Laplace-Beltrami Research Articles

Spectral analysis of causal dynamical triangulations via finite element method

We examine the dual graph representation of simplicial manifolds in causal dynamical triangulations (CDT) as a means to build observables and propose a new representation based on the finite element methods (FEM). In particular, with the application of FEM techniques, we extract the (low-lying) spectrum of the Laplace-Beltrami (LB) operator on the Sobolev space ${H}^{1}$ of scalar functions on piecewise flat manifolds and compare them with corresponding results obtained by using the dual graph representation. We show that, except for nonpathological cases in two dimensions, the dual graph spectrum and spectral dimension do not generally agree, neither quantitatively nor qualitatively, with the ones obtained from the LB operator on the continuous space. We analyze the reasons for this discrepancy and discuss its possible implications on the definition of generic observables built from the dual graph representation.

Global and local shape features of the hippocampus based on Laplace-Beltrami eigenvalues and eigenfunctions: a potential application in the lateralization of temporal lobe epilepsy.

Using magnetic resonance (MR) images to evaluate changes in the shape of the hippocampus has been an active research topic. This paper presents a new shape analysis approach to quantify and visualize deformations of the hippocampus in epilepsy. The proposed method is based on Laplace-Beltrami (LB) eigenvalues and eigenfunctions as isometric invariant shape features, and thus, the procedure does not require any image registration. In addition to the LB-based shape features, total hippocampal volume and surface area are calculated using manually segmented images. Theses shape and volumetric descriptors are used to distinguish the patients with temporal lobe epilepsy (TLE) (N = 55) from healthy control subjects (N = 12, age = 32.2 ± 9.1, sex (M/F) = 6/6) and patients with right TLE (N = 26, age = 45.1 ± 11.0, sex (M/F) = 9/17) from left TLE (N = 29, age = 45.4 ± 11.9, sex (M/F) = 10/19). Experimental results illustrate the usefulness of the proposed approach for the diagnosis and lateralization of TLE with 93.0% and 86.4% of the cases, respectively. Moreover, the proposed method outperforms the volumetric analysis in terms of both sensitivity (94.9% vs. 88.1%) and specificity (83.3% vs. 50.0%) of the lateralization. The analysis of local hippocampal thickness variations suggests significant deformation in both ipsilateral and contralateral hippocampi of epileptic patients, while there were no differences between right and left hippocampi in controls. It is anticipated that the proposed method could be advantageous in the presurgical evaluation of patients with drug-resistant epilepsy; however, further validation of the method using a larger dataset is required.

A registration-free approach for statistical process control of 3D scanned objects via FEM

Recent work in on-line Statistical Process Control (SPC) of manufactured 3-dimensional (3-D) objects has been proposed based on the estimation of the spectrum of the Laplace-Beltrami (LB) operator, a differential operator that encodes the geometrical features of a manifold and is widely used in Machine Learning (i.e., Manifold Learning). The resulting spectra are an intrinsic geometrical feature of each part, and thus can be compared between parts to assess the state of the manufacturing process while avoiding part to part registration (or “part localization”) pre-processing or the need for equal size meshes, characteristics which are required in previous approaches for SPC of 3D parts. The recent spectral SPC methods, however, are limited to monitoring surface data from objects such that the scanned meshes have no boundaries, holes or missing portions. In this paper we extend spectral methods by first considering a more accurate and general estimator of the LB spectrum that is obtained by application of Finite Element Methods (FEM) to the solution of Helmholtz's equation with boundaries. It is shown how the new spectral FEM approach, while it retains the advantages of not requiring part localization/registration or equal size datasets scanned from each part, it provides more accurate spectrum estimates, which results in faster detection of out of control conditions than earlier methods, can be applied to both mesh or volumetric (solid) scans, and furthermore, it is shown how it can be applied to partial scans that result in open meshes (surface or volumetric) with boundaries, increasing the practical applicability of the methods. The present work brings SPC methods closer to contemporary research in Computer Graphics and Manifold Learning. MATLAB code that reproduces the examples of this paper is provided in the supplementary materials.

