Matrix analysis
Think globally, fit locally: unsupervised learning of low dimensional manifolds
The Journal of Machine Learning Research
A Statistical Approach to Texture Classification from Single Images
International Journal of Computer Vision - Special Issue on Texture Analysis and Synthesis
Statistical Shape Analysis: Clustering, Learning, and Testing
IEEE Transactions on Pattern Analysis and Machine Intelligence
A Riemannian Framework for Tensor Computing
International Journal of Computer Vision
A computational approach to fisher information geometry with applications to image analysis
EMMCVPR'05 Proceedings of the 5th international conference on Energy Minimization Methods in Computer Vision and Pattern Recognition
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We present an algorithm for grouping families of probability density functions (pdfs). We exploit the fact that under the square-root re-parametrization, the space of pdfs forms a Riemannian manifold, namely the unit Hilbert sphere. An immediate consequence of this re-parametrization is that different families of pdfs form different submanifolds of the unit Hilbert sphere. Therefore, the problem of clustering pdfs reduces to the problem of clustering multiple submanifolds on the unit Hilbert sphere. We solve this problem by first learning a low-dimensional representation of the pdfs using generalizations of local nonlinear dimensionality reduction algorithms from Euclidean to Riemannian spaces. Then, by assuming that the pdfs from different groups are separated, we show that the null space of a matrix built from the local representation gives the segmentation of the pdfs. We also apply of our approach to the texture segmentation problem in computer vision.