Left-invariant evolutions of wavelet transforms on the similitude group

• Design an application appropriate SIM(2)-wavelet transform of an image
• Geometric and probabilistic interpretation of left-invariant PDEs on SIM(2) group
• Analytic approximations of Green's function of linear diffusion on SIM(2)
• Crossing preserving multi-scale flows via nonlinear diffusions on SIM(2) group

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 continuous wavelet transform on the similitude group, SIM(2)SIM(2). Our unitary transform maps the space of images onto a reproducing kernel space defined on SIM(2)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 Coherent state transforms defined on Euclidean motion group.