DOCSLIB.ORG

Principal Curvature Measures Estimation and Application to 3D Face Recognition

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Principal Curvature Measures Estimation and Application to 3D Face Recognition J Math Imaging Vis DOI 10.1007/s10851-017-0728-2 Principal Curvature Measures Estimation and Application to 3D Face Recognition Yinhang Tang1 · Huibin Li2 · Xiang Sun3 · Jean-Marie Morvan3,4 · Liming Chen1 Received: 4 August 2016 / Accepted: 27 March 2017 © Springer Science+Business Media New York 2017 Abstract This paper presents an effective 3D face keypoint ple curvature measures are employed to assign the canonical detection, description and matching framework based on directions. Similarity comparison between faces is accom- three principle curvature measures. These measures give a plished by matching all these curvature-based local shape unified definition of principle curvatures for both smooth descriptors using the sparse representation-based reconstruc- and discrete surfaces. They can be reasonably computed tion method. The proposed method was evaluated on three based on the normal cycle theory and the geometric mea- public databases, i.e. FRGC v2.0, Bosphorus, and Gavab. sure theory. The strong theoretical basis of these measures Experimental results demonstrated that the three princi- provides us a solid discrete estimation method on real 3D ple curvature measures contain strong complementarity for face scans represented as triangle meshes. Based on these 3D facial shape description, and their fusion can largely estimated measures, the proposed method can automatically improve the recognition performance. Our approach achieves detect a set of sparse and discriminating 3D facial feature rank-one recognition rates of 99.6, 95.7, and 97.9% on the points. The local facial shape around each 3D feature point is neutral subset, expression subset, and the whole FRGC v2.0 comprehensively described by histograms of these principal databases, respectively. This indicates that our method is curvature measures. To guarantee the pose invariance of these robust to moderate facial expression variations. Moreover, descriptors, three principle curvature vectors of these princi- it also achieves very competitive performance on the pose subset (over 98.6% except Yaw 90°) and the occlusion sub- B Huibin Li set (98.4%) of the Bosphorus database. Even in the case of [email protected] extreme pose variations like profiles, it also significantly out- Yinhang Tang performs the state-of-the-art approaches with a recognition [email protected] rate of 57.1%. The experiments carried out on the Gavab Xiang Sun databases further demonstrate the robustness of our method [email protected] to varies head pose variations. Jean-Marie Morvan [email protected] Keywords Principal curvature measures · 3D keypoint Liming Chen detection, description and matching · Expression, pose and [email protected] occlusion · Mesh-based 3D face recognition 1 Université de Lyon, CNRS, Ecole Centrale de Lyon, LIRIS, 69134 Lyon, France 1 Introduction 2 School of Mathematics and Statistics, Xi’an Jiaotong University, No.28, Xianning West Road, Xi’an, Shaanxi 710049, P.R. China Recently, with the rapid development of 3D imaging and 3 King Abdullah University of Science and Technologh, V.C.C. visualization techniques, 3D shape analysis and process- Research Center, Thuwal 23955-6900, Saudi Arabia ing (e.g. surface registration, 3D shape retrieval) have been 4 Université de Lyon, CNRS, Université Claude Bernard Lyon widely studied. As stated in [57], capturing 3D shape makes 1, ICJ UMR 5208, 69622 Villeurbanne, France an extremely wide array of new kinds of application possible. 123 J Math Imaging Vis For example, virtual reality (VR), augmented reality (AR), vatures (i.e. principal curvature measures) can be used for 3D biometrics, 3D human–machine interaction, 3D medi- 3D face recognition? How to estimate them on meshed 3D cal imaging, 3D remote sensing, to name just a few. Behind face scans, and are they effective and efficient for 3D face these applications, a fundamental and challenging problem description and recognition? To answer these questions, we is how to represent and characterize the geometric structures give the following contributions in this paper: of these discrete 3D shapes (surfaces or volumes). Up to now, many approaches [9,12,60,65] have been proposed to solve 1. Different from all the existing 3D face recognition this key issue. approaches, we are the first attempting to study the As a