Left-invariant evolutions of wavelet transforms on the Similitude Group

TL;DR

This paper introduces a novel wavelet transform on the similitude group to enhance elongated structures in noisy images, preserving crossings and improving over existing PDE-based methods.

Contribution

It develops a continuous wavelet transform on SIM(2) and employs left-invariant evolutions for robust, crossing-preserving image enhancement, advancing prior PDE techniques.

Findings

Improved enhancement of elongated structures in noisy images.

Better preservation of crossings compared to existing methods.

Demonstrated advantages over previous Euclidean motion group approaches.

Abstract

Enhancement of multiple-scale elongated structures in noisy image data is relevant for many biomedical applications but commonly used PDE-based enhancement techniques often fail at crossings in an image. To get an overview of how an image is composed of local multiple-scale elongated structures we construct a multiple scale orientation score, which is a continuous wavelet transform on the similitude group, SIM(2). Our unitary transform maps the space of images onto a reproducing kernel space defined on SIM(2), allowing us to robustly relate Euclidean (and scaling) invariant operators on images to left-invariant operators on the corresponding continuous wavelet transform. Rather than often used wavelet (soft-)thresholding techniques, we employ the group structure in the wavelet domain to arrive at left-invariant evolutions and flows (diffusion), for contextual crossing preserving enhancement of multiple scale elongated structures in noisy images. We present experiments that display benefits of our work compared to recent PDE techniques acting directly on the images and to our previous work on left-invariant diffusions on orientation scores defined on Euclidean motion group.