Nonisometric Surface Registration via Conformal Laplace–Beltrami Basis Pursuit

Surface registration is one of the most fundamental problems in geometry processing. Many approaches have been developed to tackle this problem in cases where the surfaces are nearly isometric. However, it is much more challenging to compute correspondence between surfaces which are intrinsically less similar. In this paper, we propose a variational model to align the Laplace-Beltrami (LB) eigensytems of two non-isometric genus zero shapes via conformal deformations. This method enables us to compute geometrically meaningful point-to-point maps between non-isometric shapes. Our model is based on a novel basis pursuit scheme whereby we simultaneously compute a conformal deformation of a ’target shape’ and its deformed LB eigensystem. We solve the model using a proximal alternating minimization algorithm hybridized with the augmented Lagrangian method which produces accurate correspondences given only a few landmark points. We also propose a re-initialization scheme to overcome some of the difficulties caused by the non-convexity of the variational problem. Intensive numerical experiments illustrate the effectiveness and robustness of the proposed method to handle non-isometric surfaces with large deformation with respect to both noises on the underlying manifolds and errors within the given landmarks or feature functions.

An Intrinsic Geometrical Approach for Statistical Process Control of Surface and Manifold Data

We present a new method for statistical process control (SPC) of a discrete part manufacturing system based on intrinsic geometrical properties of the parts, estimated from three-dimensional sensor data. An intrinsic method has the computational advantage of avoiding the difficult part registration problem, necessary in previous SPC approaches of three-dimensional geometrical data, but inadequate if noncontact sensors are used. The approach estimates the spectrum of the Laplace–Beltrami (LB) operator of the scanned parts and uses a multivariate nonparametric control chart for online process control. Our proposal brings SPC closer to computer vision and computer graphics methods aimed to detect large differences in shape (but not in size). However, the SPC problem differs in that small changes in either shape or size of the parts need to be detected, keeping a controllable false alarm rate and without completely filtering noise. An online or “Phase II” method and a scheme for starting up in the absence of prior data (“Phase I”) are presented. Comparison with earlier approaches that require registration shows the LB spectrum method to be more sensitive to rapidly detect small changes in shape and size, including the practical case when the sequence of part datasets is in the form of large, unequal size meshes. A post-alarm diagnostic method to investigate the location of defects on the surface of a part is also presented. While we focus in this article on surface (triangulation) data, the methods can also be applied to point cloud and voxel metrology data.

Data-driven metamaterial design with Laplace-Beltrami spectrum as “shape-DNA”

The design of multiscale metamaterial systems often suffers from high computational cost and incompatible boundaries between unit cells. As a result, unit cells are either assumed to be repeated (periodic) everywhere or limited to a small number of shapes. To address these limitations, this work proposes a data-driven design framework consisting of a metamaterial genome with a reduced-order geometrical representation as well as methods for the efficient design and analysis of 2D aperiodic metamaterials with compatible boundaries. To collect a large amount of designs, a set of unit cells generated by topology optimization is taken as initial seeds for the genome, and then expanded iteratively through random shape perturbations to form a rich database that covers a wide range of properties. For a reduced-order representation, the Laplace-Beltrami (LB) spectrum is adopted to describe complex unit cell shapes using a low number of descriptors, therefore significantly reducing the design dimensionality. Moreover, the physical and geometrical information contained in the LB spectrum is revealed through both quantitative and theoretical analysis. This information as well as the lower dimensionality allows the genome to be effectively leveraged to build a neural network model of structure-property relations for the rapid design of new unit cells. Finally, the combination of the metamaterial genome with an efficient optimization method based on the Markov random field (MRF) model is proposed to ensure connected boundaries between unit cells in multiscale aperiodic microstructure designs.

Temporal non-local means filtering for studies of intrinsic brain connectivity from individual resting fMRI.

Characterizing functional brain connectivity using resting functional magnetic resonance imaging (fMRI) is challenging due to the relatively small Blood-Oxygen-Level Dependent contrast and low signal-to-noise ratio. Denoising using surface-based Laplace-Beltrami (LB) or volumetric Gaussian filtering tends to blur boundaries between different functional areas. To overcome this issue, a time-based Non-Local Means (tNLM) filtering method was previously developed to denoise fMRI data while preserving spatial structure. The kernel and parameters that define the tNLM filter need to be optimized for each application. Here we present a novel Global PDF-based tNLM filtering (GPDF) algorithm that uses a data-driven kernel function based on a Bayes factor to optimize filtering for spatial delineation of functional connectivity in resting fMRI data. We demonstrate its performance relative to Gaussian spatial filtering and the original tNLM filtering via simulations. We also compare the effects of GPDF filtering against LB filtering using individual in-vivo resting fMRI datasets. Our results show that LB filtering tends to blur signals across boundaries between adjacent functional regions. In contrast, GPDF filtering enables improved noise reduction without blurring adjacent functional regions. These results indicate that GPDF may be a useful preprocessing tool for analyses of brain connectivity and network topology in individual fMRI recordings.