special application of 3D shape analysis and pro- effectiveness of the principal curvature measures for the cessing, the key issue of 3D face recognition has also practical 3D face recognition application. been widely addressed. Facial surface measurements, i.e. 2. We improved our previous work [78] and designed novel points [10], curves [63], stripes [5], regions [13,14], normals 3D facial keypoint detection and description schemes [39,77], curvatures [40,78], geodesic distance [11,42,43], based on principal curvature measures, and their corre- have been popularly used to represent and characterize 3D sponding principal curvature vectors. face shapes. Among them, surface curvatures, including 3. We conducted comprehensive experiments on three 3D Gaussian curvature, mean curvature, principal curvatures, as face databases, i.e., FRGC v2.0, Bosphorus and Gavab, well as shape index values, are the most widely used ones to evaluate the general face recognition performance and [16,20,28,39,40,42,43,67,76,78,84]. Thus, the representa- the robustness to pose and occlusion variations of the tion and characterization of 3D face surface are the most proposed approach. important techniques for an efficient 3D face recognition sys- tem, where curvatures play fundamental effect. Specifically, the proposed 3D face recognition approach However, 3D face scans are discrete surfaces and gener- consists of three modules. First of all, a group of discrim- ally represented by discontinuous point clouds, depth images inating feature points are detected automatically based on or piece-wise triangle meshes. This means that the classical principal curvature measures. Then, the histograms of three curvature estimation formulations for at least C2 smooth sur- principal curvature measures describe comprehensively the faces in the classical differential geometry are not suitable for local shape around these feature points. The descriptor is 3D face scans. Instead of working directly on face scans, a further endowed with rotation invariance by assigning canon- popular solution is to estimate the curvatures of a polynomial ical direction based on the corresponding principal curvature surface patch achieved by fitting it to the local 3D facial sur- vectors. The similarity measurement between face scans is face. Otherwise, pointwise curvatures can also be estimated finally defined as the sparse representation-based reconstruc- directly on triangle meshes. For example, the angle defect tion error. estimation [7] and the normal deviation estimation [85]are The proposed method was comprehensively evaluated on utilized to compute pointwise Gaussian curvature and mean three public 3D face databases, i.e. FRGC v2.0 [59], Bospho- curvature, respectively. However, the pointwise convergence rus [64], and Gavab [53]. On the FRGC v2.0 databases, guarantee is satisfied only for triangle meshes with special the effectiveness of each single curvature-based descriptor is structure [86]. validated, and the discriminative power of their score-level Fortunately, based on the normal cycle theory [17,58] and fusion demonstrates the strong informative complementarity geometry measure theory [18,54], Morvan et al. proposed among different principal curvature measures. Our approach the concepts of generalized curvatures, i.e. curvature mea- achieves rank-one recognition rates of 99.6% with neutral sures, for both smooth and discrete surfaces [17,54]. They probes and 95.7% with expressive probes, and a verification further proposed a surface meshing algorithm and proved that rate of 95.28% with 0.1% false accept rate (FAR). More- Gaussian curvature measure and mean curvature measure of over, when the experiments performed on the pose subsets a triangle mesh converge to the ones of its sampled C2 smooth of the Bosphorus and the Gavab databases, the proposed surface, if the mesh is well-shaped (i.e. with minimal angle approach achieves the best scores among the state-of-the- bound) and well-sized (i.e. with infinitesimal circumcircle) art ones showing its robustness to diverse pose changes. The [41]. Recently, they further generalized the second funda- outstanding recognition results of the descriptors are also mental form and asymptotic direction of smooth surface, and presented when different external occlusions are involved. proposed asymptotic cone and principal curvature measures The rest of this paper is organized as follows. Section 2 for general polyhedral surface [73]. The convergence of the gives a brief review of related works on 3D face recogni- principal curvature measures can also be guaranteed based tion and curvature estimation on discrete surfaces. Section 3 on the normal cycle theory [73] (see Sect. 3 for more details). introduces the theory of principal curvature measures of Motivated by this curvature estimation method for triangle smooth surface, triangle mesh, as well as the meshed 3D meshes, a natural idea is to see how these generalized cur- facial surface. 3D face recognition method based on prin- 123 J Math Imaging Vis cipal curvature measures and their corresponding principal [10,23,63], and expression deformation modelling based curvature vectors is introduced
