Locally Adaptive Frames in the Roto-Translation Group and Their Applications in Medical Imaging

Locally adaptive differential frames (gauge frames) are a well-known effective tool in image analysis, used in differential invariants and PDE-flows. However, at complex structures such as crossings or junctions, these frames are not well defined. Therefore, we generalize the notion of gauge frames on images to gauge frames on data representations U:ℝ^d⋊ S^d-1→ℝ defined on the extended space of positions and orientations, which we relate to data on the roto-translation group SE ( d ), d=2,3 . This allows to define multiple frames per position, one per orientation. We compute these frames via exponential curve fits in the extended data representations in SE ( d ). These curve fits minimize first- or second-order variational problems which are solved by spectral decomposition of, respectively, a structure tensor or Hessian of data on SE ( d ). We include these gauge frames in differential invariants and crossing-preserving PDE-flows acting on extended data representation U and we show their advantage compared to the standard left-invariant frame on SE ( d ). Applications include crossing-preserving filtering and improved segmentations of the vascular tree in retinal images, and new 3D extensions of coherence-enhancing diffusion via invertible orientation scores.