A review of deformable surfaces: topology, geometry and deformation

Deformable models have raised much interest and found various applications in the fields of computer vision and medical imaging. They provide an extensible framework to reconstruct shapes. Deformable surfaces, in particular, are used to represent 3D objects. They have been used for pattern recognition [Computer Vision and Image Understanding 69(2) (1998) 201; IEEE Transactions on Pattern Analysis and Machine Intelligence 19(10) (1997) 1115], computer animation [ACM Computer Graphics (SIGGRAPH'87) 21(4) (1987) 205], geometric modelling [61][Computer Aided Design (CAD) 24(4) (1992) 178], simulation [Visual Computer 16(8) (2000) 437], boundary tracking [ACM Computer Graphics (SIGGRAPH'94) (1994) 185], image segmentation [Computer Integrated Surgery, Technology and Clinical Applications (1996) 59; IEEE Transactions on Medical Imaging 14 (1995) 442; Joint Conference on Computer Vision, Virtual Reality and Robotics in Medicine (CVRMed-MRCAS'97) 1205 (1997) 13; Medical Image Computing and Computer-Assisted Intervention (MICCAI'99) 1679 (1999) 176; Medical Image Analysis 1(1) (1996) 19], etc. In this paper we propose a survey on deformable surfaces. Many surface representations have been proposed to meet different 3D reconstruction problem requirements. We classify the main representations proposed in the literature and we study the influence of the representation on the model evolution behavior, revealing some similarities between different approaches.

[1]  Gabriel Taubin,et al.  Distance approximations for rasterizing implicit curves , 1994, TOGS.

[2]  David C. Hogg,et al.  Wormholes in shape space: tracking through discontinuous changes in shape , 1998, Sixth International Conference on Computer Vision (IEEE Cat. No.98CH36271).

[3]  Andrew Zisserman,et al.  Measurement of Brain Structures Based on Statistical and Geometrical 3D Segmentation , 1998, MICCAI.

[4]  Jerry L. Prince,et al.  Adaptive active contour algorithms for extracting and mapping thick curves , 1993, Proceedings of IEEE Conference on Computer Vision and Pattern Recognition.

[5]  Lennart Thurfjell,et al.  A 3D Deformable Surface Model to Segment the Brain in MR Images , 1999 .

[6]  Andrew J. Bulpitt,et al.  An Efficient 3D Deformable Model with a Self-Optimising Topology , 1995, BMVC.

[7]  Laurent D. Cohen,et al.  A Hybrid Hyperquadric Model for 2-D and 3-D Data Fitting , 1996, Comput. Vis. Image Underst..

[8]  Andrew Blake,et al.  Affine-invariant contour tracking with automatic control of spatiotemporal scale , 1993, 1993 (4th) International Conference on Computer Vision.

[9]  Laurent D. Cohen,et al.  Tracking and motion analysis of the left ventricle with deformable superquadrics , 1996, Medical Image Anal..

[10]  Guido Gerig,et al.  Segmentation of 2-D and 3-D objects from MRI volume data using constrained elastic deformations of flexible Fourier contour and surface models , 1996, Medical Image Anal..

[11]  Song Han,et al.  Using hyperquadrics for shape recovery from range data , 1993, ICCV.

[12]  Demetri Terzopoulos,et al.  Medical Image Segmentation Using Topologically Adaptable Snakes , 1995, CVRMed.

[13]  Tom Lyche,et al.  Curves and Surfaces , 2014, Lecture Notes in Computer Science.

[14]  Russell H. Taylor,et al.  Computer-Integrated Surgery: Technology and Clinical Applications , 1995 .

[15]  J. Sethian.  Level set methods : evolving interfaces in geometry, fluid mechanics, computer vision, and materials science , 1996 .

[16]  Nicholas Ayache,et al.  A scheme for automatically building three-dimensional morphometric anatomical atlases: application to a skull atlas , 1998, Medical Image Anal..

[17]  Johan Montagnat,et al.  Globally constrained deformable models for 3D object reconstruction , 1998, Signal Process..