Load more

Recommended publications
  • DISCRETE DIFFERENTIAL GEOMETRY: an APPLIED INTRODUCTION Keenan Crane • CMU 15-458/858 LECTURE 15: CURVATURE
    DISCRETE DIFFERENTIAL GEOMETRY: AN APPLIED INTRODUCTION Keenan Crane • CMU 15-458/858 LECTURE 15: CURVATURE DISCRETE DIFFERENTIAL GEOMETRY: AN APPLIED INTRODUCTION Keenan Crane • CMU 15-458/858 Curvature—Overview • Intuitively, describes “how much a shape bends” – Extrinsic: how quickly does the tangent plane/normal change? – Intrinsic: how much do quantities differ from flat case? N T B Curvature—Overview • Driving force behind wide variety of physical phenomena – Objects want to reduce—or restore—their curvature – Even space and time are driven by curvature… Curvature—Overview • Gives a coordinate-invariant description of shape – fundamental theorems of plane curves, space curves, surfaces, … • Amazing fact: curvature gives you information about global topology! – “local-global theorems”: turning number, Gauss-Bonnet, … Curvature—Overview • Geometric algorithms: shape analysis, local descriptors, smoothing, … • Numerical simulation: elastic rods/shells, surface tension, … • Image processing algorithms: denoising, feature/contour detection, … Thürey et al 2010 Gaser et al Kass et al 1987 Grinspun et al 2003 Curvature of Curves Review: Curvature of a Plane Curve • Informally, curvature describes “how much a curve bends” • More formally, the curvature of an arc-length parameterized plane curve can be expressed as the rate of change in the tangent Equivalently: Here the angle brackets denote the usual dot product, i.e., . Review: Curvature and Torsion of a Space Curve •For a plane curve, curvature captured the notion of “bending” •For a space curve we also have torsion, which captures “twisting” Intuition: torsion is “out of plane bending” increasing torsion Review: Fundamental Theorem of Space Curves •The fundamental theorem of space curves tells that given the curvature κ and torsion τ of an arc-length parameterized space curve, we can recover the curve (up to rigid motion) •Formally: integrate the Frenet-Serret equations; intuitively: start drawing a curve, bend & twist at prescribed rate.
    [Show full text]
  • Lines of Curvature on Surfaces, Historical Comments and Recent Developments
    S˜ao Paulo Journal of Mathematical Sciences 2, 1 (2008), 99–143 Lines of Curvature on Surfaces, Historical Comments and Recent Developments Jorge Sotomayor Instituto de Matem´atica e Estat´ıstica, Universidade de S˜ao Paulo, Rua do Mat˜ao 1010, Cidade Universit´aria, CEP 05508-090, S˜ao Paulo, S.P., Brazil Ronaldo Garcia Instituto de Matem´atica e Estat´ıstica, Universidade Federal de Goi´as, CEP 74001-970, Caixa Postal 131, Goiˆania, GO, Brazil Abstract. This survey starts with the historical landmarks leading to the study of principal configurations on surfaces, their structural sta- bility and further generalizations. Here it is pointed out that in the work of Monge, 1796, are found elements of the qualitative theory of differential equations ( QTDE ), founded by Poincar´ein 1881. Here are also outlined a number of recent results developed after the assimilation into the subject of concepts and problems from the QTDE and Dynam- ical Systems, such as Structural Stability, Bifurcations and Genericity, among others, as well as extensions to higher dimensions. References to original works are given and open problems are proposed at the end of some sections. 1. Introduction The book on differential geometry of D. Struik [79], remarkable for its historical notes, contains key references to the classical works on principal curvature lines and their umbilic singularities due to L. Euler [8], G. Monge [61], C. Dupin [7], G. Darboux [6] and A. Gullstrand [39], among others (see The authors are fellows of CNPq and done this work under the project CNPq 473747/2006-5. The authors are grateful to L.
