Holistic Processing of Colour Images Using Novel Quaternion-Valued Wavelets on the Plane

TL;DR

The paper addresses the challenge of exploiting inter-channel correlations in colour images by embedding RGB(-NIR) data into quaternions and applying a Mallat-style quaternion-valued wavelet transform on the plane. It introduces a quaternion-valued wavelet framework constructed via a feasibility approach, detailing a scaling function and three wavelets that form a non-separable, orthonormal ensemble, and demonstrates decomposition/reconstruction in a colour image context. Key contributions include formalising quaternion-based colour embedding, establishing perfect reconstruction, and showcasing end-to-end processing for compression, enhancement, edge detection, and denoising within a holistic framework. The work suggests notable potential benefits in energy compaction and memory efficiency, with future directions spanning optimization, comparisons to channel-wise methods, and integration with advanced regularisation and quaternion neural networks.

Abstract

Recently, novel quaternion-valued wavelets on the plane were constructed using an optimisation approach. These wavelets are compactly supported, smooth, orthonormal, non-separable and truly quaternionic. However, they have not been tested in application. In this paper, we introduce a methodology for decomposing and reconstructing colour images using quaternionic wavelet filters associated to recently developed quaternion-valued wavelets on the plane. We investigate its applicability in compression, enhancement, segmentation, and denoising of colour images. Our results demonstrate these wavelets as promising tools for an end-to-end quaternion processing of colour images.

Figures (12)

• Figure 1: One step of a Douglas--Rachford fixed-point iteration which follows a simple reflect-reflect-average scheme. Starting with a point $x_0$, the algorithm performs a reflection with respect to $K_1$ to obtain the point $R_{K_1}(x_0)$, followed by another reflection with respect to $K_2$ to obtain the point $R_{K_2}(R_{K_1}(x_0))$. Averaging $x_0$ and $R_{K_2}(R_{K_1}(x_0))$ yields the point corresponding to the next iterate $x_1$.
• Figure 2: An example of a wavelet ensemble generated from a solution of the quaternionic wavelet feasibility problem. For each plot, the height of a point on the graph corresponds to the modulus of the quaternion, and the intensities of RGB colour of the point represent the imaginary parts in the polar form of the quaternion.
• Figure 3: Colour image processing steps using quaternion-valued wavelets on the plane. The basic layout of a one-level discrete wavelet transform (with scaling filter $H$, wavelet filters $G_1, G_2, G_3$, and their respective inverse filters $\tilde{H}, \tilde{G}_1, \tilde{G}_2, \tilde{G}_3$) also includes the downsampling ($\downarrow 2$) and upsampling ($\uparrow 2$) steps.
• Figure 4: Example RGB-NIR image. This also coincides with the reconstructed image when no alternations were made in the wavelet coefficients.
• Figure 5: Example of a one-level wavelet decomposition of an RGB-NIR image (with detail coefficient values inverted and accentuated for illustrative purposes). The colour image on the left displays the imaginary parts of the quaternionic wavelet decomposition treated as RGB, while the greyscale image on the right is the scalar part of the decomposition.