Smooth invariant interpolation of rotations
ACM Transactions on Graphics (TOG)
Eigenfaces vs. Fisherfaces: Recognition Using Class Specific Linear Projection
IEEE Transactions on Pattern Analysis and Machine Intelligence
Hilbert-Schmidt Lower Bounds for Estimators on Matrix Lie Groups for ATR
IEEE Transactions on Pattern Analysis and Machine Intelligence
Spherical averages and applications to spherical splines and interpolation
ACM Transactions on Graphics (TOG)
Means and Averaging in the Group of Rotations
SIAM Journal on Matrix Analysis and Applications
The De Casteljau Algorithm on Lie Groups and Spheres
Journal of Dynamical and Control Systems
Optimal Linear Representations of Images for Object Recognition
IEEE Transactions on Pattern Analysis and Machine Intelligence
Energy-minimizing splines in manifolds
ACM SIGGRAPH 2004 Papers
A Differential Geometric Approach to the Geometric Mean of Symmetric Positive-Definite Matrices
SIAM Journal on Matrix Analysis and Applications
Proceedings of the 2005 ACM symposium on Solid and physical modeling
A Riemannian Framework for Tensor Computing
International Journal of Computer Vision
A two-step algorithm of smooth spline generation on Riemannian manifolds
Journal of Computational and Applied Mathematics
Tracking Deforming Objects Using Particle Filtering for Geometric Active Contours
IEEE Transactions on Pattern Analysis and Machine Intelligence
Bézier curves and C2 interpolation in Riemannian manifolds
Journal of Approximation Theory
On the Geometry of Rolling and Interpolation Curves on Sn, SOn, and Grassmann Manifolds
Journal of Dynamical and Control Systems
Rate-invariant recognition of humans and their activities
IEEE Transactions on Image Processing
Nonstationary Shape Activities: Dynamic Models for Landmark Shape Change and Applications
IEEE Transactions on Pattern Analysis and Machine Intelligence
Deform PF-MT: particle filter with mode tracker for tracking nonaffine contour deformations
IEEE Transactions on Image Processing
Shape Analysis of Elastic Curves in Euclidean Spaces
IEEE Transactions on Pattern Analysis and Machine Intelligence
A Gradient-Descent Method for Curve Fitting on Riemannian Manifolds
Foundations of Computational Mathematics
IEEE Transactions on Image Processing
Image and Vision Computing
Polynomial regression on riemannian manifolds
ECCV'12 Proceedings of the 12th European conference on Computer Vision - Volume Part III
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We address the problem of estimating full curves/paths on certain nonlinear manifolds using only a set of time-indexed points, for use in interpolation, smoothing, and prediction of dynamic systems. These curves are analogous to smoothing splines in Euclidean spaces as they are optimal under a similar objective function, which is a weighted sum of a fitting-related (data term) and a regularity-related (smoothing term) cost functions. The search for smoothing splines on manifolds is based on a Palais metric-based steepest-decent algorithm developed in Samir et al. [38]. Using three representative manifolds: the rotation group for pose tracking, the space of symmetric positive-definite matrices for DTI image analysis, and Kendall's shape space for video-based activity recognition, we demonstrate the effectiveness of the proposed algorithm for optimal curve fitting. This paper derives certain geometrical elements, namely the exponential map and its inverse, parallel transport of tangents, and the curvature tensor, on these manifolds, that are needed in the gradient-based search for smoothing splines. These ideas are illustrated using experimental results involving both simulated and real data, and comparing the results to some current algorithms such as piecewise geodesic curves and splines on tangent spaces, including the method by Kume et al. [24].