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Hypercomplex Fourier transforms of color images
Todd A Ell1, Stephen J Sangwine
1Goodrich Sensor Systems, Burnsville, MN 55306, USA. t.ell@ieee.org
Summary
This study introduces a novel Fourier transform for color images using hypercomplex numbers (quaternions). This new method enables holistic analysis and spectral visualization of color image data.
Area of Science:
- Image Processing
- Applied Mathematics
- Computer Vision
Background:
- Fourier transforms are essential for signal and image analysis.
- Existing methods lack a holistic approach for color image Fourier transforms.
- Hypercomplex numbers offer a potential framework for extending Fourier analysis.
Purpose of the Study:
- To define a holistic Fourier transform for color images using quaternions.
- To explore the properties and computational aspects of this novel transform.
- To develop methods for spectral visualization and analysis of color image data.
Main Methods:
- Utilizing hypercomplex numbers (quaternions) to formulate the color image Fourier transform.
- Developing the mathematical properties of the quaternion-based Fourier transform.
- Implementing the transform using two standard complex fast Fourier transforms.
- Introducing a hypercomplex axis concept for spectral interpretation.
Main Results:
- A novel quaternion-based Fourier transform for color images is successfully defined.
- The transform can be efficiently computed using existing fast Fourier transform algorithms.
- The spectral representation is explained using phase, modulus, and a new hypercomplex axis.
- A color graphics visualization method for the spectrum is presented.
Conclusions:
- The proposed quaternion Fourier transform provides a unified framework for color image analysis.
- This method offers new insights into image spectral properties and facilitates visualization.
- The transform has potential applications in image processing and computer vision tasks.
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