Computational Information Geometry Wonderland

Medical imaging has been one of the first field to consider non-Euclidean distances (and geometries) for image processing applications. This was a natural move because the L2 norm simply does not work, and those people were also confronted with multi-modal registration (Eg, CT with ultrasound, etc.)

So, checking the literature these past years on leading medical imaging projects such as ASCEPLIOS (http://www-sop.inria.fr/asclepios/) of INRIA Sophia-Antipolis is a golden source of information for my objective of better understanding ``distances''. I came across a series of interesting papers dealing with Riemannian processing.

For example, Log-Euclidean Metrics for Fast and Simple Calculus on Diffusion Tensors Magnetic Resonance in Medicine, Vol. 56, No. 2. (August 2006), pp. 411-421. http://www.citeulike.org/group/2854/article/1646596

makes use of log-Euclidean metrics that in its simplest form is defined as the square root of the Trace of the squared difference of the log matrix (provided a tensor given from Diffusion-Tensor MRI, or DTI for short). The log of a positive definite matrix is simply computed from the spectral decomposition of matrices in M=U Lambda U^T, where Lambda is the diagonal eigenvalue matrix (all positive). Therefore, M=U log(Lambda) U^T

Ok, so good for today. I should try to set the Latex greasemonkey script (http://www.citeulike.org/group/2854/article/1646596) that allows to write tex maths for now!