Singular Value Decompositions of Third-Order Reduced Biquaternion Tensors in Color Video Processing

المفاهيم الأساسية

The author introduces the applications of third-order reduced biquaternion tensors in color video processing, focusing on the singular value decomposition (SVD) and Moore-Penrose inverse. The approach aims to compress color videos efficiently using these tensor operations.

الملخص

The content discusses the application of third-order reduced biquaternion tensors in color video processing, emphasizing SVD computation and Moore-Penrose inverse definitions. It explores theoretical applications and solutions to tensor equations, highlighting their significance in signal processing and data analysis.

الإحصائيات

A = A(c),1e1 + A(c),2e2 ∈ Hm×n_c.
B = B(c),1e1 + B(c),2e2 ∈ Hm×l_c.
∥A∥F = 1/√n3∥diag(ˆA)∥F.
∥A∥∞ = max(1≤i≤n1n3)(Pn2j=1(|cij| + |dij|)).
X = Fold(1/√n3(F*n3 ⊗ In2)X).

اقتباسات

"The experimental data shows that our method is faster than the compared scheme."
"In this paper, we call it 'reduced biquaternion' (or RB for short)."
"Reduced biquaternion Hc has numerous applications in many areas."

الرؤى الأساسية المستخلصة من

Singular value decompositions of third-order reduced biquaternion tensors

by Cui-E Yu,Xin... في arxiv.org 03-05-2024

https://arxiv.org/pdf/2403.01690.pdf

استفسارات أعمق

What are the practical implications of using reduced biquaternion tensors in color video compression beyond speed

Reduced biquaternion tensors have practical implications in color video compression beyond just speed. By utilizing reduced biquaternions, the color video processing can benefit from improved accuracy and quality of compressed videos. Reduced biquaternion tensors offer a more efficient representation of color data compared to traditional methods, leading to better preservation of image details and colors during compression. This results in higher-quality compressed videos with reduced artifacts and distortions, making them suitable for applications where visual fidelity is crucial, such as medical imaging or surveillance systems.

How do reduced biquaternions compare to other tensor decomposition methods in terms of efficiency and accuracy

When comparing reduced biquaternions to other tensor decomposition methods in terms of efficiency and accuracy, several factors come into play. Reduced biquaternion tensors provide a unique approach that combines the benefits of quaternion algebra with commutativity properties similar to real numbers. This allows for simpler computations and faster algorithms while maintaining accuracy in representing complex color data.
In terms of efficiency, reduced biquaternion tensor decompositions offer advantages such as faster computation times due to their simplified operations compared to traditional quaternions or complex numbers. The Ht-product used in reduced biquaternion tensors enables efficient calculations for singular value decompositions (SVDs) and Moore-Penrose inverses.
Regarding accuracy, reduced biquaternions excel at preserving information-rich structures like color images or videos due to their ability to handle multidimensional data effectively. Their unique properties make them well-suited for applications requiring precise representations without sacrificing computational speed.

How can the concept of reduced biquaternions be applied to other fields outside of color video processing

The concept of reduced biquaternions can be applied beyond color video processing to various fields where multi-way data analysis is essential. Some potential applications include:

Signal Processing: Reduced biquaternion tensors can be utilized in signal processing tasks such as audio signal analysis or radar signal processing. Their ability to capture multidimensional relationships makes them valuable for extracting meaningful insights from complex signals.
Neuroscience: In neuroscience research, reduced biquaternion tensors could help analyze brain imaging data efficiently by capturing intricate patterns within neural networks.
Data Mining: Reduced bi-quaternions may find applications in analyzing large datasets with multiple dimensions, enabling researchers to uncover hidden patterns or correlations within the data.
Computer Vision: In computer vision tasks like object recognition or image classification, reduced bi-quaternions could enhance feature extraction processes by preserving spatial relationships among pixels accurately.
By leveraging the unique properties of reduced bi-quaternions across diverse domains, researchers can explore innovative solutions for complex data analysis challenges outside the realm of color video processing.