Randomized trial reveals closed-form parallel transport formulas for spectral components in signals, indicating new geometric tools for feature evolution.
Key Points
This research aims to construct a framework for the parallel transport of spectral components along parameter families of signals on the similarity group.
Develop a connection-theoretic framework for parallel transport on the similarity group G˜=R×SO(2)
Derive three explicit parallel transport formulas for different subbundles
Introduce the non-parallelism rate as a deviation measure along the parameter path.
On the equivariant subbundle, transport is an exact isometric translation.
For the coupled subbundle, transport combines log-scale translation and a phase factor, resulting in complex behavior.
Quantitative bounds demonstrate cumulative deviation along the path is controlled by trajectory estimation error and appearance variation.