A Synthetic Feature Skull Descriptor for 3D Skull Similarity Measurement

3D skull similarity measurement is a challenging and meaningful task in the fields of archaeology, forensic science, and anthropology. However, it is difficult to correctly and directly measure the similarity between 3D skulls which are geometric models with multiple border holes and complex topologies. In this paper, based on the synthetic feature method, we propose a novel 3D skull descriptor, synthetic wave kernel distance distribution (SWKDD) constructed by the laplace–beltrami operator. By defining SWKDD, we obtain a concise global skull representation method and transform the complex 3D skull similarity measurement into a simple 1D vector similarity measurement. First, we give the definition and calculation of SWKDD and analyse its properties. Second, we represent a framework for 3D skull similarity measurement using the SWKDD of 3D skulls and details of the calculation steps involved. Finally, we validate the effectiveness of our proposed method by calculating the similarity measurement of 3D skulls based on the real craniofacial database.

Riemannian metric optimization on surfaces (RMOS) for intrinsic brain mapping in the Laplace-Beltrami embedding space.

Surface mapping methods play an important role in various brain imaging studies from tracking the maturation of adolescent brains to mapping gray matter atrophy patterns in Alzheimer's disease. Popular surface mapping approaches based on spherical registration, however, have inherent numerical limitations when severe metric distortions are present during the spherical parameterization step. In this paper, we propose a novel computational framework for intrinsic surface mapping in the Laplace-Beltrami (LB) embedding space based on Riemannian metric optimization on surfaces (RMOS). Given a diffeomorphism between two surfaces, an isometry can be defined using the pullback metric, which in turn results in identical LB embeddings from the two surfaces. The proposed RMOS approach builds upon this mathematical foundation and achieves general feature-driven surface mapping in the LB embedding space by iteratively optimizing the Riemannian metric defined on the edges of triangular meshes. At the core of our framework is an optimization engine that converts an energy function for surface mapping into a distance measure in the LB embedding space, which can be effectively optimized using gradients of the LB eigen-system with respect to the Riemannian metrics. In the experimental results, we compare the RMOS algorithm with spherical registration using large-scale brain imaging data, and show that RMOS achieves superior performance in the prediction of hippocampal subfields and cortical gyral labels, and the holistic mapping of striatal surfaces for the construction of a striatal connectivity atlas from substantia nigra.

Modified Virtual Grid Difference for Discretizing the Laplace--Beltrami Operator on Point Clouds

We propose a new and simple discretization, named the modified virtual grid difference (MVGD), for numerical approximation of the Laplace--Beltrami (LB) operator on manifolds sampled by point clouds. The key observation is that both the manifold and a function defined on it can both be parametrized in a local Cartesian coordinate system and approximated using least squares. Based on the above observation, we first introduce a local virtual grid with a scale adapted to the sampling density centered at each point. Then we propose a modified finite difference scheme on the virtual grid to discretize the LB operator. Instead of using the local least squares values on all virtual grid points like the typical finite difference method, we use the function value explicitly at the grid located at the center (coinciding with the data point). The new discretization provides more diagonal dominance to the resulting linear system and improves its conditioning. We show that the linear system can be robustly, efficiently and accurately solved by existing fast solver such as the algebraic multigrid (AMG) method. We will present numerical tests and comparison with other existing methods to demonstrate the effectiveness and the performance of the proposed approach.

