In applications such as social, energy, transportation, sensor, and neuronal networks, high-dimensional data naturally reside on the vertices of weighted graphs. The emerging field of signal processing on graphs merges algebraic and spectral graph theoretic concepts with computational harmonic analysis to process such signals on graphs. In this tutorial overview, we outline the main challenges of the area, discuss different ways to define graph spectral domains, which are the analogs to the classical frequency domain, and highlight the importance of incorporating the irregular structures of graph data domains when processing signals on graphs. We then review methods to generalize fundamental operations such as filtering, translation, modulation, dilation, and downsampling to the graph setting and survey the localized, multiscale transforms that have been proposed to efficiently extract information from high-dimensional data on graphs. We conclude with a brief discussion of open issues and possible extensions.

The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains

Convolutional Neural Networks are extremely efficient architectures in image and audio recognition tasks, thanks to their ability to exploit the local translational invariance of signal classes over their domain. In this paper we consider possible generalizations of CNNs to signals defined on more general domains without the action of a translation group. In particular, we propose two constructions, one based upon a hierarchical clustering of the domain, and another based on the spectrum of the graph Laplacian. We show through experiments that for low-dimensional graphs it is possible to learn convolutional layers with a number of parameters independent of the input size, resulting in efficient deep architectures.

Spectral Networks and Locally Connected Networks on Graphs

Digital watermarking is a key ingredient to copyright protection. It provides a solution to illegal copying of digital material and has many other useful applications such as broadcast monitoring and the recording of electronic transactions. Now, for the first time, there is a book that focuses exclusively on this exciting technology. Digital Watermarking covers the crucial research findings in the field: it explains the principles underlying digital watermarking technologies, describes the requirements that have given rise to them, and discusses the diverse ends to which these technologies are being applied. As a result, additional groundwork is laid for future developments in this field, helping the reader understand and anticipate new approaches and applications.

Digital Watermarking

This paper proposes a unified and consistent set of flexible tools to approximate important geometric attributes, including normal vectors and curvatures on arbitrary triangle meshes. We present a consistent derivation of these first and second order differential properties using averaging Voronoi cells and the mixed Finite-Element/Finite-Volume method, and compare them to existing formulations. Building upon previous work in discrete geometry, these operators are closely related to the continuous case, guaranteeing an appropriate extension from the continuous to the discrete setting: they respect most intrinsic properties of the continuous differential operators. We show that these estimates are optimal in accuracy under mild smoothness conditions, and demonstrate their numerical quality. We also present applications of these operators, such as mesh smoothing, enhancement, and quality checking, and show results of denoising in higher dimensions, such as for tensor images.

/pdf/discrete-differential-geometry-operators-for-triangulated-2-w8vp3jjpy1.pdf

Discrete Differential-Geometry Operators for Triangulated 2-Manifolds

In this paper, we develop methods to rapidly remove rough features from irregularly triangulated data intended to portray a smooth surface. The main task is to remove undesirable noise and uneven edges while retaining desirable geometric features. The problem
arises mainly when creating high-fidelity computer graphics objects using imperfectly-measured data from the real world.

Our approach contains three novel features: an implicit integration method to achieve efficiency, stability, and large time-steps; a scale-dependent Laplacian operator to improve the diffusion process; and finally, a robust curvature flow operator that achieves a smoothing of the shape itself, distinct from any parameterization.
Additional features of the algorithm include automatic exact volume preservation, and hard and soft constraints on the positions of the points in the mesh.

We compare our method to previous operators and related algorithms, and prove that our curvature and Laplacian operators have several mathematically-desirable qualities that improve the appearance of the resulting surface. In consequence, the user can easily select the appropriate operator according to the desired type of fairing.
Finally, we provide a series of examples to graphically and
numerically demonstrate the quality of our results.

/pdf/implicit-fairing-of-irregular-meshes-using-diffusion-and-ywwff809ar.pdf

Implicit fairing of irregular meshes using diffusion and curvature flow

We generalize basic signal processing tools such as downsampling, upsampling, and filters to irregular connectivity triangle meshes. This is accomplished through the design of a non-uniform relaxation procedure whose weights depend on the geometry and we
show its superiority over existing schemes whose weights depend only on connectivity. This is combined with known mesh simplification methods to build subdivision and pyramid algorithms. We demonstrate the power of these algorithms through a number of application examples including smoothing, enhancement, editing, and texture mapping.

/pdf/multiresolution-signal-processing-for-meshes-zyjbixsofy.pdf

Multiresolution signal processing for meshes

Normal meshes are new fundamental surface descriptions inspired by differential geometry. A normal mesh is a multiresolution mesh where each level can be written as a normal offset from a coarser version. Hence the mesh can be stored with a single float per vertex. We present an algorithm to approximate any surface arbitrarily closely with a normal semi-regular mesh. Normal meshes can be useful in numerous applications such as compression, filtering, rendering, texturing, and modeling.

Normal meshes

Meshes obtained from laser scanner data often contain topological noise due to inaccuracies in the scanning and merging process. This topological noise complicates subsequent operations such as remeshing, parameterization and smoothing. We introduce an approach that removes unnecessary nontrivial topology from meshes. Using a local wave front traversal, we discover the local topologies of the mesh and identify features such as small tunnels. We then identify non-separating cuts along which we cut and seal the mesh, reducing the genus and thus the topological complexity of the mesh.

/pdf/topological-noise-removal-10qpcraa6i.pdf

Topological noise removal

We introduce a novel method for approximate alignment of point-based surfaces. Our approach is based on detecting a set of salient feature points using a scale-space representation. For each feature point we compute a signature vector that is approximately invariant under rigid transformations. We use the extracted signed feature set in order to obtain approximate alignment of two surfaces. We apply our method for the automatic alignment of multiple scans using both scan-to-scan and scan-to-model matching capabilities.

Multi-scale features for approximate alignment of point-based surfaces

In this article we review techniques for building and analysing wavelets on irregular point sets in one and two dimensions. We discuss current results both on the practical and theoretical side. In particular, we focus on subdivision schemes and commutation rules. Several examples are included.

/pdf/wavelets-on-irregular-point-sets-3p58lqwmde.pdf

Igor Guskov

Papers

Multiresolution signal processing for meshes

Normal meshes

Topological noise removal

Multi-scale features for approximate alignment of point-based surfaces

Wavelets on irregular point sets