Vector Extension of Monogenic Wavelets for Geometric Representation of Color Images

Abstract

Monogenic wavelets offer a geometric representation of grayscale images through an AM-FM model allowing invariance of coefficients to translations and rotations. The underlying concept of local phase includes a fine contour analysis into a coherent unified framework. Starting from a link with structure tensors, we propose a nontrivial extension of the monogenic framework to vector-valued signals to carry out a nonmarginal color monogenic wavelet transform. We also give a practical study of this new wavelet transform in the contexts of sparse representations and invariant analysis, which helps to understand the physical interpretation of coefficients and validates the interest of our theoretical construction.

A Sinusoidal Image Model Derived from the Circular Harmonic Vector

“…Of particular relevance is that the RT is norm-preserving, R f = f , and invertible, R −n {R n f }(z) = f (z), if f (z) has zero mean. These properties allow the generation of monogenic wavelets [17,36] in multiple dimensions [54] and for colour images [48], as well as monogenic versions of existing quadrature wavelets that give a directional decomposition into amplitude and phase components [50].…”

Color monogenic wavelet transform for multichannel image denoising

“…Then the extension to the wavelet domain is direct with the marginal Riesz features described above. The isotropic poly-harmonic spline wavelet defined in Soulard et al (2013) is used as new smoothing kernel. The following sub-bands decompositions of color monogenic wavelet are given by:…”

“…1. The color monogenic wavelet transform (defined in Soulard et al (2013)) is implemented by using two perfect reconstruction filter-banks separately in parallel: a poly-harmonic wavelet transform and a Riesz wavelet transform. At the analysis stage, H (z) and G(z) (defined in Unser et al (2008)) are low-pass filter and high-pass filters respectively, with down-sampling by a factor of two.…”

“…Demarcq et al (2011) generalized construction of the monogenic signal to mappings with values in the vector part of the Clifford algebra. Soulard et al (2013) proposed non-marginal color monogenic wavelet transform by extension of the monogenic framework to vector-valued signals which has shown to be shift invariant and sparse representation. Elad and Aharon (2006) developed an image denoising algorithm based on sparse and redundant representations of over-learned dictionaries (KSVD).…”

On locality of Radon to Riesz transform

“…The Riesz transform is also the way to construct the monogenic signal in several dimensions, which is the natural extension of the one dimensional analytic signal [9,7,8]. The monogenic signal, as well as the Riesz transform, have many applications in image processing or computer vision, like the demodulation of 2D fringe patterns [11], the extraction of local features in 2D signals [23,16,21,10,4], the demodulation of holograms [17], or the analysis of color images [19].…”