Extraction and prediction of indices for monsoon intraseasonal oscillations: an approach based on nonlinear Laplacian spectral analysis

An improved index for real-time monitoring and forecast verification of monsoon intraseasonal oscillations (MISOs) is introduced using the recently developed nonlinear Laplacian spectral analysis (NLSA) technique. Using NLSA, a hierarchy of Laplace–Beltrami (LB) eigenfunctions are extracted from unfiltered daily rainfall data from the Global Precipitation Climatology Project over the south Asian monsoon region. Two modes representing the full life cycle of the northeastward-propagating boreal summer MISO are identified from the hierarchy of LB eigenfunctions. These modes have a number of advantages over MISO modes extracted via extended empirical orthogonal function analysis including higher memory and predictability, stronger amplitude and higher fractional explained variance over the western Pacific, Western Ghats, and adjoining Arabian Sea regions, and more realistic representation of the regional heat sources over the Indian and Pacific Oceans. Real-time prediction of NLSA-derived MISO indices is demonstrated via extended-range hindcasts based on NCEP Coupled Forecast System version 2 operational output. It is shown that in these hindcasts the NLSA MISO indices remain predictable out to $$\sim$$ 3 weeks.

Multiscale Nonrigid Point Cloud Registration Using Rotation-Invariant Sliced-Wasserstein Distance via Laplace--Beltrami Eigenmap

In this work, we propose computational models and algorithms for point cloud registration with nonrigid transformation. First, point clouds sampled from manifolds originally embedded in some Euclidean space are transformed to new point clouds embedded in $\mathbb{R}^n$ by the Laplace--Beltrami (LB) eigenmap, which is invariant under isometric transformation, using the first $n$ leading eigenvalues and corresponding eigenfunctions of the LB operator. Then we develop computational models and algorithms for registration of the transformed point clouds in a distribution/probability sense based on optimal transport, which provides both generality and flexibility for point cloud registration. In particular, we propose to use a rotation-invariant sliced-Wasserstein distance to achieve computation efficiency and handle ambiguities introduced by LB eigenmaps. By going from smaller $n$, which provides a quick and robust registration in coarse scale as well as a good initial guess for registration in finer scale, to a larger $n$, our method provides an efficient and robust multiscale nonrigid point cloud registration.

Holistic Mapping of Striatum Surfaces in the Laplace-Beltrami Embedding Space.

In brain shape analysis, the striatum is typically divided into three parts: the caudate, putamen, and accumbens nuclei for its analysis. Recent connectivity and animal studies, however, indicate striatum-cortical inter-connections do not always follow such subdivisions. For the holistic mapping of striatum surfaces, conventional spherical registration techniques are not suitable due to the large metric distortions in spherical parameterization of striatal surfaces. To overcome this difficulty, we develop a novel striatal surface mapping method using the recently proposed Riemannian metric optimization techniques in the Laplace-Beltrami (LB) embedding space. For the robust resolution of sign ambiguities in the LB spectrum, we also devise novel anatomical contextual features to guide the surface mapping in the embedding space. In our experimental results, we compare with spherical registration tools from FreeSurfer and FSL to demonstrate that our novel method provides a superior solution to the striatal mapping problem. We also apply our method to map the striatal surfaces from 211 subjects of the Human Connectome Project (HCP), and use the surface maps to construct a cortical connectivity atlas. Our atlas results show that the striato-cortical connectivity is not distinctive according to traditional structural subdivision of the striatum, and further confirms the holistic approach for mapping striatal surfaces.

Laplace Beltrami eigen value based classification of normal and Alzheimer MR images using parametric and non-parametric classifiers

Automated study of brain sub-anatomic region like Corpus Callosum (CC) is challenging due to its complex topology and varying shape. The development of reliable Computer Aided Diagnosis (CAD) systems would help in the early detection of Alzheimer's Disease (AD) and to perform drug trails to palliate the effect of AD. In this work, an attempt has been made to analyse the shape changes of CC using shape based Laplace Beltrami (LB) eigen value features and machine learning techniques. CC from the normal and AD T1-weighted magnetic resonance images are segmented using Reaction Diffusion (RD) level set method and the obtained results are validated against the Ground Truth (GT) images. Ten LB eigen values are extracted from the segmented CC images. LB eigen values are positive sequence of infinite series that describe the intrinsic geometry of objects. These values capture the shape information of CC by solving the eigen value problem of LB operator on the triangular meshes. The significant features are selected based on Information Gain (IG) ranking and subjected to classification using K-Nearest Neighbour (KNN), Support Vector Machine (SVM) and Naïve Bayes (NB). The performance of LB eigen values in the AD diagnosis is evaluated using classifiers’ accuracy, specificity and sensitivity measures.Results show that, RD level set is able to segment CC in normal and AD images with high percentage of similarity with GT. The extracted LB eigen values are found to show high difference in the mean values between normal and AD subjects with high statistical significance. The LB eigen modes λ2, λ7 and λ8 are identified as prominent features by IG based ranking. KNN is able to give maximum classification accuracy of 93.37% compared to linear SVM and NB classifiers. This value is observed to be high than the results obtained using geometric features. The proposed CAD system focuses solely on the geometric variations of CC extracted using LB eigen value spectrum. The extraction of eigen modes in the LB spectrum is easy to compute, does not involve too many parameters and less time consuming. Thus this CAD study seems to be clinically significant in the shape investigation of brain structures for AD diagnosis.

