Polar Wavelets in Space
This work provides a practical tool for signal processing and imaging applications requiring steerable wavelets, but the improvements are incremental over existing frameworks.
The authors derived closed-form expressions for polar wavelets in space, enabling fast transforms and frame representations of the Laplace operator. They demonstrated practical benefits in signal estimation from non-uniform samples and reconstruction over lower-dimensional sub-manifolds, achieving improved accuracy.
Recent work introduced a unified framework for steerable and directional wavelets in two and three dimensions that ensures many desirable properties, such as a multi-scale structure, fast transforms, and a flexible angular localization. We show that, for an appropriate choice for the radial window function, these wavelets also have closed form expressions for, among other things, the spatial representation, the filter taps for the fast transform, and the frame representation of the Laplace operator. The numerical practicality and benefits of our work are demonstrated using signal estimation from non-uniform, point-wise samples, as required for example in ray tracing, and for reconstructing a signal over a lower-dimensional sub-manifold, with applications for instance in medical imaging.