Abstract
We present a novel approach for the derivation of PDE modeling curvature-driven flows for matrix-valued data. This approach is based on the Riemannian geometry of the manifold of symmetric positive-definite matrices \(\mathcal{P}(n)\). The differential geometric attributes of \(\mathcal{P}(n)\) −such as the bi-invariant metric, the covariant derivative and the Christoffel symbols− allow us to extend scalar-valued mean curvature and snakes methods to the tensor data setting. Since the data live on \(\mathcal{P}(n)\), these methods have the natural property of preserving positive definiteness of the initial data. Experiments on three-dimensional real DT-MRI data show that the proposed methods are highly robust.
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Zéraï, M., Moakher, M. (2007). Riemannian Curvature-Driven Flows for Tensor-Valued Data. In: Sgallari, F., Murli, A., Paragios, N. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2007. Lecture Notes in Computer Science, vol 4485. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72823-8_51
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DOI: https://doi.org/10.1007/978-3-540-72823-8_51
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