Stephen J. Sangwine

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

The colour image processing handbook

Preface. Acknowledgements. 1. The present state and the future of colour image processing. Part One: Colour. 2. Colour vision W. McIlhagga. 3. Colour science M. Ronnier Luo. 4. Colour spaces H. Palus. Part Two: Image Acquisition. 5. Colour video systems and signals R.E.N. Horne. 6. Image sources C. Connolly. 7. Practical system considerations C. Connolly, H. Palus. Part Three: Processing. 8. Noise removal and contrast enhancement J.M. Gauch. 9. Segmentation and edge detection J.M. Gauch. 10. Vector filtering K.N. Plataniotis, A.N. Venetsanopoulos. 11. Morphological operations M.L. Comer, E.J. Delp. 12. Frequency domain methods S.J. Sangwine, A.L. Thornton. 13. Compression M. Domanski, M. Bartkowiak. Part Four: Applications. 14. Colour management for the textile industry P.A. Rhodes. 15. Colour management for the graphic arts J. Morovic. 16. Medical imaging case study B.F. Jones, P. Plassmann. 17. Industrial colour inspection case studies C. Connolly. References. Index.

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 principal component analysis of color images

In this paper, we present quaternion matrix algebra techniques that can be used to process the eigen analysis of a color image. Applications of principal component analysis (PCA) in image processing are numerous, and the proposed tools aim to give material for color image processing, that take into account their particular nature. For this purpose, we use the quaternion model for color images and introduce the extension of two classical techniques to their quaternionic case: singular value decomposition (SVD) and Karhunen-Loeve transform (KLT). For the quaternionic version of the KLT, we also introduce the problem of eigenvalue decomposition (EVD) of a quaternion matrix. We give the properties of these quaternion tools for color images and present their behavior on natural images. We also present a method to compute the decompositions using complex matrix algebra. Finally, we start a discussion on possible applications of the proposed techniques in color images processing.

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.

Fast Complexified Quaternion Fourier Transform

In this paper, we consider the extension of the Fourier transform to biquaternion-valued signals. We introduce a transform that we call the biquaternion Fourier transform (BiQFT). After giving some general properties of this transform, we show how it can be used to generalize the notion of analytic signal to complex-valued signals. We introduce the notion of hyperanalytic signal. We also study the Hermitian symmetries of the BiQFT and their relation to the geometric nature of a biquaternion-valued signal. Finally, we present a fast algorithm for the computation of the BiQFT. This algorithm is based on a (complex) change of basis and four standard complex FFTs.

Quaternion singular value decomposition based on bidiagonalization to a real or complex matrix using quaternion Householder transformations

We present a practical and efficient means to compute the singular value decomposition (SVD) of a real or complex quaternion matrix A based on bidiagonalization of A to a real or complex bidiagonal matrix B using quaternionic Householder transformations. Computation of the SVD of B using an existing subroutine library such as lapack provides the singular values of A. The singular vectors of A are obtained trivially from the product of the Householder transformations and the real or complex singular vectors of B. We show in the paper that left and right quaternionic Householder transformations are different because of the non-commutative multiplication of quaternions and we present formulae for computing the Householder vector and matrix in each case.

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