanced sampling methods can accelerate these simulations, they rely on pre-defined collective variables that are often difficult to identify, restricting their ability to model complex switching mechanisms between metastable states. A recent generative model, LD-FPG, demonstrated that this problem could be bypassed by learning to sample the static equilibrium ensemble as all-atom deformations from

This paper describes a novel video coding scheme based on a three-dimensional Matching Pursuit algorithm. In addition to good compression performance at low bit rate, the proposed coder allows for flexible spatial, temporal and rate scalability thanks to its progressive coding structure. The Matching Pursuit algorithm generates a sparse decomposition of a video sequence in a series of spatio-temporal atoms, taken from an overcomplete dictionary of three-dimensional basis functions. The dictionary is generat

Pierre Vandergheynst

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Redundancy in Non-Orthogonal Transforms

Wavelets on the 2-Sphere: A Group-Theoretical Approach

Compressed Sensing and Adaptive Graph Total Variation for Tomographic Reconstructions

Sparse decomposition over multi-component redundant dictionaries

Matrix Completion on Graphs

Unequal Error Protection of Atomic Image Streams

A Matching Pursuit Full Search Algorithm for Image Approximations

Scalable Motion-Adaptive Video Coding with Redundant Representations

Interpretable statistical representations of neural population dynamics and geometry

A Posteriori Quantization of Progressive Matching Pursuit Streams

Plenoptic based super-resolution for omnidirectional image sequences

The continuous wavelet transform on conic sections

Interconnected Network of Cameras

Multiresolution Segmentation of Natural Images: From linear to Non-Linear Scale-Space Representations

Non-Gaussianity analysis on local morphological measures of WMAP data

Advection-Diffusion on Graphs: A Bakry-Émery Laplacian for Spectral Graph Neural Networks

Graph-based denoising for time-varying point clouds

An optimal first-order solver for the TV-$L_1$ optical flow problem

Beyond Ensembles: Simulating All-Atom Protein Dynamics in a Learned Latent Space

MP3D: Highly Scalable Video Coding Scheme Based on Matching Pursuit