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Locally adaptive frames in the roto-translation group and their applications in medical imaging. (English) Zbl 1350.42050

Summary: 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 \(U:\mathbb {R}^{d} \rtimes S^{d-1} \to \mathbb {R}\) defined on the extended space of positions and orientations, which we relate to data on the roto-translation group \(\mathrm{SE}(d)\), \(d=2,3\). 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 \(\mathrm{SE}(d)\). 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 \(\mathrm{SE}(d)\). 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 \(\mathrm{SE}(d)\). 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.

MSC:

42C15 General harmonic expansions, frames
92C55 Biomedical imaging and signal processing
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
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