Finsler Active Contours

Abstract

In this paper, we propose an image segmentation technique based on augmenting the conformal (or geodesic) active contour framework with directional information. In the isotropic case, the euclidean metric is locally multiplied by a scalar conformal factor based on image information such that the weighted length of curves lying on points of interest (typically edges) is small. The conformal factor that is chosen depends only upon position and is in this sense isotropic. Although directional information has been studied previously for other segmentation frameworks, here, we show that if one desires to add directionality in the conformal active contour framework, then one gets a welldefined minimization problem in the case that the factor defines a Finsler metric. Optimal curves may be obtained using the calculus of variations or dynamic programming-based schemes. Finally, we demonstrate the technique by extracting roads from aerial imagery, blood vessels from medical angiograms, and neural tracts from diffusion-weighted magnetic resonance imagery.

A Riemannian scalar measure for diffusion tensor images

“…This model is insufficient in voxels that contain two or more bundles of axonal fibers with different orientation. To be able to model more complex shapes of diffusion profiles, we use Finsler geometry [25,26,27,28,29,30], which is a general framework that also includes Riemann geometry as a special case. Instead of DTI, we consider high angular resolution diffusion images (HARDI) [31] that contain more angular measurements than the DTI, although it is possible to use (typically up to 6th order) Finsler-model also on DTI.…”

“…Then, all we need to do is to convexify this profile. This can be done for example as suggested in [28], or by representing the diffusion profile as an nth order polynomial and then taking a nth root as suggested in [30]. Whatever the method, as soon as we have a (strongly) convex [25] modification of the spherical diffusion profile, we can compute directional metric tensors g ij (y) that locally approximate the profile [26,30].…”

A New Tensorial Framework for Single-Shell High Angular Resolution Diffusion Imaging

“…The generalized counterpart of (24) suggests a Finsler [6,7,34] rather than Riemannian [5,16,31,32,38,39] framework for tractography and connectivity analysis, whereby one replaces the contravariant rank-2 (dual metric) tensor…”

Riemann-Finsler Geometry for Diffusion Weighted Magnetic Resonance Imaging

“…Riemann-Finsler geometry appears to be ideally suited for this purpose, as has already been hinted upon in previous work [16,30,31,32]. However, foregoing work is either driven by heuristics or merely scratches the surface of Riemann-Finsler geometry.…”