Active shape models—their training and application
Computer Vision and Image Understanding
Computable elastic distances between shapes
SIAM Journal on Applied Mathematics
Nonlinear Modeling of Scattered Multivariate Data and Its Application to Shape Change
IEEE Transactions on Pattern Analysis and Machine Intelligence
Group Actions, Homeomorphisms, and Matching: A General Framework
International Journal of Computer Vision - Special issue on statistical and computational theories of vision: Part II
Automatic Construction of 3D Statistical Deformation Models Using Non-rigid Registration
MICCAI '01 Proceedings of the 4th International Conference on Medical Image Computing and Computer-Assisted Intervention
A level set algorithm for minimizing the Mumford-Shah functional in image processing
VLSM '01 Proceedings of the IEEE Workshop on Variational and Level Set Methods (VLSM'01)
Approximations of Shape Metrics and Application to Shape Warping and Empirical Shape Statistics
Foundations of Computational Mathematics
A Theoretical and Computational Framework for Isometry Invariant Recognition of Point Cloud Data
Foundations of Computational Mathematics
Shape-Based Approach to Robust Image Segmentation using Kernel PCA
CVPR '06 Proceedings of the 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition - Volume 1
Geometric modeling in shape space
ACM SIGGRAPH 2007 papers
Numerical Geometry of Non-Rigid Shapes
Numerical Geometry of Non-Rigid Shapes
An Elasticity Approach to Principal Modes of Shape Variation
SSVM '09 Proceedings of the Second International Conference on Scale Space and Variational Methods in Computer Vision
Shape Metrics Based on Elastic Deformations
Journal of Mathematical Imaging and Vision
Geodesics in Shape Space via Variational Time Discretization
EMMCVPR '09 Proceedings of the 7th International Conference on Energy Minimization Methods in Computer Vision and Pattern Recognition
A Nonlinear Elastic Shape Averaging Approach
SIAM Journal on Imaging Sciences
Similarity metrics for groupwise non-rigid registration
MICCAI'07 Proceedings of the 10th international conference on Medical image computing and computer-assisted intervention
Statistics of shape via principal geodesic analysis on lie groups
CVPR'03 Proceedings of the 2003 IEEE computer society conference on Computer vision and pattern recognition
Construction of a 4D statistical atlas of the cardiac anatomy and its use in classification
MICCAI'05 Proceedings of the 8th international conference on Medical image computing and computer-assisted intervention - Volume Part II
Statistical shape models using elastic-string representations
ACCV'06 Proceedings of the 7th Asian conference on Computer Vision - Volume Part I
Comparative analysis of kernel methods for statistical shape learning
CVAMIA'06 Proceedings of the Second ECCV international conference on Computer Vision Approaches to Medical Image Analysis
IEEE Transactions on Image Processing
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We introduce the covariance of a number of given shapes if they are interpreted as boundary contours of elastic objects. Based on the notion of nonlinear elastic deformations from one shape to another, a suitable linearization of geometric shape variations is introduced. Once such a linearization is available, a principal component analysis can be investigated. This requires the definition of a covariance metric--an inner product on linearized shape variations. The resulting covariance operator robustly captures strongly nonlinear geometric variations in a physically meaningful way and allows to extract the dominant modes of shape variation. The underlying elasticity concept represents an alternative to Riemannian shape statistics. In this paper we compare a standard L 2-type covariance metric with a metric based on the Hessian of the nonlinear elastic energy. Furthermore, we explore the dependence of the principal component analysis on the type of the underlying nonlinear elasticity. For the built-in pairwise elastic registration, a relaxed model formulation is employed which allows for a non-exact matching. Shape contours are approximated by single well phase fields, which enables an extension of the method to a covariance analysis of image morphologies. The model is implemented with multilinear finite elements embedded in a multi-scale approach. The characteristics of the approach are demonstrated on a number of illustrative and real world examples in 2D and 3D.