Manifold-valued image processing with SPD matrices

International audienceSymmetric positive definite (SPD) matrices are geometric data that appear in many applications. They are used in particular in Diffusion Tensor Imaging (DTI) as a simple model of the anisotropic diffusion of water in the tissues. This chapter extends the Riemannian computing statistical estimation framework of chapters 1 and 2 to manifold-valued images with the example of SPD matrices. SPD matrices constitute a smooth but incomplete manifold with the classical Euclidean metric on matrices. This creates important computational problems for image processing since we easily pass the boundaries to end-up with negative eigenvalues.This chapter describes how we can give this space more interesting properties thanks to invariance: asking the Riemannian metric on covariance matrices to be invariant by affine changes of coordinates of the space leads to a one-parameter family of affine-invariant metrics that share the same affine connection. The geodesics are thus the same for all these metrics, even if the distance in the direction of scaling is different. With this structure, SPD matrices with null eigenvalues are at an infinite distance of any SPD matrix, so that they cannot be reached in finite time. The space obtained is a typical example of a Hadamard space of nonpositive curvature. It is also a symmetric Riemannian space of nonconstant curvature. This can be seen thanks to the explicit computation of the Riemannian, sectional and Ricci curvatures.Thanks to the relative simplicity of the geodesic equation, we can work out in detail the algorithms for statistical estimation, for instance, the Fréchet mean computation. Moreover, the knowledge of the Ricci curvature allows us to see clearly what is controlling the difference between tangent PCA and Principal Geodesic Analysis (PGA). Building on the reformulation of the mean value in manifolds as the minimization of an intrinsic functional, we also generalize many important image processing algorithms such as interpolation, filtering, diffusion and restoration of missing data to manifold-valued images.We illustrate this framework on diffusion tensor image processing and on the modeling of the brain variability from a dataset of lines on the cerebral cortex. We also discuss alternative choices to the affine-invariant metrics. Log-Euclidean metrics, for instance, provide a very fast approximation when the data are concentrated with respect to the curvature. Other metrics give to the SPD space a positive curvature with a boundary at finite distance for rank-deficient matrices. Since the metric determines the main properties of the space, we should carefully review the assumptions that we can do on our data. Once the metric is chosen, the implementation of the geodesics (Exp and Log) provides once again the basis for all our manifold-valued image processing algorithms