    [Show full text]
  • On the Patterns of Principal Curvature Lines Around a Curve of Umbilic Points
    Anais da Academia Brasileira de Ciências (2005) 77(1): 13–24 (Annals of the Brazilian Academy of Sciences) ISSN 0001-3765 www.scielo.br/aabc On the Patterns of Principal Curvature Lines around a Curve of Umbilic Points RONALDO GARCIA1 and JORGE SOTOMAYOR2 1Instituto de Matemática e Estatística, Universidade Federal de Goiás Caixa Postal 131 – 74001-970 Goiânia, GO, Brasil 2Instituto de Matemática e Estatística, Universidade de São Paulo Rua do Matão 1010, Cidade Universitária, 05508-090 São Paulo, SP, Brasil Manuscript received on June 15, 2004; accepted for publication on October 10, 2004; contributed by Jorge Sotomayor* ABSTRACT In this paper is studied the behavior of principal curvature lines near a curve of umbilic points of a smooth surface. Key words: Umbilic point, principal curvature lines, principal cycles. 1 INTRODUCTION The study of umbilic points on surfaces and the patterns of principal curvature lines around them has attracted the attention of generation of mathematicians among whom can be named Monge, Darboux and Carathéodory. One aspect – concerning isolated umbilics – of the contributions of these authors, departing from Darboux (Darboux 1896), has been elaborated and extended in several directions by Garcia, Sotomayor and Gutierrez, among others. See (Gutierrez and Sotomayor, 1982, 1991, 1998), (Garcia and Sotomayor, 1997, 2000) and (Garcia et al. 2000, 2004) where additional references can be found. In (Carathéodory 1935) Carathéodory mentioned the interest of non isolated umbilics in generic surfaces pertinent to Geometric Optics. In a remarkably concise study he established that any local analytic regular arc of curve in R3 is a curve of umbilic points of a piece of analytic surface.
    [Show full text]
  • Riemannian Submanifolds: a Survey
    RIEMANNIAN SUBMANIFOLDS: A SURVEY BANG-YEN CHEN Contents Chapter 1. Introduction .............................. ...................6 Chapter 2. Nash’s embedding theorem and some related results .........9 2.1. Cartan-Janet’s theorem .......................... ...............10 2.2. Nash’s embedding theorem ......................... .............11 2.3. Isometric immersions with the smallest possible codimension . 8 2.4. Isometric immersions with prescribed Gaussian or Gauss-Kronecker curvature .......................................... ..................12 2.5. Isometric immersions with prescribed mean curvature. ...........13 Chapter 3. Fundamental theorems, basic notions and results ...........14 3.1. Fundamental equations ........................... ..............14 3.2. Fundamental theorems ............................ ..............15 3.3. Basic notions ................................... ................16 3.4. A general inequality ............................. ...............17 3.5. Product immersions .............................. .............. 19 3.6. A relationship between k-Ricci tensor and shape operator . 20 3.7. Completeness of curvature surfaces . ..............22 Chapter 4. Rigidity and reduction theorems . ..............24 4.1. Rigidity ....................................... .................24 4.2. A reduction theorem .............................. ..............25 Chapter 5. Minimal submanifolds ....................... ...............26 arXiv:1307.1875v1 [math.DG] 7 Jul 2013 5.1. First and second variational formulas
    [Show full text]
  • Basics of the Differential Geometry of Surfaces
    Chapter 20 Basics of the Differential Geometry of Surfaces 20.1 Introduction The purpose of this chapter is to introduce the reader to some elementary concepts of the differential geometry of surfaces. Our goal is rather modest: We simply want to introduce the concepts needed to understand the notion of Gaussian curvature, mean curvature, principal curvatures, and geodesic lines. Almost all of the material presented in this chapter is based on lectures given by Eugenio Calabi in an upper undergraduate differential geometry course offered in the fall of 1994. Most of the topics covered in this course have been included, except a presentation of the global Gauss–Bonnet–Hopf theorem, some material on special coordinate systems, and Hilbert’s theorem on surfaces of constant negative curvature. What is a surface? A precise answer cannot really be given without introducing the concept of a manifold. An informal answer is to say that a surface is a set of points in R3 such that for every point p on the surface there is a small (perhaps very small) neighborhood U of p that is continuously deformable into a little flat open disk. Thus, a surface should really have some topology. Also,locally,unlessthe point p is “singular,” the surface looks like a plane. Properties of surfaces can be classified into local properties and global prop- erties.Intheolderliterature,thestudyoflocalpropertieswascalled geometry in the small,andthestudyofglobalpropertieswascalledgeometry in the large.Lo- cal properties are the properties that hold in a small neighborhood of a point on a surface. Curvature is a local property. Local properties canbestudiedmoreconve- niently by assuming that the surface is parametrized locally.
