## Hypercomplex fourier transforms of color images (2001)

Venue: | in Proc. ICIP |

Citations: | 24 - 1 self |

### BibTeX

@INPROCEEDINGS{Ell01hypercomplexfourier,

author = {Todd A. Ell and Senior Member and Stephen J. Sangwine and Senior Member},

title = {Hypercomplex fourier transforms of color images},

booktitle = {in Proc. ICIP},

year = {2001},

pages = {137--140}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract—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. Index Terms—Color image processing, convolution, correlation, Fourier transforms, hypercomplex numbers, quaternions. I.

### Citations

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Citation Context ...3-D pixels (regardless of the color space in which they are represented). The solution presented here is to utilize hypercomplex numbers, in particular, quaternions, as discovered by Hamilton in 1843 =-=[1]-=-, and to represent color image pixels using quaternions. The first definition of a quaternion Fourier transform was that of Ell [2], [3], which is discussed later in this paper, and the first Manuscri... |

34 | Hypercomplex spectral signal representation for the processing and analysis of images. Thesis - Bülow - 1999 |

12 |
Hypercomplex spectral transforms
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Citation Context ...mbers, in particular, quaternions, as discovered by Hamilton in 1843 [1], and to represent color image pixels using quaternions. The first definition of a quaternion Fourier transform was that of Ell =-=[2]-=-, [3], which is discussed later in this paper, and the first Manuscript received March 17, 2005; revised June 6, 2006. This work was supported in part by the U.K. Engineering and Physical Sciences Res... |

11 | Hypercomplex signals—a novel extension of the analytic signal to the multidimensional case
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- 2001
(Show Context)
Citation Context ...ficiency of the computation of quaternion Fourier transforms and on linear quaternion filters [10]. Bülow and Sommer have published work on hypercomplex Fourier transforms applied to grayscale images =-=[5]-=-–[8]. They utilized a hypercomplex Fourier transform because of its symmetry properties when applied to real-valued images (in the same way that the complex Fourier transform exhibits symmetry when ap... |

9 | Multi-dimensional signal processing using an algebraically extended signal representation - Bülow, Sommer - 1997 |

9 |
Hypercomplex correlation techniques for vector images
- Moxey, Sangwine, et al.
(Show Context)
Citation Context ...rms to color. In particular, using the more recent transform definition [9], the validity of hypercomplex autoand cross-correlation, and vector phase correlation have been demonstrated by the authors =-=[15]-=-, [16], [19]–[22]. Two years later, Pei et al., using the transform defined in [9], published work on the efficiency of the computation of quaternion Fourier transforms and on linear quaternion filter... |

7 |
Efficient implementation of quaternion Fourier transform convolution and correlation by 2-D
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- 2001
(Show Context)
Citation Context ...[16], [19]–[22]. Two years later, Pei et al., using the transform defined in [9], published work on the efficiency of the computation of quaternion Fourier transforms and on linear quaternion filters =-=[10]-=-. Bülow and Sommer have published work on hypercomplex Fourier transforms applied to grayscale images [5]–[8]. They utilized a hypercomplex Fourier transform because of its symmetry properties when ap... |

6 |
Fourier transforms of colour images using quaternion or hypercomplex
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Citation Context ...tp://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIP.2006.884955 1057-7149/$25.00 © 2006 IEEE application of a quaternion Fourier transform to color images was reported in 1996 by Sangwine =-=[4]-=- using a discrete version of Ell’s transform. These definitions of hypercomplex Fourier transforms have shown their applicability by allowing generalizations of many image processing techniques depend... |

6 |
Colour image filters based on hypercomplex convolution
- Sangwine, Ell
(Show Context)
Citation Context ...ons where the image is uniform in color and shows colored variation where the image has sharp color transitions. This filter is a linear vector filter and cannot be implemented using color separation =-=[12]-=-. To be more precise, although the filter may be applied separately to the red, green, and blue color components, the filtering operation applied in each case is required to be a vector operation. For... |