[18]  Mubarak Shah,et al.  A Fast algorithm for active contours and curvature estimation , 1992, CVGIP Image Underst..

[19]  Jérôme Declerck,et al.  Automatic registration and alignment on a template of cardiac stress and rest reoriented SPECT images , 1997, IEEE Transactions on Medical Imaging.

[20]  Rebecca Fahrig,et al.  Segmentation of medical images using an active-contour model incorporating region-based image features , 1994, Other Conferences.

[21]  James A. Sethian,et al.  A real-time algorithm for medical shape recovery , 1998, Sixth International Conference on Computer Vision (IEEE Cat. No.98CH36271).

[22]  Chitra Dorai,et al.  COSMOS - A Representation Scheme for 3D Free-Form Objects , 1997, IEEE Trans. Pattern Anal. Mach. Intell..

[23]  Alex Pentland,et al.  Closed-Form Solutions for Physically Based Shape Modeling and Recognition , 1991, IEEE Trans. Pattern Anal. Mach. Intell..

[24]  Laurent D. Cohen,et al.  Finite-Element Methods for Active Contour Models and Balloons for 2-D and 3-D Images , 1993, IEEE Trans. Pattern Anal. Mach. Intell..

[25]  Laurent D. Cohen,et al.  On active contour models and balloons , 1991, CVGIP Image Underst..

[26]  David J. Kriegman,et al.  Parameterized Families of Polynomials for Bounded Algebraic Curve and Surface Fitting , 1994, IEEE Trans. Pattern Anal. Mach. Intell..

[27]  Dimitris N. Metaxas,et al.  Constrained deformable superquadrics and nonrigid motion tracking , 1991, Proceedings. 1991 IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[28]  Robert T. Schultz,et al.  Segmentation and Measurement of the Cortex from 3D MR Images , 1998, MICCAI.

[29]  Sabine Coquillart,et al.  Extended free-form deformation: a sculpturing tool for 3D geometric modeling , 1990, SIGGRAPH.

[30]  C. Nastar,et al.  Classification of nonrigid motion in 3D images using physics-based vibration analysis , 1994, Proceedings of IEEE Workshop on Biomedical Image Analysis.

[31]  Timothy F. Cootes,et al.  The Use of Active Shape Models for Locating Structures in Medical Images , 1993, IPMI.

[32]  Baba C. Vemuri,et al.  Multiresolution stochastic hybrid shape models with fractal priors , 1994, TOGS.

[33]  John C. Platt,et al.  Elastically deformable models , 1987, SIGGRAPH.

[34]  Jean-Philippe Thirion,et al.  Image matching as a diffusion process: an analogy with Maxwell's demons , 1998, Medical Image Anal..

[35]  Demetri Terzopoulos,et al.  Deformable models in medical image analysis: a survey , 1996, Medical Image Anal..

[36]  Peter Schröder,et al.  Interactive multiresolution mesh editing , 1997, SIGGRAPH.

[37]  Fred L. Bookstein,et al.  Principal Warps: Thin-Plate Splines and the Decomposition of Deformations , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[38]  Xavier Pennec.  L'incertitude dans les problèmes de reconnaissance et de recalage - Applications en imagerie médicale et biologie moléculaire , 1996 .

[39]  Jos Stam,et al.  Exact evaluation of Catmull-Clark subdivision surfaces at arbitrary parameter values , 1998, SIGGRAPH.

[40]  Hervé Delingette,et al.  Intrinsic Stabilizers of Planar Curves , 1994, ECCV.

[41]  Philippe Cinquin,et al.  From Splines and Snakes to SNAKE SPLINES , 1991, Geometric Reasoning for Perception and Action.

[42]  Ramesh C. Jain,et al.  Using Dynamic Programming for Solving Variational Problems in Vision , 1990, IEEE Trans. Pattern Anal. Mach. Intell..