    [Show full text]
  • Metrics of Positive Ricci Curvature on Connected Sums: Projective Spaces, Products, and Plumbings
    METRICS OF POSITIVE RICCI CURVATURE ON CONNECTED SUMS: PROJECTIVE SPACES, PRODUCTS, AND PLUMBINGS by BRADLEY LEWIS BURDICK ADISSERTATION Presented to the Department of Mathematics and the Graduate School of the University of Oregon in partial fulfillment of the requirements for the degree of Doctor of Philosophy June 2019 DISSERTATION APPROVAL PAGE Student: Bradley Lewis Burdick Title: Metrics of Positive Ricci Curvature on Connected Sums: Projective Spaces, Products, and Plumbings This dissertation has been accepted and approved in partial fulfillment of the requirements for the Doctor of Philosophy degree in the Department of Mathematics by: Boris Botvinnik Chair NicholasProudfoot CoreMember Robert Lipshitz Core Member Micah Warren Core Member Graham Kribs Institutional Representative and Janet Woodru↵-Borden Vice Provost & Dean of the Graduate School Original approval signatures are on file with the University of Oregon Graduate School. Degree awarded June 2019 ii c 2019 Bradley Lewis Burdick This work is licensed under a Creative Commons Attribution 4.0 International License iii DISSERTATION ABSTRACT Bradley Lewis Burdick Doctor of Philosophy Department of Mathematics June 2019 Title: Metrics of Positive Ricci Curvature on Connected Sums: Projective Spaces, Products, and Plumbings The classification of simply connected manifolds admitting metrics of positive scalar curvature of initiated by Gromov-Lawson, at its core, relies on a careful geometric construction that preserves positive scalar curvature under surgery and, in particular, under connected sum. For simply connected manifolds admitting metrics of positive Ricci curvature, it is conjectured that a similar classification should be possible, and, in particular, there is no suspected obstruction to preserving positive Ricci curvature under connected sum.
    [Show full text]
  • Differential Geometry
    ALAGAPPA UNIVERSITY [Accredited with ’A+’ Grade by NAAC (CGPA:3.64) in the Third Cycle and Graded as Category–I University by MHRD-UGC] (A State University Established by the Government of Tamilnadu) KARAIKUDI – 630 003 DIRECTORATE OF DISTANCE EDUCATION III - SEMESTER M.Sc.(MATHEMATICS) 311 31 DIFFERENTIAL GEOMETRY Copy Right Reserved For Private use only Author: Dr. M. Mullai, Assistant Professor (DDE), Department of Mathematics, Alagappa University, Karaikudi “The Copyright shall be vested with Alagappa University” All rights reserved. No part of this publication which is material protected by this copyright notice may be reproduced or transmitted or utilized or stored in any form or by any means now known or hereinafter invented, electronic, digital or mechanical, including photocopying, scanning, recording or by any information storage or retrieval system, without prior written permission from the Alagappa University, Karaikudi, Tamil Nadu. SYLLABI-BOOK MAPPING TABLE DIFFERENTIAL GEOMETRY SYLLABI Mapping in Book UNIT -I INTRODUCTORY REMARK ABOUT SPACE CURVES 1-12 13-29 UNIT- II CURVATURE AND TORSION OF A CURVE 30-48 UNIT -III CONTACT BETWEEN CURVES AND SURFACES . 49-53 UNIT -IV INTRINSIC EQUATIONS 54-57 UNIT V BLOCK II: HELICES, HELICOIDS AND FAMILIES OF CURVES UNIT -V HELICES 58-68 UNIT VI CURVES ON SURFACES UNIT -VII HELICOIDS 69-80 SYLLABI Mapping in Book 81-87 UNIT -VIII FAMILIES OF CURVES BLOCK-III: GEODESIC PARALLELS AND GEODESIC 88-108 CURVATURES UNIT -IX GEODESICS 109-111 UNIT- X GEODESIC PARALLELS 112-130 UNIT- XI GEODESIC
    [Show full text]
  • Robust Principal Curvatures on Multiple Scales
