Left-invariant evolutions of wavelet transforms on the similitude group

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 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 leftinvariant 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.

Design and Processing of Invertible Orientation Scores of 3D Images

“…Methods that deal with crossings by applying coherenceenhancing diffusion via 2D orientation scores have been developed for 2D data [37,64]. Here, we propose an extension Fig.…”

“…In these cases it is useful to work with multi-orientation representations of image data. Such multi-orientation representations can be made using various techniques, such as invertible orientation scores (which is obtained via a coherent state transform) [3,5,10,30,36,42], continuous wavelet transforms [10,28,30,64], orientation lifts [13,71], or orientation channel representations [35]. Here we focus on constructing an invertible orientation score.…”

“…Let us nevertheless mention that our work fits in a larger Lie group theoretical framework, see, for example, [3,7,25,26,39] that has many applications in image processing. Besides the special cases of the Heisenberg group [6,31,57], the 2D Euclidean motion group [5,9,21,22,30,53,59], the similitude group [4,56,64] and the 3D rotation group [52], we now consider invertible orientation scores on the 3D Euclidean motion group (in which the coupled space of positions and orientations is embedded). The invertible orientation scores relate to coherent states from physics [42] for n = 3, with a specific (semi-)analytic design for our image processing purposes.…”

“…In [37,64] the orientation scores were used to perform crossing-preserving coherenceenhancing diffusions. These diffusions greatly reduce the noise in the data, while preserving the elongated crossing structures.…”

Approximative Coding Methods for Channel Representations

“…Log-polar coordinate systems have been applied to images before, e.g., for filter design in the Fourier domain [12] and similitude group invariant transforms, both globally [9] and locally [28].…”