Steerable Wavelet Frames Based on the Riesz Transform

Abstract

We consider an extension of the 1-D concept of analytical wavelet to n-D which is by construction compatible with rotations. This extension, called a monogenic wavelet, yields a decomposition of the wavelet coefficients into amplitude, phase, and phase direction. The monogenic wavelet is based on the hypercomplex monogenic signal which is defined using Riesz transforms and perfectly isotropic wavelets frames. Employing the new concept of Clifford frames, we can show that the monogenic wavelet generates a wavelet frame. Furthermore, this approach yields wavelet frames that are steerable with respect to direction. Applications to descreening and contrast enhancement illustrate the versatility of this approach to image analysis and reconstruction.

Unsupervised texture segmentation using monogenic curvelets and the Potts model

“…They behave like the classical curvelets of [17] at fine scales and like the monogenic wavelets of [23,24] at coarse scales. We give a short description of monogenic curvelets.…”

Coherence Probe Microscopy Imaging and Analysis for Fiber-Reinforced Polymers

“…Therefore, a further image enhancement step comprehends a correction for the non-uniformly shaped background. We have used both, a method based on a morphological opening approach, the well-established so-called rolling ball algorithm as introduced by Sternberg [11], and a method exploiting an (only) phasebased reconstruction scheme applying monogenic wavelets for an equalization of brightness, as recently suggested in [12].…”

“…3. Background correction exemplified on the fiber-reinforced polymer: (a) the demodulated amplitude image as used for analysis, applying in (c) the morphological correction method and (e) the monogenic wavelet-based equalization of brightness approach [12]. The corresponding binary versions are shown in (b), (d), and (f).…”

“…For computing the monogenic orientation and equalization of brightness we have applied both wavelet toolboxes: MonogenicJ [15] and Monogenic Wavelet Toolbox [12].…”

Optimal Isotropic Wavelets for Localized Tight Frame Representations

“…Assumption 2), then we can take advantage of a fast filterbank implementation [3,Section IV]. A general construction for wavelets of this type can be found in [4], and commonly used versions include the following: Held [5], Papadakis [1], and Simoncelli [6] (which was given as an example in [4]). An overview of these can be found in [3].…”