    Eurographics Symposium on Geometry Processing (2006) Konrad Polthier, Alla Sheffer (Editors) Robust Principal Curvatures on Multiple Scales Yong-Liang Yang1 Yu-Kun Lai1 Shi-Min Hu1 Helmut Pottmann2 1Tsinghua University, Beijing 2Vienna University of Technology. Abstract Geometry processing algorithms often require the robust extraction of curvature information. We propose to achieve this with principal component analysis (PCA) of local neighborhoods, defined via spherical kernels cen- tered on the given surface Φ. Intersection of a kernel ball Br or its boundary sphere Sr with the volume bounded by Φ leads to the so-called ball and sphere neighborhoods. Information obtained by PCA of these neighborhoods turns out to be more robust than PCA of the patch neighborhood Br ∩Φ previously used. The relation of the quan- tities computed by PCA with the principal curvatures of Φ is revealed by an asymptotic analysis as the kernel radius r tends to zero. This also allows us to define principal curvatures “at scale r” in a way which is consistent with the classical setting. The advantages of the new approach are discussed in a comparison with results obtained by normal cycles and local fitting; whereas the former method somewhat lacks in robustness, the latter does not achieve a consistent behavior at features on coarse scales. As to applications, we address computing principal curves and feature extraction on multiple scales. 1. Introduction Differential geometry plays a central role in the analysis of curves and surfaces. Local investigations frequently use dif- ferential invariants such as curvatures, but also the global un- derstanding of shapes can benefit from differential geomet- ric entities.
    [Show full text]
  • Connections and Curvature
    C Connections and Curvature Introduction In this appendix we present results in differential geometry that serve as a useful background for material in the main body of the book. Material in 1 on connections is somewhat parallel to the study of the natural connec- tion§ on a Riemannian manifold made in 11 of Chapter 1, but here we also study the curvature of a connection. Material§ in 2 on second covariant derivatives is connected with material in Chapter 2 on§ the Laplace operator. Ideas developed in 3 and 4, on the curvature of Riemannian manifolds and submanifolds, make§§ contact with such material as the existence of com- plex structures on two-dimensional Riemannian manifolds, established in Chapter 5, and the uniformization theorem for compact Riemann surfaces and other problems involving nonlinear, elliptic PDE, arising from studies of curvature, treated in Chapter 14. Section 5 on the Gauss-Bonnet theo- rem is useful both for estimates related to the proof of the uniformization theorem and for applications to the Riemann-Roch theorem in Chapter 10. Furthermore, it serves as a transition to more advanced material presented in 6–8. In§§ 6 we discuss how constructions involving vector bundles can be de- rived§ from constructions on a principal bundle. In the case of ordinary vector fields, tensor fields, and differential forms, one can largely avoid this, but it is a very convenient tool for understanding spinors. The principal bundle picture is used to construct characteristic classes in 7. The mate- rial in these two sections is needed in Chapter 10, on the index§ theory for elliptic operators of Dirac type.
    [Show full text]
  • Computational Assessment of Curvatures and Principal Directions
    Computational assessment of curvatures and principal directions of implicit surfaces from 3D scalar data Eric Albin, Ronnie Knikker, Shihe Xin, Christian Oliver Paschereit, Yves d’Angelo To cite this version: Eric Albin, Ronnie Knikker, Shihe Xin, Christian Oliver Paschereit, Yves d’Angelo. Computational assessment of curvatures and principal directions of implicit surfaces from 3D scalar data. Lecture Notes in Computer Science, Springer, 2017, Mathematical Methods for Curves and Surfaces, 10521, pp.1-22. 10.1007/978-3-319-67885-6_1. hal-01486547 HAL Id: hal-01486547 https://hal.archives-ouvertes.fr/hal-01486547 Submitted on 16 Jun 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Computational assessment of curvatures and principal directions of implicit surfaces from 3D scalar data Eric Albin1, Ronnie Knikker1, Shihe Xin1, Christian Oliver Paschereit2, and Yves D'Angelo3 1 Universit´ede Lyon, CNRS Universit´eLyon 1, F-69622, France INSA-Lyon, CETHIL, UMR5008, F-69621, Villeurbanne, France [email protected], 2 Technische Universit¨atBerlin, Institute of Fluid Dynamics and Technical Acoustics, Hermann-F¨ottinger-Institut.M¨uller-Breslau-Str.8 10623 Berlin, Germany 3 Laboratoire de Math´ematiques J.A.
