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:Rd⋊Sd−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