Locally adaptive frames in the roto-translation group and their applications in medical imaging

Abstract

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 \u3cbr/\u3eU:Rd⋊Sd−1→R\u3cbr/\u3e defined on the extended space of positions and orientations, which we relate to data on the roto-translation group SE(d), d=2,3 \u3cbr/\u3ed=2,3\u3cbr/\u3e. 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