    [Show full text]
  • Principal Curvature Ridges and Geometrically Salient Regions of Parametric B-Spline Surfaces
    Principal Curvature Ridges and Geometrically Salient Regions of Parametric B-Spline Surfaces Suraj Musuvathya,∗, Elaine Cohenb, James Damonc, Joon-Kyung Seongd aSchool of Computing, University of Utah bSchool of Computing, University of Utah cDepartment of Mathematics, University of North Carolina at Chapel Hill dKorea Advanced Institute of Science and Technology Abstract Ridges are characteristic curves of a surface that mark salient intrinsic fea- tures of its shape and are therefore valuable for shape matching, surface quality control, visualization and various other applications. Ridges are loci of points on a surface where one of the principal curvatures attain a critical value in its respective principal direction. We present a new algorithm for accurately extracting ridges on B-Spline surfaces and define a new type of salient region corresponding to major ridges that characterize geometrically significant regions on surfaces. Ridges exhibit complex behavior near umbil- ics on a surface, and may also pass through certain turning points causing added complexity for ridge computation. We present a new numerical tracing algorithm for extracting ridges that also accurately captures ridge behavior at umbilics and ridge turning points. The algorithm traverses ridge segments by detecting ridge points while advancing and sliding in principal directions on a surface in a novel manner, thereby computing connected curves of ridge points. The output of the algorithm is a set of curve segments, some or all of which, may be selected for other applications such as those mentioned above. The results of our technique are validated by comparison with results from previous research and with a brute-force domain sampling technique.
    [Show full text]
  • Ricci Curvature and Minimal Submanifolds
    Pacific Journal of Mathematics RICCI CURVATURE AND MINIMAL SUBMANIFOLDS Thomas Hasanis and Theodoros Vlachos Volume 197 No. 1 January 2001 PACIFIC JOURNAL OF MATHEMATICS Vol. 197, No. 1, 2001 RICCI CURVATURE AND MINIMAL SUBMANIFOLDS Thomas Hasanis and Theodoros Vlachos The aim of this paper is to find necessary conditions for a given complete Riemannian manifold to be realizable as a minimal submanifold of a unit sphere. 1. Introduction. The general question that served as the starting point for this paper was to find necessary conditions on those Riemannian metrics that arise as the induced metrics on minimal hypersurfaces or submanifolds of hyperspheres of a Euclidean space. There is an abundance of complete minimal hypersurfaces in the unit hy- persphere Sn+1. We recall some well known examples. Let Sm(r) = {x ∈ Rn+1, |x| = r},Sn−m(s) = {y ∈ Rn−m+1, |y| = s}, where r and s are posi- tive numbers with r2 + s2 = 1; then Sm(r) × Sn−m(s) = {(x, y) ∈ Rn+2, x ∈ Sm(r), y ∈ Sn−m(s)} is a hypersurface of the unit hypersphere in Rn+2. As is well known, this hypersurface has two distinct constant principal cur- vatures: One is s/r of multiplicity m, the other is −r/s of multiplicity n − m. This hypersurface is called a Clifford hypersurface. Moreover, it is minimal only in the case r = pm/n, s = p(n − m)/n and is called a Clifford minimal hypersurface. Otsuki [11] proved that if M n is a com- pact minimal hypersurface in Sn+1 with two distinct principal curvatures of multiplicity greater than 1, then M n is a Clifford minimal hypersur- face Sm(pm/n) × Sn−m(p(n − m)/n), 1 < m < n − 1.
    [Show full text]