6 |
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Citation Context ...pper left: original, modulus, phase, and axis. where is the largest modulus in the image. The phase is also easily visualized, and we choose to use a color representation based on the IHS color space =-=[29]-=-. We create an RGB value from an IHS value, where is the phase, is mid-range, and is unity, utilizing simple clipping if the result is outside the valid range for the RGB value. This means that a phas... |

6 |
Color-sensitive edge detection using hypercomplex filters
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- 2000
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Citation Context ...welve. As with the one-sided convolution, the six mask transforms can be precomputed for use on multiple images. It should be noted that many of the linear vector filters designed to date (e.g., [11]–=-=[31]-=-) are single-axis invariant. Hence, when decomposing the masks, two of the components vanish, e.g., . In this situation, the image is split only once along the mask’s axis. The conversion from bi-conv... |

5 |
Decomposition of 2D hypercomplex fourier transforms into pairs of complex fourier transforms
- Ell, Sangwine
(Show Context)
Citation Context ... McCabe et al. [26]–[28]). (7) Any coefficient of the Fourier spectrum may be separated into two components parallel and perpendicular to , using the decomposition discussed earlier and in Appendix B =-=[24]-=- where and . Hence, the parallel component is the simplex part of and the perpendicular component is a vector (pure quaternion) composed of the perplex part postmultiplied by . We can express the para... |

4 |
Hypercomplex auto- and cross-correlation of colour images
- Sangwine, Ell
- 1999
(Show Context)
Citation Context .... In particular, using the more recent transform definition [9], the validity of hypercomplex autoand cross-correlation, and vector phase correlation have been demonstrated by the authors [15], [16], =-=[19]-=-–[22]. Two years later, Pei et al., using the transform defined in [9], published work on the efficiency of the computation of quaternion Fourier transforms and on linear quaternion filters [10]. Bülo... |

4 | Hypercomplex Wiener-Khintchine theorem with applications to color image correlation - Ell, Sangwine - 2000 |

3 |
The discrete Fourier transform of a colour image
- Sangwine, Ell
- 2000
(Show Context)
Citation Context ...orms have shown their applicability by allowing generalizations of many image processing techniques dependent on Fourier transforms to color. In particular, using the more recent transform definition =-=[9]-=-, the validity of hypercomplex autoand cross-correlation, and vector phase correlation have been demonstrated by the authors [15], [16], [19]–[22]. Two years later, Pei et al., using the transform def... |

3 |
Colour image edge detector based on quaternion convolution
- Sangwine
- 1998
(Show Context)
Citation Context ... linear vector filters which cannot be implemented in a component-separated approach. Two examples have been published to date. The first was a linear vector edge-detecting filter devised by Sangwine =-=[11]-=- in which convolution with a pair of quaternion-valued masks produces an almost grayscale image in regions where the image is uniform in color and shows colored variation where the image has sharp col... |

2 |
Vector Phase correlation
- Sangwine, Ell, et al.
(Show Context)
Citation Context ... produces a useful result despite this difference between the images. Using vector correlation based on transforms presented in this paper, phase correlation has also been generalized to color images =-=[14]-=-, [15], and direct color-grayscale correlation is possible [16]. This paper proceeds in Section II to explain how and why a color image is represented using quaternions. In Section III, we then presen... |

2 |
A novel block truncation coding of color images by using quaternion-moment preserving principle
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Citation Context ...ilters. The representation of color image pixels by pure quaternions has been conceived independently at least twice, by Sangwine [4], and Pei and Cheng, working on color image compression algorithms =-=[17]-=-, [18]. III. DEFINITION OF TRANSFORM Ell’s original quaternion Fourier transform [2], [3] was defined as follows: with inverse defined as In 1996. Sangwine [4] was the first to demonstrate the applica... |

2 |
novel block truncation coding of color images using a quaternion-moment-preserving principle
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Citation Context .... The representation of color image pixels by pure quaternions has been conceived independently at least twice, by Sangwine [4], and Pei and Cheng, working on color image compression algorithms [17], =-=[18]-=-. III. DEFINITION OF TRANSFORM Ell’s original quaternion Fourier transform [2], [3] was defined as follows: with inverse defined as In 1996. Sangwine [4] was the first to demonstrate the applicability... |

