Draft:Commutative Quaternion Model

1. REDIRECT Commutative quaternion model

Commutative Quaternion Model

The commutative quaternion model, also known as the (2,2)-model of quaternions, is an alternative quaternion algebra that allows for commutative multiplication, in contrast to the traditional non-commutative quaternion algebra. This model has been developed for applications in signal and image processing, particularly in color image enhancement, where it simplifies convolution operations and reduces computational complexity.

Overview

The traditional quaternion algebra, often referred to as the (1,3)-model, is defined by a set of imaginary units $\mathbf {i,j,k}$ that follow non-commutative multiplication rules:

$\mathbf {i} ^{2}=\mathbf {j} ^{2}=\mathbf {k} ^{2}=\mathbf {ijk} =-1$

This non-commutativity introduces complexities in defining operations such as convolution and the Fourier transform in quaternion space, making it challenging to apply to image processing tasks.

In contrast, the (2,2)-model of quaternions, introduced by Grigoryan in 2022, is based on a pair of complex numbers, ensuring commutative multiplication:

$q=[a_{1},a_{2}],\quad a_{1},a_{2}\in \mathbb {C}$

where each quaternion is represented as a pair of complex numbers instead of a triplet of imaginary components.

Mathematical Formulation

In the (2,2)-model, the sum of two quaternions is defined component-wise:

$q_{1}+q_{2}=[a_{1}+b_{1},a_{2}+b_{2}]$

whereas multiplication is defined as:

$q_{1}q_{2}=[a_{1}b_{1}-a_{2}b_{2},a_{1}b_{2}+a_{2}b_{1}]$

This ensures that quaternion multiplication remains commutative:

$q_{1}q_{2}=q_{2}q_{1}$

which is not the case in the traditional (1,3)-model.

Quaternion Fourier Transform in the (2,2)-Model

A key advantage of the (2,2)-model is the simplification of the Quaternion Discrete Fourier Transform (QDFT). In this model, the QDFT is uniquely defined due to the limited number of valid quaternion exponentials, unlike in the (1,3)-model, where an infinite number of exponentials exist.

The basis functions for the QDFT are derived from the exponential functions:

$e^{\mu \varphi }=\cos \varphi +\mu \sin \varphi$

where $\mu$ is one of two specific quaternion units, reducing ambiguity in transformation selection.

This property ensures that linear convolution in the (2,2)-model can be reduced to multiplication in the frequency domain, making it computationally efficient for image processing applications.

Applications in Image Processing

One of the primary applications of the (2,2)-model is color image enhancement. Traditional grayscale and color image enhancement techniques often process color channels separately, leading to color distortions and artifacts. The (2,2)-model allows for processing all color components of a pixel as a single quaternion unit, preserving color consistency.

Techniques such as alpha-rooting, a Fourier-based enhancement method, have been adapted to the (2,2)-model, demonstrating superior performance in contrast enhancement and detail preservation.

Key Benefits

• Unique Convolution Definition: Reduces to simple multiplication in the frequency domain.
• Computational Efficiency: Simplifies quaternion-based image processing.
• Color Preservation: Processes all color channels as a single entity, avoiding distortions.

Comparison with the (1,3)-Model

Feature	(1,3)-Model (Traditional)	(2,2)-Model (Commutative)
Commutativity	No	Yes
Convolution	Multiple definitions	Unique definition
QDFT	Infinite possible versions	Only two valid versions

Conclusion

The (2,2)-model of quaternions represents a significant advancement in quaternion algebra, particularly for applications requiring efficient and robust transformations, such as image processing. By ensuring commutative multiplication and unique convolution, it provides a practical alternative to traditional quaternion algebra, with direct benefits for Fourier-based techniques like alpha-rooting.

References

References

• Grigoryan, A. M.; Gomez, A. A. (2025). "Commutative Quaternion Algebra with Quaternion Fourier Transform-Based Alpha-Rooting Color Image Enhancement". Computers. 14 (37). doi:10.3390/computers14020037.{{cite journal}}: CS1 maint: unflagged free DOI (link)
• Grigoryan, A. M.; Gomez, A. A. (2024). Quaternion Fourier Transform-Based Alpha-Rooting Color Image Enhancement in 2 Algebras: Commutative and Non-Commutative. Proc. of SPIE Vol. 13033. doi:10.1117/12.3017692.