Manifold-valued image processing and Diffusion tensor imaging
Diffusion tensor imaging (DTI) is an MRI modality that measures at each point of the image the covariance matrix of the Brownian motion of water. Symmetric positive-definite matrices form a cone in the space of symmetric matrices and many usual operations (like averaging) are stable in this space. However, negative eigenvalues appear as soon as gradient descent or more complex algorithms are performed, which was raising important problems for image estimation and smoothing.
Manifold-valued image processing with an affine invariant metric on SPD matrices
In parallel with other groups, I proposed to use an affine-invariant Riemannian metric which rejects the edges of the cone at an infinite distance. However, the Riemannian computing methods previously developed were a key to go much beyond the sole use of the distance and to generalize to manifold-valued images numerous image processing algorithms like interpolation, convolution filtering, anisotropic diffusion and image restoration. The breadth of algorithms considered here was a disruption that grounded the domain of manifold-valued image processing.
The first principle consists in reformulating the weighted averages as the minimization of the weighted Fréchet distance, whose optima are reached by the solutions of partial differential equations. This principle of convolution in manifolds is central to some geometric deep learning techniques that have rediscovered it independently. The second principle is the use of very simple intrinsic numerical schemes based on the exponential map to calculate the gradient and the Laplace-Beltrami operator.
Xavier Pennec, Pierre Fillard, and Nicholas Ayache. A Riemannian Framework for Tensor Computing. International Journal of Computer Vision, 66(1):41-66, Jan. 2006. [PDF] Keywords: Affine invariant metric on SPD matrices, Manifold-valued image processing.
[1700 citations].
The affine invariant metric proposed above gives a Hadamard structure to the space of SPD matrices. This is very convenient as one can use a single chart and the Fréchet mean is unique thanks to the non-positive curvature. However, the curvature lead to iterative algorithms that are obviously more expensive than the explicit solutions found in Euclidean spaces. By trying to put a group structure on the space of SPD matrices, we came out with Vincent Arsigny with a method to flatten this space: because the matrix exponential and logarithm realize a diffeomorphism from symmetric to SPD matrices, we can pull back any Euclidean metric on symmetric matrices to the space of SPD matrices, hence the name of log-Euclidean metrics. This very simple observation radically simplified the implementation of the algorithms and led to very wide adoption of these techniques in the diffusion imaging community.
Estimating and processing diffusion tensor imaging with low SNR
In a clinical environment, Diffusion tensor MRI (DTI) images have to be acquired rapidly, often at the expense of the image quality. With Pierre Fillard, we propose in this paper a new variational framework to improve the estimation of DT-MRI in this low signal to noise ratio (SNR) context. We first propose a maximum likelihood strategy that fully exploits the assumption of a Rician noise in order to avoid the shrinking effect of simpler Gaussian or log-Gaussian noise assumptions that are usually done in higher SNR contexts. To further reduce the influence of the noise, we also propose to use an anisotropic spatial prior based on the regularity of the tensor field itself. Optimizing both terms in a maximum a posteriori (MAP) framework leads to a highly non-linear criterion requiring adapted tools for tensor computing. We show that Log-Euclidean metrics provide a very efficient optimization that makes our framework very easy to use in practice. Results on real clinical data demonstrate the truthfulness of the proposed approach even with very low SNR and show promising improvements of fiber tracking in the brain and the spinal cord.
This method was used in several clinical research studies and its exploitation was initially one of the goals of the Therapixel spin-off created by Olivier Clatz and Pierre Fillard.