Gabriel Peyré

Surface compression with geometric bandelets

This paper describes the construction of second generation bandelet bases and their application to 3D geometry compression. This new coding scheme is orthogonal and the corresponding basis functions are regular. In our method, surfaces are decomposed in a bandelet basis with a fast bandeletization algorithm that removes the geometric redundancy of orthogonal wavelet coefficients. The resulting transform coding scheme has an error decay that is asymptotically optimal for geometrically regular surfaces. We then use these bandelet bases to perform geometry image and normal map compression. Numerical tests show that for complex surfaces bandelets bring an improvement of 1.5dB to 2dB over state of the art compression schemes.

Sparse Modeling of Textures

This paper presents a generative model for textures that uses a local sparse description of the image content. This model enforces the sparsity of the expansion of local texture patches on adapted atomic elements. The analysis of a given texture within this framework performs the sparse coding of all the patches of the texture into the dictionary of atoms. Conversely, the synthesis of a new texture is performed by solving an optimization problem that seeks for a texture whose patches are sparse in the dictionary. This paper explores several strategies to choose this dictionary. A set of hand crafted dictionaries composed of edges, oscillations, lines or crossings elements allows to synthesize synthetic images with geometric features. Another option is to define the dictionary as the set of all the patches of an input exemplar. This leads to computer graphics methods for synthesis and shares some similarities with non-local means filtering. The last method we explore learns the dictionary by an optimization process that maximizes the sparsity of a set of exemplar patches. Applications of all these methods to texture synthesis, inpainting and classification shows the efficiency of the proposed texture model.

Best Basis Compressed Sensing

This paper proposes a best basis extension of compressed sensing recovery. Instead of regularizing the compressed sensing inverse problem with a sparsity prior in a fixed basis, our framework makes use of sparsity in a tree-structured dictionary of orthogonal bases. A new iterative thresholding algorithm performs both the recovery of the signal and the estimation of the best basis. The resulting reconstruction from compressive measurements optimizes the basis to the structure of the sensed signal. Adaptivity is crucial to capture the regularity of complex natural signals. Numerical experiments on sounds and geometrical images indeed show that this best basis search improves the recovery with respect to fixed sparsity priors.

Manifold models for signals and images

This article proposes a new class of models for natural signals and images. These models constrain the set of patches extracted from the data to analyze to be close to a low-dimensional manifold. This manifold structure is detailed for various ensembles suitable for natural signals, images and textures modeling. These manifolds provide a low-dimensional parameterization of the local geometry of these datasets. These manifold models can be used to regularize inverse problems in signal and image processing. The restored signal is represented as a smooth curve or surface traced on the manifold that matches the forward measurements. A manifold pursuit algorithm computes iteratively a solution of the manifold regularization problem. Numerical simulations on inpainting and compressive sensing inversion show that manifolds models bring an improvement for the recovery of data with geometrical features.

A review of Bandlet methods for geometrical image representation

This article reviews bandlet approaches to geometric image representations. Orthogonal bandlets using an adaptive segmentation and a local geometric flow well suited to capture the anisotropic regularity of edge structures. They are constructed with a “bandletization” which is a local orthogonal transformation applied to wavelet coefficients. The approximation in these bandlet bases exhibits an asymptotically optimal decay for images that are regular outside a set of regular edges. These bandlets can be used to perform image compression and noise removal. More flexible orthogonal bandlets with less vanishing moments are constructed with orthogonal grouplets that group wavelet coefficients alon a multiscale association field. Applying a translation invariant grouplet transform over a translation invariant wavelet frame leads to state of the art results for image denoising and super-resolution.

Image Processing with Non-local Spectral Bases

This article studies regularization schemes that are defined using a lifting of the image pixels in a high dimensional space. For some specific classes of geometric images, this discrete set of points is sampled along a low dimensional smooth manifold. The construction of differential operators on this lifted space allows one to compute PDE flows and perform variational optimizations. All these schemes lead to regularizations that exploit the manifold structure of the lifted image. Depending on the specific definition of the lifting, one recovers several well-known semi-local and non-local denoising algorithms that can be interpreted as local estimators over a semi-local or a non-local manifold. This framework also allows one to define thresholding operators in adapted orthogonal bases. These bases are eigenvectors of the discrete Laplacian on a manifold adapted to the geometry of the image. Numerical results compare the efficiency of PDE flows, energy minimizations and thresholdings in the semi-local and non-local settings. The superiority of the non-local computations is studied through the performance of non-linear approximation in orthogonal bases.

Optimal Transport with Proximal Splitting

This article reviews the use of first order convex optimization schemes to solve the discretized dynamic optimal transport problem, initially proposed by Benamou and Brenier. We develop a staggered grid discretization that is well adapted to the computation of the

Texture Synthesis with Grouplets

L^2

Surface segmentation using geodesic centroidal tesselation

optimal transport geodesic between distributions defined on a uniform spatial grid. We show how proximal splitting schemes can be used to solve the resulting large scale convex optimization problem. A specific instantiation of this method on a centered grid corresponds to the initial algorithm developed by Benamou and Brenier. We also show how more general cost functions can be taken into account and how to extend the method to perform optimal transport on a Riemannian manifold.

Learning the Morphological Diversity

This paper proposes a new method to synthesize and inpaint geometric textures. The texture model is composed of a geometric layer that drives the computation of a new grouplet transform. The geometry is an orientation flow that follows the patterns of the texture to analyze or synthesize. The grouplet transform extends the original construction of Mallat and is adapted to the modeling of natural textures. Each grouplet atoms is an elongated stroke located along the geometric flow. These atoms exhibit a wide range of lengths and widths, which is important to match the variety of structures present in natural images. Statistical modeling and sparsity optimization over these grouplet coefficients enable the synthesis of texture patterns along the flow. This paper explores texture inpainting and texture synthesis, which both require the joint optimization of the geometric flow and the grouplet coefficients.