[43]  Geir Storvik,et al.  A Bayesian Approach to Dynamic Contours Through Stochastic Sampling and Simulated Annealing , 1994, IEEE Trans. Pattern Anal. Mach. Intell..

[44]  Dimitris N. Metaxas,et al.  Dynamic 3D Models with Local and Global Deformations: Deformable Superquadrics , 1991, IEEE Trans. Pattern Anal. Mach. Intell..

[45]  R. Wasserman,et al.  Biomedical imaging modalities: a tutorial. , 1995, Computerized medical imaging and graphics : the official journal of the Computerized Medical Imaging Society.

[46]  Timothy F. Cootes,et al.  A Non-linear Generalisation of PDMs using Polynomial Regression , 1994, BMVC.

[47]  Baba C. Vemuri,et al.  Shape Modeling with Front Propagation: A Level Set Approach , 1995, IEEE Trans. Pattern Anal. Mach. Intell..

[48]  Christopher J. Taylor,et al.  3D point distribution models of the cortical sulci , 1998, Sixth International Conference on Computer Vision (IEEE Cat. No.98CH36271).

[49]  Jacques-Olivier Lachaud,et al.  Deformable meshes with automated topology changes for coarse-to-fine three-dimensional surface extraction , 1999, Medical Image Anal..

[50]  Patrick Clarysse,et al.  Tracking geometrical descriptors on 3-D deformable surfaces: application to the left-ventricular surface of the heart , 1997, IEEE Transactions on Medical Imaging.

[51]  Christopher J. Taylor,et al.  Model-Based Interpretation of 3D Medical Images , 1993, BMVC.

[52]  Johan Montagnat,et al.  Volumetric medical images segmentation using shape constrained deformable models , 1997, CVRMed.

[53]  Michael Isard,et al.  3D position, attitude and shape input using video tracking of hands and lips , 1994, SIGGRAPH.

[54]  Demetri Terzopoulos,et al.  Constraints on Deformable Models: Recovering 3D Shape and Nonrigid Motion , 1988, Artif. Intell..

[55]  Jyrki Lötjönen,et al.  Model extraction from magnetic resonance volume data using the deformable pyramid , 1999, Medical Image Anal..

[56]  Davi Geiger,et al.  Resolution-appropriate shape representation , 1998, Sixth International Conference on Computer Vision (IEEE Cat. No.98CH36271).

[57]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[58]  Baba C. Vemuri,et al.  From global to local, a continuum of shape models with fractal priors , 1993, Proceedings of IEEE Conference on Computer Vision and Pattern Recognition.

[59]  James S. Duncan,et al.  Deformable Fourier models for surface finding in 3-D images , 1992, Other Conferences.

[60]  Andrew Blake,et al.  Visual Reconstruction , 1987, Deep Learning for EEG-Based Brain–Computer Interfaces.

[61]  Kaleem Siddiqi,et al.  Hyperbolic "Smoothing" of shapes , 1998, Sixth International Conference on Computer Vision (IEEE Cat. No.98CH36271).

[62]  H. Saunders,et al.  Finite element procedures in engineering analysis , 1982 .

[63]  Demetri Terzopoulos,et al.  A finite element model for 3D shape reconstruction and nonrigid motion tracking , 1993, 1993 (4th) International Conference on Computer Vision.

[64]  Lisa M. Brown,et al.  A survey of image registration techniques , 1992, CSUR.

[65]  Tony DeRose,et al.  Piecewise smooth surface reconstruction , 1994, SIGGRAPH.

[66]  David G. Luenberger,et al.  Linear and nonlinear programming , 1984 .

[67]  Alex Pentland,et al.  Generalized Image Matching: Statistical Learning of Physically-Based Deformations , 1996, ECCV.

[68]  Matthias Eck,et al.  Automatic reconstruction of B-spline surfaces of arbitrary topological type , 1996, SIGGRAPH.

[69]  Jyrki Lötjönen,et al.  Automatic Reconstruction of 3D Geometry Using Projections and a Geometric Prior Model , 1999, MICCAI.

