3D Shape Analysis Based On Spectral Methods

Shape analysis is a fundamental digital geometry processing task.Spectral method is an effective tool of shape analysis,and it has outstanding performance in denoising,dimension reduction and intrinsic feature extraction.Spectral methods have been developed to solve a diversity of problems in shape analysis,including shape description,symmetry detection,segmentation,skeleton extraction,shape correspondence and matching,shape retrieval,and mesh saliency.This article focuses on non-rigid 3D retrieval and mesh saliency based on spectral methods.The main workes are summarized as follows:(1)We propose a non-rigid 3D retrieval framework based on mean normalized Laplace-Beltrami spectrum descriptor.In addition,we analyze the matching mechanism of spectrum descriptor.Inspired by the definition of contribution rate in principal component analysis(PCA),we interpret spectral descriptor normalization by the idea of contribution rate,and compare our normalization method against other two classical ones using the interpretation from two aspects of theory and experiment.To improve the discrimination ability of the high-frequency part of Laplace-Beltrami spectrum to local details,weighted function filter method is used.Moreover,we employ multi-resolution fusion method to improve retrieval precision.For the determination of number of eigenvalue,we estimate a interval by rank estimation method.The number of eigenvalue can be randomly selected in the estimation interval.Experiments show that mean normalized Laplace-Beltrami spectral descriptor is robust to a variety of different transformations,including isometry,holes,sampling,scale,localscale,affine and noise.(2)We propose a non-rigid 3D shape retrieval method based on Laplace-Beltrami spectra and a improved extreme learning machine.In our method,many neural networks are trained in a training set by classification learning.Using optimization algorithm,multiple trained neural networks are selected.In shape retrieval stage,shape descriptors are inputed those selected networks as input layer,and the output layers of those networks are reassembled as new shape descriptors.The new shape descriptors are used to compute the similarity between the query shape and inquiry targets.For a higher computational efficiency,USUA(Upper-layer-solution-unaware algorithm),a improved extreme learning machine,is adopted as neural networks learning algorithm.To ensure the stability of USUA,we give the theoretical analysis for USUA.In our method,unprocessed Laplace-Beltrami spectra,without any normalization and fliter,are the input of the network,and USUA can independently mine intrinsic information of Laplace-Beltrami spectra.This method avoids the determination of number of eigenvalues and the normalization of Laplace-Beltrami spectra.In addition,Multi-networks optimum algorithm can avoid weak generalization ability of USUA caused by improper initial weights.(3)We summarize and put forward the principles of global correlation spectral signature and multi-level spectral mean signature.Using the two principles,two new spectral signatures,multi-resolution fusion best rank-k approximation signature(MRF-BRKAS)and multi-level spectral mean wave kernel signature(ML-MWKS),are defined.Furthermore,two new spectral signatures are used to detect mesh saliency.To highlight salient feature of local details,high peaks suppression and staircase function weighted method are used in the constructing process of the two spectral signatures.The salient regions detected by MRF-BRKAS and ML-MWKS have their own characteristic,respectively.ML-MWKS can detect Large convex features,and MRF-BRKAS is good at local details detection.The combination of two spectral signatures is further improved the detection performance of mesh saliency.