2 |
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Citation Context ... for the case of complex images where the spectral coefficients do not have conjugate symmetry. Significant progress with understanding the complex case for chromatic images was made by McCabe et al. =-=[26]-=-–[28]). (7) Any coefficient of the Fourier spectrum may be separated into two components parallel and perpendicular to , using the decomposition discussed earlier and in Appendix B [24] where and . He... |

2 |
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Citation Context ...the case of complex images where the spectral coefficients do not have conjugate symmetry. Significant progress with understanding the complex case for chromatic images was made by McCabe et al. [26]–=-=[28]-=-). (7) Any coefficient of the Fourier spectrum may be separated into two components parallel and perpendicular to , using the decomposition discussed earlier and in Appendix B [24] where and . Hence, ... |

1 | transforms for analysis of two-dimensional linear time-invariant partial-differential systems - “Quaternion-Fourier - 1993 |

1 |
Linear colour-dependent image filtering based on vector decomposition
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Citation Context ...not be implemented as three separated scalar filters applied to the separated components. A second example is a linear vector color-sensitive averaging filter which selectively smooths a chosen color =-=[13]-=-. Again, this filter cannot be implemented using scalar filters applied to separated color components, although, being linear, it can be applied separately to thesELL AND SANGWINE: HYPERCOMPLEX FOURIE... |

1 |
grayscale image registration using hypercomplex phasecorrelation
- “Color
(Show Context)
Citation Context ... color. In particular, using the more recent transform definition [9], the validity of hypercomplex autoand cross-correlation, and vector phase correlation have been demonstrated by the authors [15], =-=[16]-=-, [19]–[22]. Two years later, Pei et al., using the transform defined in [9], published work on the efficiency of the computation of quaternion Fourier transforms and on linear quaternion filters [10]... |

1 |
Vector correlation of color images
- Moxey, Ell, et al.
(Show Context)
Citation Context ...erplex part has a fixed phase . Finally, we note that the visualization discussed here is equally applicable to the representation of the results of the hypercomplex correlation, as has been shown in =-=[21]-=-. VI. HYPERCOMPLEX SPECTRUM A partial interpretation of the hypercomplex Fourier transform of (5), previously published by the authors [25], is presented in this section with an expanded interpretatio... |

1 |
operators and vector correlation
- “Hypercomplex
(Show Context)
Citation Context ...particular, using the more recent transform definition [9], the validity of hypercomplex autoand cross-correlation, and vector phase correlation have been demonstrated by the authors [15], [16], [19]–=-=[22]-=-. Two years later, Pei et al., using the transform defined in [9], published work on the efficiency of the computation of quaternion Fourier transforms and on linear quaternion filters [10]. Bülow and... |

1 |
Online]. Available: http://www.fftw.org
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(Show Context)
Citation Context ...written quaternion code, it means that existing complex FFT code can be utilized, such as the library found online and made available by Frigo and Johnson at the Massachusetts Institute of Technology =-=[23]-=-. Aside from code being readily available, it is also significant for understanding the structure of the spatial frequency domain representation of the image. The steps necessary to implement the quat... |

1 | A Theory of Spatio-Chromatic Image Encoding - McCabe - 1999 |

1 |
M’79–SM’04) was born in Minot, ND, in 1959. He received the B.S. (magna cum laude) and M.S. degrees in electrical engineering from the University of North Dakota, Grand Forks, in 1982 and 1983, respectively, and the Ph.D. degree in control sciences and dy
- Norwell
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(Show Context)
Citation Context ...ith quaternions may wish at this point to read Appendix A, where a review of the mathematical concepts of quaternions pertinent to this paper is given. For a more detailed treatment, the book by Ward =-=[32]-=- is recommended. Color image pixels have three components, and they can be represented in quaternion form using pure quaternions. For example, a pixel at image coordinates in an RGB image can be repre... |