Draft:Commutative Quaternion Model

1. REDIRECT Quaternion

Commutative Quaternion Model

The commutative quaternion model, also known as the (2,2)-model of quaternions, is an alternative quaternion algebra that ensures commutative multiplication. This contrasts with the traditional non-commutative quaternion algebra, referred to as the (1,3)-model. The (2,2)-model was pioneered by Artyom M. Grigoryan in 2022 and has been developed for applications in mathematics, physics, and engineering, particularly in signal processing, image enhancement, and Fourier analysis, where it provides computational advantages over the traditional model.

Historical Background

Quaternions were introduced by Sir William Rowan Hamilton in 1843 as an extension of complex numbers into four dimensions. The traditional quaternion model follows a non-commutative multiplication rule, making it distinct from complex numbers. However, in 2022, Artyom M. Grigoryan introduced the (2,2)-model, a new quaternion arithmetic in which multiplication is commutative. This allows for a unique definition of operations like convolution and Fourier transforms, making it more applicable to computational tasks.

Mathematical Definition

The traditional quaternion algebra is based on imaginary units $\mathbf {i,j,k}$ that satisfy the relations:

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

A quaternion $q=a+(bi+cj+dk)$ is a number with one real part $a$ and three-component imaginary part $bi+cj+kd$. Therefore, this algebra can be referred to as the (1,3)-model. The above multiplication rule makes this quaternion algebra non-commutative, meaning that for two quaternions $q_{1}$ and $q_{2}$:

$q_{1}q_{2}\neq q_{2}q_{1}$

The (2,2)-model of quaternions, in contrast, represents quaternions as pairs of complex numbers:

$q=[a_{1},a_{2}]=[(a_{1,1},a_{1,2}),(a_{2,1},a_{2,2})],\quad a_{1,1},a_{1,2},a_{2,1},a_{2,2}\in \mathbb {R}$

where 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 commutative multiplication, meaning that:

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

Properties

Some key properties of the (2,2)-model include:

• Commutativity: Unlike traditional quaternions, multiplication in the (2,2)-model follows the rule $q_{1}q_{2}=q_{2}q_{1}$.
• Well-defined Convolution: In the (2,2)-model, convolution is uniquely defined and simplifies to multiplication in the Fourier domain.
• Limited Exponential Basis Functions: The (2,2)-model has only two valid quaternion exponentials, unlike the infinite variations found in the (1,3)-model.
• Quaternion Conjugation: The conjugate of a quaternion $q=[a_{1},a_{2}]$ is given by ${\bar {q}}=[a_{1},-a_{2}]=[(a_{1,1},-a_{1,2}),(a_{2,1},-a_{2,2})]$.

Applications

The (2,2)-model of quaternions has applications in several fields, including:

• Color Image Processing: Used in alpha-rooting and Fourier-based enhancement techniques, where processing entire color channels as quaternions prevents distortions.
• Signal Processing: Improves convolution operations, making it useful for filtering and data compression.
• Neural Networks: Helps define activation functions and transformations that work on quaternion-based deep learning models.
• Physics and Quantum Mechanics: Provides an alternative to traditional quaternion mathematics in simulations and physical modeling.

Key Benefits

• Unique Convolution Definition: Reduces to simple multiplication in the frequency domain.
• Computational Efficiency: Simplifies quaternion-based transformations.
• Avoids Color Distortions: Especially useful in color image enhancement applications.
• More Defined Fourier Analysis: Avoids ambiguities present in traditional quaternion Fourier transforms.

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 versions	Only two valid versions

References

Sources

• Grigoryan, A. M.; Gomez, A. A. (2025). "Commutative Quaternion Algebra with Quaternion Fourier Transform-Based Alpha-Rooting Color Image Enhancement". Computers. 14 (37): 37. doi:10.3390/computers14020037.
• 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.
• Grigoryan, A. M.; Agaian, S. S. (2022). "Commutative quaternion algebra and DSP fundamental properties: Quaternion convolution and Fourier transform". Signal Processing. 196. Bibcode:2022SigPr.19608533G. doi:10.1016/j.sigpro.2022.108533.
• Gonzalez, R. C.; Woods, R. E. (2018). Digital Image Processing (4th ed.). New York, NY: Pearson. ISBN 9780133356724.
• Grigoryan, A. M.; Agaian, S. S. (2018). Quaternion and Octonion Color Image Processing with MATLAB. Bellingham, WA: SPIE Press. ISBN 978-1510611351.
• Gauss, C. F. (1900). Brendel, M. (ed.). Mutationen des Raumes. Carl Friedrich Gauss Werke. Vol. 8. Stuttgart, Germany: Teubner. pp. 357–361.