Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations
Journal of Computational Physics
Active vision
Region-based strategies for active contour models
International Journal of Computer Vision
Shape Modeling with Front Propagation: A Level Set Approach
IEEE Transactions on Pattern Analysis and Machine Intelligence
Computable elastic distances between shapes
SIAM Journal on Applied Mathematics
CONDENSATION—Conditional Density Propagation forVisual Tracking
International Journal of Computer Vision
The Geometry of Algorithms with Orthogonality Constraints
SIAM Journal on Matrix Analysis and Applications
The velocity snake: deformable contour for tracking in Spatio-Velocity space
Computer Vision and Image Understanding
DEFORMOTION: Deforming Motion, Shape Average and the Joint Registration and Segmentation of Images
ECCV '02 Proceedings of the 7th European Conference on Computer Vision-Part III
Gradient flows and geometric active contour models
ICCV '95 Proceedings of the Fifth International Conference on Computer Vision
ICCV '95 Proceedings of the Fifth International Conference on Computer Vision
ICCV '95 Proceedings of the Fifth International Conference on Computer Vision
A Statistical Approach to Snakes for Bimodal and Trimodal Imagery
ICCV '99 Proceedings of the International Conference on Computer Vision-Volume 2 - Volume 2
Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms
International Journal of Computer Vision
Motion Competition: A Variational Approach to Piecewise Parametric Motion Segmentation
International Journal of Computer Vision
Approximations of Shape Metrics and Application to Shape Warping and Empirical Shape Statistics
Foundations of Computational Mathematics
CVPR '05 Proceedings of the 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05) - Volume 2 - Volume 02
Conformal Metrics and True "Gradient Flows" for Curves
ICCV '05 Proceedings of the Tenth IEEE International Conference on Computer Vision (ICCV'05) Volume 1 - Volume 01
Designing Spatially Coherent Minimizing Flows for Variational Problems Based on Active Contours
ICCV '05 Proceedings of the Tenth IEEE International Conference on Computer Vision - Volume 2
Generalized Gradients: Priors on Minimization Flows
International Journal of Computer Vision
International Journal of Computer Vision
Geometric modeling in shape space
ACM SIGGRAPH 2007 papers
A Variational Technique for Time Consistent Tracking of Curves and Motion
Journal of Mathematical Imaging and Vision
Coarse-to-Fine Segmentation and Tracking Using Sobolev Active Contours
IEEE Transactions on Pattern Analysis and Machine Intelligence
Geometric Observers for Dynamically Evolving Curves
IEEE Transactions on Pattern Analysis and Machine Intelligence
Nonlinear Dynamical Shape Priors for Level Set Segmentation
Journal of Scientific Computing
View Point Tracking of Rigid Objects Based on Shape Sub-manifolds
ECCV '08 Proceedings of the 10th European Conference on Computer Vision: Part III
New Possibilities with Sobolev Active Contours
International Journal of Computer 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
Deform PF-MT: particle filter with mode tracker for tracking nonaffine contour deformations
IEEE Transactions on Image Processing
Elastic-string models for representation and analysis of planar shapes
CVPR'04 Proceedings of the 2004 IEEE computer society conference on Computer vision and pattern recognition
VLSM'05 Proceedings of the Third international conference on Variational, Geometric, and Level Set Methods in Computer Vision
Calculus of Nonrigid Surfaces for Geometry and Texture Manipulation
IEEE Transactions on Visualization and Computer Graphics
IEEE Transactions on Image Processing
IEEE Transactions on Image Processing
A fully implicit framework for Sobolev active contours and surfaces
DAGM'11 Proceedings of the 33rd international conference on Pattern recognition
Weakly convex coupling continuous cuts and shape priors
SSVM'11 Proceedings of the Third international conference on Scale Space and Variational Methods in Computer Vision
A Reparameterisation Based Approach to Geodesic Constrained Solvers for Curve Matching
International Journal of Computer Vision
Time-Discrete Geodesics in the Space of Shells
Computer Graphics Forum
Multiple object tracking via prediction and filtering with a sobolev-type metric on curves
ECCV'12 Proceedings of the 12th international conference on Computer Vision - Volume Part I
Modelling Convex Shape Priors and Matching Based on the Gromov-Wasserstein Distance
Journal of Mathematical Imaging and Vision
Global structure constrained local shape prior estimation for medical image segmentation
Computer Vision and Image Understanding
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We define a novel metric on the space of closed planar curves which decomposes into three intuitive components. According to this metric, centroid translations, scale changes, and deformations are orthogonal, and the metric is also invariant with respect to reparameterizations of the curve. While earlier related Sobolev metrics for curves exhibit some general similarities to the novel metric proposed in this work, they lacked this important three-way orthogonal decomposition, which has particular relevance for tracking in computer vision. Another positive property of this new metric is that the Riemannian structure that is induced on the space of curves is a smooth Riemannian manifold, which is isometric to a classical well-known manifold. As a consequence, geodesics and gradients of energies defined on the space can be computed using fast closed-form formulas, and this has obvious benefits in numerical applications. The obtained Riemannian manifold of curves is ideal for addressing complex problems in computer vision; one such example is the tracking of highly deforming objects. Previous works have assumed that the object deformation is smooth, which is realistic for the tracking problem, but most have restricted the deformation to belong to a finite-dimensional group—such as affine motions—or to finitely parameterized models. This is too restrictive for highly deforming objects such as the contour of a beating heart. We adopt the smoothness assumption implicit in previous work, but we lift the restriction to finite-dimensional motions/deformations. We define a dynamical model in this Riemannian manifold of curves and use it to perform filtering and prediction to infer and extrapolate not just the pose (a finitely parameterized quantity) of an object but its deformation (an infinite-dimensional quantity) as well. We illustrate these ideas using a simple first-order dynamical model and show that it can be effective even on image sequences where existing methods fail.