Fast and Simple Calculus on Tensors in the Log- Euclidean Framework Vincent Arsigny, Pierre Fillard, Xavier Pennec, Nicholas Ayache. Research Project/Team EPIDAURE/ASCLEPIOS INRIA, Sophia-Antipolis, France. 8th International Conference on Medical Image Computing and Computer Assisted Intervention, Oct 26 to 30, 2005.
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Fast and Simple Calculus on Tensors in the Log-Euclidean Framework
8th International Conference on M edical I mage C omputing and C omputer A ssisted I ntervention, Oct 26 to 30, 2005. Fast and Simple Calculus on Tensors in the Log-Euclidean Framework. Vincent Arsigny, Pierre Fillard, Xavier Pennec, Nicholas Ayache. - PowerPoint PPT Presentation
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Fast and Simple Calculus on Tensors in the Log-Euclidean
Framework
Vincent Arsigny, Pierre Fillard,
Xavier Pennec, Nicholas Ayache.
Research Project/Team EPIDAURE/ASCLEPIOSINRIA, Sophia-Antipolis, France.
8th International Conference on Medical Image Computing and Computer Assisted Intervention, Oct 26 to 30, 2005.
October 27th, 2005 Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05
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What are ‘tensors’?
• In general: all multilinear applications.
• In this talk: symmetric positive-definite matrices. – Typically : covariance matrices.
October 27th, 2005 Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05
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Diffusion Tensor MRI
• Diffusion-weighted MR images
• Diffusion Tensor: local covariance of diffusion [Basser, 94].
• Generalization of vector processing tools (filtering, statistics, etc.) to tensors?
October 27th, 2005 Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05
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Outline
1. Presentation
2. Euclidean and Affine-Invariant Calculus
3. Log-Euclidean Framework
4. Experimental Results
5. Conclusions and Perspectives
October 27th, 2005 Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05
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Euclidean calculus
• DTs: 3x3 symmetric matrices, thus belong to a vector space.
• Simple, but: – unphysical negative eigenvalues appear– ‘swelling effect’: more diffusion than originally.
October 27th, 2005 Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05
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Remedies in the literature
• First family:1. process features from tensors
2. propagate processing to tensors.
• Example: regularization– dominant directions of diffusion [Coulon, IPMI’01]– orientations and eigenvalues separately [Tschumperlé,
IJCV, 02, Chefd’hotel JMIV, 04].
• Drawback: some information left behind.
October 27th, 2005 Vincent Arsigny et al., Log-Euclidean Framework, MICCAI'05