[70]  C Knoll,et al.  Deformation simulation algorithms of elastic tissues in "real-time" based in elasticity theory. , 1999, Studies in health technology and informatics.

[71]  Richard Szeliski,et al.  Fast Surface Interpolation Using Hierarchical Basis Functions , 1990, IEEE Trans. Pattern Anal. Mach. Intell..

[72]  Johan Montagnat,et al.  New Algorithms for Controlling Active Contours Shape and Topology , 2000, ECCV.

[73]  Stephane Cotin,et al.  A hybrid elastic model for real-time cutting, deformations, and force feedback for surgery training and simulation , 2000, The Visual Computer.

[74]  Laurent D. Cohen,et al.  A hyperquadric model for 2-D and 3-D data fitting , 1994, Proceedings of the 12th IAPR International Conference on Pattern Recognition, Vol. 3 - Conference C: Signal Processing (Cat. No.94CH3440-5).

[75]  Nicholas Ayache,et al.  Frequency-Based Nonrigid Motion Analysis: Application to Four Dimensional Medical Images , 1996, IEEE Trans. Pattern Anal. Mach. Intell..

[76]  Isabelle E. Magnin,et al.  Modeling 3D Deformable Object with The Active Pyramid , 1997, Int. J. Pattern Recognit. Artif. Intell..

[77]  Timothy F. Cootes,et al.  Active Shape Models-Their Training and Application , 1995, Comput. Vis. Image Underst..

[78]  Shang-Hong Lai,et al.  Fast numerical algorithms for fitting multiresolution hybrid shape models to brain MRI , 1997, Medical Image Anal..

[79]  Mathieu Desbrun,et al.  Smoothed particles: a new paradigm for animating highly deformable bodies , 1996 .

[80]  Guido Gerig,et al.  Segmentation of 3D Objects from MRI Volume Data Using Constrained Elastic Deformations of Flexible Fourier Surface Models , 1995, CVRMed.

[81]  Timothy F. Cootes,et al.  Use of active shape models for locating structures in medical images , 1994, Image Vis. Comput..

[82]  Thomas W. Sederberg,et al.  Free-form deformation of solid geometric models , 1986, SIGGRAPH.

[83]  Alok Gupta,et al.  Dynamic Programming for Detecting, Tracking, and Matching Deformable Contours , 1995, IEEE Trans. Pattern Anal. Mach. Intell..

[84]  Richard Szeliski,et al.  Surface modeling with oriented particle systems , 1992, SIGGRAPH.

[85]  Jean-Laurent Mallet,et al.  Discrete smooth interpolation in geometric modelling , 1992, Comput. Aided Des..

[86]  Mark S. Nixon,et al.  Parameterizing Arbitrary Shapes via Fourier Descriptors for Evidence-Gathering Extraction , 1998, Comput. Vis. Image Underst..

[87]  Alexandre Moreau-Gaudry,et al.  Active Model Based Carotid Ultrasonic Data Segmentation , 1999, MICCAI.

[88]  Michael I. Miller,et al.  Deformable templates using large deformation kinematics , 1996, IEEE Trans. Image Process..

[89]  Baba C. Vemuri,et al.  Snake pedals: geometric models with physics-based control , 1998, Sixth International Conference on Computer Vision (IEEE Cat. No.98CH36271).

[90]  Nicholas Ayache,et al.  Smoothing and Matching of 3-D Space Curves , 1992, ECCV.

[91]  Paul J. Besl,et al.  A Method for Registration of 3-D Shapes , 1992, IEEE Trans. Pattern Anal. Mach. Intell..

[92]  D. Louis Collins,et al.  Model-based 3-D segmentation of multiple sclerosis lesions in magnetic resonance brain images , 1995, IEEE Trans. Medical Imaging.

[93]  Demetri Terzopoulos,et al.  Adaptive meshes and shells: irregular triangulation, discontinuities, and hierarchical subdivision , 1992, Proceedings 1992 IEEE Computer Society Conference on Computer Vision and Pattern Recognition.