Making Laplacians commute

In this paper, we construct multimodal spectral geometry by finding a pair of closest commuting operators (CCO) to a given pair of Laplacians. The CCOs are jointly diagonalizable and hence have the same eigenbasis. Our construction naturally extends classical data analysis tools based on spectral geometry, such as diffusion maps and spectral clustering. We provide several synthetic and real examples of applications in dimensionality reduction, shape analysis, and clustering, demonstrating that our method better captures the inherent structure of multi-modal data

Spectral Graph Transformer Networks for Brain Surface Parcellation

The analysis of the brain surface modeled as a graph mesh is a challenging task. Conventional deep learning approaches often rely on data lying in the Euclidean space. As an extension to irregular graphs, convolution operations are defined in the Fourier or spectral domain. This spectral domain is obtained by decomposing the graph Laplacian, which captures relevant shape information. However, the spectral decomposition across different brain graphs causes inconsistencies between the eigenvectors of individual spectral domains, causing the graph learning algorithm to fail. Current spectral graph convolution methods handle this variance by separately aligning the eigenvectors to a reference brain in a slow iterative step. This paper presents a novel approach for learning the transformation matrix required for aligning brain meshes using a direct data-driven approach. Our alignment and graph processing method provides a fast analysis of brain surfaces. The novel Spectral Graph Transformer (SGT) network proposed in this paper uses very few randomly sub-sampled nodes in the spectral domain to learn the alignment matrix for multiple brain surfaces. We validate the use of this SGT network along with a graph convolution network to perform cortical parcellation. Our method on 101 manually-labeled brain surfaces shows improved parcellation performance over a no-alignment strategy, gaining a significant speed (1400 fold) over traditional iterative alignment approaches.Comment: Equal contribution of R. He and K. Gopinat

Data-driven shape analysis and processing

Data-driven methods serve an increasingly important role in discovering geometric, structural, and semantic relationships between shapes. In contrast to traditional approaches that process shapes in isolation of each other, data-driven methods aggregate information from 3D model collections to improve the analysis, modeling and editing of shapes. Through reviewing the literature, we provide an overview of the main concepts and components of these methods, as well as discuss their application to classification, segmentation, matching, reconstruction, modeling and exploration, as well as scene analysis and synthesis. We conclude our report with ideas that can inspire future research in data-driven shape analysis and processing

From spline wavelet to sampling theory on circulant graphs and beyond– conceiving sparsity in graph signal processing

Graph Signal Processing (GSP), as the field concerned with the extension of classical signal processing concepts to the graph domain, is still at the beginning on the path toward providing a generalized theory of signal processing. As such, this thesis aspires to conceive the theory of sparse representations on graphs by traversing the cornerstones of wavelet and sampling theory on graphs. Beginning with the novel topic of graph spline wavelet theory, we introduce families of spline and e-spline wavelets, and associated filterbanks on circulant graphs, which lever- age an inherent vanishing moment property of circulant graph Laplacian matrices (and their parameterized generalizations), for the reproduction and annihilation of (exponen- tial) polynomial signals. Further, these families are shown to provide a stepping stone to generalized graph wavelet designs with adaptive (annihilation) properties. Circulant graphs, which serve as building blocks, facilitate intuitively equivalent signal processing concepts and operations, such that insights can be leveraged for and extended to more complex scenarios, including arbitrary undirected graphs, time-varying graphs, as well as associated signals with space- and time-variant properties, all the while retaining the focus on inducing sparse representations. Further, we shift from sparsity-inducing to sparsity-leveraging theory and present a novel sampling and graph coarsening framework for (wavelet-)sparse graph signals, inspired by Finite Rate of Innovation (FRI) theory and directly building upon (graph) spline wavelet theory. At its core, the introduced Graph-FRI-framework states that any K-sparse signal residing on the vertices of a circulant graph can be sampled and perfectly reconstructed from its dimensionality-reduced graph spectral representation of minimum size 2K, while the structure of an associated coarsened graph is simultaneously inferred. Extensions to arbitrary graphs can be enforced via suitable approximation schemes. Eventually, gained insights are unified in a graph-based image approximation framework which further leverages graph partitioning and re-labelling techniques for a maximally sparse graph wavelet representation.Open Acces