Todd A. Ell

Hypercomplex Fourier Transforms of Color Images

Fourier transforms are a fundamental tool in signal and image processing, yet, until recently, there was no definition of a Fourier transform applicable to color images in a holistic manner. In this paper, hypercomplex numbers, specifically quaternions, are used to define a Fourier transform applicable to color images. The properties of the transform are developed, and it is shown that the transform may be computed using two standard complex fast Fourier transforms. The resulting spectrum is explained in terms of familiar phase and modulus concepts, and a new concept of hypercomplex axis. A method for visualizing the spectrum using color graphics is also presented. Finally, a convolution operational formula in the spectral domain is discussed

Hypercomplex correlation techniques for vector images

Correlation techniques have been applied to almost every area of signal processing over the past century, yet their use has, in general, been limited to scalar signals. While there have been implementations in multichannel applications, these can be characterized as a combination of single channel processes. True vector correlation techniques, with global and interchannel measures, have only recently been demonstrated and are still in their infancy by comparison. This paper describes our work on vector correlation based on the use of hypercomplex Fourier transforms and presents, for the first time, a unified theory behind the information contained in the peak of a vector correlation response. By using example applications for color images, we also demonstrate some of the practical implications, together with our latest results.

Quaternion involutions and anti-involutions

An involution or anti-involution is a self-inverse linear mapping. In this paper we study quaternion involutions and anti-involutions. We review formal axioms for such involutions and anti-involutions. We present two mappings, one a quaternion involution and one an anti-involution, and a geometric interpretation of each as reflections. We present results on the composition of these mappings and show that the quaternion conjugate may be expressed using three mutually perpendicular anti-involutions. Finally, we show that projection of a vector or quaternion can be expressed concisely using three mutually perpendicular anti-involutions.

Hypercomplex auto- and cross-correlation of color images

Autocorrelation and cross-correlation have been defined and utilized in signal and image processing for many years, but not for color or vector images. In this paper we present for the first time a definition of correlation applicable to color images, based on quaternions or hypercomplex numbers. We have devised a visualization of the result using the polar form of a quaternion in which color denotes quaternion eigenaxis and phase, and a grayscale image represents the modulus.

Hypercomplex Wiener-Khintchine theorem with application to color image correlation

The auto-correlation and cross-correlation function have been generalized to color images, but only by explicit evaluation of a double summation over the whole image for each pixel of the correlation result. Practical use of any correlation method on images of reasonable spatial resolution requires realization using a Fourier transform. A formula for computing the cross-correlation of a pair of images (and hence the auto-correlation of a single image) is presented, based on a previously published hypercomplex Fourier transform.

Hypercomplex Fourier transforms of color images

Hypercomplex Fourier transforms based on quaternions have been proposed by several groups for use in image processing, particularly of color images. So far, however, there has not been a coherent explanation of what the spectral domain coefficients produced by a hypercomplex Fourier transform represent and this paper attempts to present such an explanation for the first time making use of the polar form of a quaternion and a separation of a quaternion spectral coefficient into components parallel and perpendicular to the hypercomplex exponentials in the transform (the basis functions).Fourier transforms are a fundamental tool in signal and image processing, yet, until recently, there was no definition of a Fourier transform applicable to color images in a holistic manner. In this paper, hypercomplex numbers, specifically quaternions, are used to define a Fourier transform applicable to color images. The properties of the transform are developed, and it is shown that the transform may be computed using two standard complex fast Fourier transforms. The resulting spectrum is explained in terms of familiar phase and modulus concepts, and a new concept of hypercomplex axis. A method for visualizing the spectrum using color graphics is also presented. Finally, a convolution operational formula in the spectral domain is discussed

Hypercomplex color-sensitive smoothing filters

In this paper we present a new color-sensitive low pass filter that may he tuned to smooth the color image component in the direction of the color C/sub 1/. A new method for detecting the boundaries of C/sub 1/-colored objects using the color-sensitive smoothing filter is also presented. We demonstrate the color-sensitive smoothing filter and edge detection procedure on both synthetic and natural color images.

Color-grayscale image registration using hypercomplex phase correlation

Phase correlation has been used for the registration of grayscale images for the past 25 years. A previously published hypercomplex generalization of phase correlation has been applied to the registration of vector, or color, images. This paper demonstrates phase-correlation between a grayscale and a color image using the two possibilities for encoding the grayscale image in a quaternion image format. By implementing both the scalar and vector forms of the grayscale image this paper compares, for the first time, vector-vector to scalar-vector hypercomplex correlation. While both are found to give a similar performance, for this application, the analysis of the hypercomplex response finds that the inherent geometry of the quaternionic encoding hides the underlying processing and the phase and axis information. The choice of which method to employ in such applications should therefore be considered with due respect to the encoding of the data in either case.

Quaternion Fourier Transforms for Signal and Image Processing: Ell/Quaternion Fourier Transforms for Signal and Image Processing

Based on updates to signal and image processing technology made in the last two decades, this text examines the most recent research results pertaining to Quaternion Fourier Transforms. QFT is a central component of processing color images and complex valued signals. The books attention to mathematical concepts, imaging applications, and Matlab compatibility render it an irreplaceable resource for students, scientists, researchers, and engineers.

Fundamental Representations and Algebraic Properties of Biquaternions or Complexified Quaternions

The fundamental properties of biquaternions (complexified quaternions) are presented including several different representations, some of them new, and definitions of fundamental operations such as the scalar and vector parts, conjugates, semi-norms, polar forms, and inner products. The notation is consistent throughout, even between representations, providing a clear account of the many ways in which the component parts of a biquaternion may be manipulated algebraically.