A Method to Differentiate Mild Cognitive Impairment and Alzheimer in MR Images using Eigen Value Descriptors.

Automated analysis and differentiation of mild cognitive impairment and Alzheimer's condition using MR images is clinically significant in dementic disorder. Alzheimer's Disease (AD) is a fatal and common form of dementia that progressively affects the patients. Shape descriptors could better differentiate the morphological alterations of brain structures and aid in the development of prospective disease modifying therapies. Ventricle enlargement is considered as a significant biomarker in the AD diagnosis. In this work, a method has been proposed to differentiate MCI from the healthy normal and AD subjects using Laplace-Beltrami (LB) eigen value shape descriptors. Prior to this, Reaction Diffusion (RD) level set is used to segment the ventricles in MR images and the results are validated against the Ground Truth (GT). LB eigen values are infinite series of spectrum that describes the intrinsic geometry of objects. Most significant LB shape descriptors are identified and their performance is analysed using linear Support Vector Machine (SVM) classifier. Results show that, the RD level set is able to segment the ventricles. The segmented ventricles are found to have high correlation with GT. The eigen values in the LB spectrum could show distinction in the feature space better than the geometric features. High accuracy is observed in the classification results of linear SVM. The proposed automated system is able to distinctly separate the MCI from normal and AD subjects. Thus this pipeline of work seems to be clinically significant in the automated analysis of dementic subjects.

Spectral Laplace-Beltrami wavelets with applications in medical images.

The spectral graph wavelet transform (SGWT) has recently been developed to compute wavelet transforms of functions defined on non-Euclidean spaces such as graphs. By capitalizing on the established framework of the SGWT, we adopt a fast and efficient computation of a discretized Laplace-Beltrami (LB) operator that allows its extension from arbitrary graphs to differentiable and closed 2-D manifolds (smooth surfaces embedded in the 3-D Euclidean space). This particular class of manifolds are widely used in bioimaging to characterize the morphology of cells, tissues, and organs. They are often discretized into triangular meshes, providing additional geometric information apart from simple nodes and weighted connections in graphs. In comparison with the SGWT, the wavelet bases constructed with the LB operator are spatially localized with a more uniform "spread" with respect to underlying curvature of the surface. In our experiments, we first use synthetic data to show that traditional applications of wavelets in smoothing and edge detectio can be done using the wavelet bases constructed with the LB operator. Second, we show that multi-resolutional capabilities of the proposed framework are applicable in the classification of Alzheimer's patients with normal subjects using hippocampal shapes. Wavelet transforms of the hippocampal shape deformations at finer resolutions registered higher sensitivity (96%) and specificity (90%) than the classification results obtained from the direct usage of hippocampal shape deformations. In addition, the Laplace-Beltrami method requires consistently a smaller number of principal components (to retain a fixed variance) at higher resolution as compared to the binary and weighted graph Laplacians, demonstrating the potential of the wavelet bases in adapting to the geometry of the underlying manifold.

Scale Space for Camera Invariant Features.

In this paper we propose a new approach to compute the scale space of any central projection system, such as catadioptric, fisheye or conventional cameras. Since these systems can be explained using a unified model, the single parameter that defines each type of system is used to automatically compute the corresponding Riemannian metric. This metric, is combined with the partial differential equations framework on manifolds, allows us to compute the Laplace-Beltrami (LB) operator, enabling the computation of the scale space of any central projection system. Scale space is essential for the intrinsic scale selection and neighborhood description in features like SIFT. We perform experiments with synthetic and real images to validate the generalization of our approach to any central projection system. We compare our approach with the best-existing methods showing competitive results in all type of cameras: catadioptric, fisheye, and perspective.

Improved Statistical Power with a Sparse Shape Model in Detecting an Aging Effect in the Hippocampus and Amygdala.

