On the optimality of shape and data representation in the spectral domain

A proof of the optimality of the eigenfunctions of the Laplace-Beltrami operator (LBO) in representing smooth functions on surfaces is provided and adapted to the field of applied shape and data analysis. It is based on the Courant-Fischer min-max principle adapted to our case. % The theorem we present supports the new trend in geometry processing of treating geometric structures by using their projection onto the leading eigenfunctions of the decomposition of the LBO. Utilisation of this result can be used for constructing numerically efficient algorithms to process shapes in their spectrum. We review a couple of applications as possible practical usage cases of the proposed optimality criteria. % We refer to a scale invariant metric, which is also invariant to bending of the manifold. This novel pseudo-metric allows constructing an LBO by which a scale invariant eigenspace on the surface is defined. We demonstrate the efficiency of an intermediate metric, defined as an interpolation between the scale invariant and the regular one, in representing geometric structures while capturing both coarse and fine details. Next, we review a numerical acceleration technique for classical scaling, a member of a family of flattening methods known as multidimensional scaling (MDS). There, the optimality is exploited to efficiently approximate all geodesic distances between pairs of points on a given surface, and thereby match and compare between almost isometric surfaces. Finally, we revisit the classical principal component analysis (PCA) definition by coupling its variational form with a Dirichlet energy on the data manifold. By pairing the PCA with the LBO we can handle cases that go beyond the scope defined by the observation set that is handled by regular PCA

Analysis of surface folding patterns of diccols using the GPU-Optimized geodesic field estimate

Localization of cortical regions of interests (ROIs) in the human brain via analysis of Diffusion Tensor Imaging (DTI) data plays a pivotal role in basic and clinical neuroscience. In recent studies, 358 common cortical landmarks in the human brain, termed as Dense Indi- vidualized and Common Connectivity-based Cortical Landmarks (DICCCOLs), have been identified. Each of these DICCCOL sites has been observed to possess fiber connection patterns that are consistent across individuals and populations and can be regarded as predictive of brain function. However, the regularity and variability of the cortical surface fold patterns at these DICCCOL sites have, thus far, not been investigated. This paper presents a novel approach, based on intrinsic surface geometry, for quantitative analysis of the regularity and variability of the cortical surface folding patterns with respect to the structural neural connectivity of the human brain. In particular, the Geodesic Field Estimate (GFE) is used to infer the relationship between the structural and connectional DTI features and the complex surface geometry of the human brain. A parallel algorithm, well suited for implementation on Graphics Processing Units (GPUs), is also proposed for efficient computation of the shortest geodesic paths between all cortical surface point pairs. Based on experimental results, a mathematical model for the morphological variability and regularity of the cortical folding patterns in the vicinity of the DICCCOL sites is proposed. It is envisioned that this model could be potentially applied in several human brain image registration and brain mapping applications

Geometric and Photometric Data Fusion in Non-Rigid Shape Analysis

In this paper, we explore the use of the diffusion geometry framework for the fusion of geometric and photometric information in local and global shape descriptors. Our construction is based on the definition of a diffusion process on the shape manifold embedded into a high-dimensional space where the embedding coordinates represent the photometric information. Experimental results show that such data fusion is useful in coping with different challenges of shape analysis where pure geometric and pure photometric methods fai

PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures

Persistence diagrams, the most common descriptors of Topological Data Analysis, encode topological properties of data and have already proved pivotal in many different applications of data science. However, since the (metric) space of persistence diagrams is not Hilbert, they end up being difficult inputs for most Machine Learning techniques. To address this concern, several vectorization methods have been put forward that embed persistence diagrams into either finite-dimensional Euclidean space or (implicit) infinite dimensional Hilbert space with kernels. In this work, we focus on persistence diagrams built on top of graphs. Relying on extended persistence theory and the so-called heat kernel signature, we show how graphs can be encoded by (extended) persistence diagrams in a provably stable way. We then propose a general and versatile framework for learning vectorizations of persistence diagrams, which encompasses most of the vectorization techniques used in the literature. We finally showcase the experimental strength of our setup by achieving competitive scores on classification tasks on real-life graph datasets