The sparse regression framework has been widely used in medical image processing and analysis. However, it has been rarely used in anatomical studies. We present a sparse shape modeling framework using the Laplace-Beltrami (LB) eigenfunctions of the underlying shape and show its improvement of statistical power. Traditionally, the LB-eigenfunctions are used as a basis for intrinsically representing surface shapes as a form of Fourier descriptors. To reduce high frequency noise, only the first few terms are used in the expansion and higher frequency terms are simply thrown away. However, some lower frequency terms may not necessarily contribute significantly in reconstructing the surfaces. Motivated by this idea, we present a LB-based method to filter out only the significant eigenfunctions by imposing a sparse penalty. For dense anatomical data such as deformation fields on a surface mesh, the sparse regression behaves like a smoothing process, which will reduce the error of incorrectly detecting false negatives. Hence the statistical power improves. The sparse shape model is then applied in investigating the influence of age on amygdala and hippocampus shapes in the normal population. The advantage of the LB sparse framework is demonstrated by showing the increased statistical power.

Multiscale Multiphysics and Multidomain Models I: Basic Theory.

This work extends our earlier two-domain formulation of a differential geometry based multiscale paradigm into a multidomain theory, which endows us the ability to simultaneously accommodate multiphysical descriptions of aqueous chemical, physical and biological systems, such as fuel cells, solar cells, nanofluidics, ion channels, viruses, RNA polymerases, molecular motors and large macromolecular complexes. The essential idea is to make use of the differential geometry theory of surfaces as a natural means to geometrically separate the macroscopic domain of solvent from the microscopic domain of solute, and dynamically couple continuum and discrete descriptions. Our main strategy is to construct energy functionals to put on an equal footing of multiphysics, including polar (i.e., electrostatic) solvation, nonpolar solvation, chemical potential, quantum mechanics, fluid mechanics, molecular mechanics, coarse grained dynamics and elastic dynamics. The variational principle is applied to the energy functionals to derive desirable governing equations, such as multidomain Laplace-Beltrami (LB) equations for macromolecular morphologies, multidomain Poisson-Boltzmann (PB) equation or Poisson equation for electrostatic potential, generalized Nernst-Planck (NP) equations for the dynamics of charged solvent species, generalized Navier-Stokes (NS) equation for fluid dynamics, generalized Newton's equations for molecular dynamics (MD) or coarse-grained dynamics and equation of motion for elastic dynamics. Unlike the classical PB equation, our PB equation is an integral-differential equation due to solvent-solute interactions. To illustrate the proposed formalism, we have explicitly constructed three models, a multidomain solvation model, a multidomain charge transport model and a multidomain chemo-electro-fluid-MD-elastic model. Each solute domain is equipped with distinct surface tension, pressure, dielectric function, and charge density distribution. In addition to long-range Coulombic interactions, various non-electrostatic solvent-solute interactions are considered in the present modeling. We demonstrate the consistency between the non-equilibrium charge transport model and the equilibrium solvation model by showing the systematical reduction of the former to the latter at equilibrium. This paper also offers a brief review of the field.

Spatially aggregating spectral descriptors for nonrigid 3D shape retrieval: a comparative survey

This paper presents a comprehensive review and analysis of recent spectral shape descriptors for nonrigid 3D shape retrieval. More specifically, we compare the latest spectral descriptors based on the Laplace---Beltrami (LB) operator, including ShapeDNA, heat kernel signature, scale invariant heat kernel signature, heat mean signature, wave kernel signature, and global point signature. We also include the eigenvalue descriptor (EVD), which is a geodesic distance-based shape signature. The global descriptors ShapeDNA and EVD are compared via the chi-squared distance, while all local descriptors are compared using the codebook model. Moreover, we investigate the ambiguity modeling of codebook for the densely distributed low-level shape descriptors. Inspired by the ability of spatial cues to improve discrimination between shapes, we also propose to adopt the isocontours of the second eigenfunction of the LB operator to perform surface partition, which can significantly ameliorate the retrieval performance of the time-scaled local descriptors. In addition, we introduce an intrinsic spatial pyramid matching approach in a bid to further enhance the retrieval accuracy. Extensive experiments are carried out on two 3D shape benchmarks to assess the performance of the spectral descriptors. Our proposed approach is shown to provide the best performance.