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研究生:蕭毓哲
研究生(外文):Yu-Zhe Hsiao
論文名稱:應用四元數與退化四元數代數之彩色影像處理演算法
論文名稱(外文):Color Image Processing Algorithms by Applying Quaternions and Reduced Biquaternions Algebra
指導教授:貝蘇章
口試委員:丁建均黃文良吳家麟鍾國亮
口試日期:2016-07-15
學位類別:博士
校院名稱:國立臺灣大學
系所名稱:電信工程學研究所
學門:工程學門
學類:電資工程學類
論文種類:學術論文
論文出版年:2016
畢業學年度:104
語文別:英文
論文頁數:192
中文關鍵詞:四元數退化四元數彩色影像處理
外文關鍵詞:QuaternionsReduced BiquaternionsColor Image Procesing
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本論文的目標是要展示四元數與退化四元數可應用於多維訊號處理,特別專注於彩色影像處理的部分。
近年來,歸功於四元數代數、退化四元數代數以及現代電腦科技的發展,許多以前被認為緩慢以及不切實際的彩色影像處理技巧變得十分地普及。因為彩色影像由三個色彩成份所組成 (即:紅色、綠色以及藍色),我們可將它的所有像素使用四元數或退化四元數編碼,並將整張彩色影像視為二維的四元數或退化四元數影像。許多彩色影像處理的工作,例如三維幾何轉換,色彩匹配,去雜訊以及陰影移除,與在傳統RGB色域相比,皆可在四元數與退化四元數領域更輕易地被處理。
在本論文中,首先我們會先回顧基本四元數代數、退化四元數代數的概念以及它們在頻率域的轉換。接著,我們會介紹作為離散四元數傅立葉轉換(DQFT)以及離散退化四元數傅立葉轉換(DRBQFT)的埃根函數,也就是二維埃爾米特-高斯函數(2D-HGF)。對於三種離散四元數傅立葉轉換以及兩種離散退化四元數傅立葉轉換,我們也推導了二維埃爾米特-高斯函數對這些轉換的埃根值。此外,二維埃爾米特-高斯函數與高斯-拉蓋爾圓諧函數之間的數學關係也會做介紹。有了前述的數學關係以及一些推導過程,高斯-拉蓋爾圓諧函數可被證明也是離散四元數傅立葉轉換以及離散退化四元數傅立葉轉換的埃根函數,其埃根值亦會做出彙整。
這些高斯-拉蓋爾圓諧函數可以作為彩色影像展開的基底函數。這些展開的係數可以被用來重建原始的彩色影像並作為旋轉不變特徵。高斯-拉蓋爾圓諧函數也可被應用於色彩匹配。我們也提出了需多創新的四元數與退化四元數的彩色影像處理技巧,例如用於紋理以及雜訊移除的四元數疊代濾波技術、用於空間仿射轉換的四元數分數延遲、用於彩色濾光片陣列解碼及色彩校正的四元數演算法以及基於四元數旋轉操作之亮度不變影像色彩校正演算法。


The objective of this dissertation is to demonstrate how quaternion and reduced biquaternion algebra can be applied to multi-dimensional signal processing, in particular color image processing.
Thanks for the development of quaternion algebra, reduced biquaternion (RB) algebra, and modern computer technologies, many color image processing techniques that are considered slow and unrealistic become very popular in recent years. Since a color image has three components (red, green, and blue), we can encode its pixels to quaternions (or RB) and consider the whole color image as a two dimensional quaternion or RB image. Many tasks of color image processing, such as three dimensional geometrical transformations, color matching, and denoising can be done more easily in quaternion or RB domain rather than in RGB domain.
In this dissertation, the basic concepts of quaternion, RB algebra and their frequency domain transformations are reviewed. Then, we introduce the two dimensional Hermite-Gaussian functions (2D-HGFs) as the eigenfunction of discrete quaternion Fourier transform (DQFT) and discrete reduced biquaternion Fourier transform (DRBQFT). The eigenvalues of 2D-HGF for three types of DQFT and two types of DRBQFT are derived. After that, the mathematical relation between 2D-HGF and Gauss-Laguerre circular harmonic function (GLCHF) is given. From the aforementioned relation and some derivations, the GLCHF can be proved as the eigenfunction of DQFT/DRBQFT and its eigenvalues are summarized.
These GLCHFs can be used as the basis to perform color image expansion. The expansion coefficients can be used to reconstruct the original color image and as a rotation invariant feature. The GLCHFs can also be applied to color matching applications. We also proposed many novel quaternion and RB based color image processing techniques, such as quaternion iterative filtering for texture and noise removal, quaternion fractional delay for spatial Affine transformations, quaternion algorithm for color filtering array (CFA) demosaicing, quaternion based color correction method, and luminance-invariant color correction method based on quaternion rotation for color image.


Chapter 1 Introduction ……………………………………………………………...1
Chapter 2 The Quaternions …………………………………………………………5
2.1 Introduction …………………………………………………………………..5
2.2 The algebra of quaternions …………………………………………..…..….6
2.2.1 The Cayley-Dickson form ………………………………………….…..7
2.2.2 The norm and conjugate ………………..……………………………8
2.2.3 Pure quaternions, unit quaternions, and unit pure quaternions …..9
2.2.4 The addition and the subtraction operations……………………. …9
2.2.5 The multiplication operation…… …………………………………...9
2.3 The geometrical interpretation ………………..…………….…………….10
2.3.1 Vector algebra and quaternions ……………………………………10
2.4 The polar form………………...…………………………………………….12
2.5 Three-dimensional rotation and Affine transformations.. …………...18
2.5.1 Three-dimensional rotation ………………………………………...18
2.5.2 Affine transformations of quaternions ……………….……………22
2.6 Conclusion ……………………………………………………………….….25
Chapter 3 The Reduced Biquaternions …………………………………………...26
3.1 Introduction ………………………………………………………………….26
3.2 The reduced biquaternions algebra ………………………….……………..26
3.2.1 form of RBs and complexity analysis ………………………...27
3.2.2 Norm and conjugate of RBs …………………………….…………..29
3.3 The reduced biquaternions polar form …………………………………….32
3.4 Conclusions …………………………………………………………………..34
Chapter 4 Quaternion and Reduced Biquaternion Fourier Transform………....35
4.1 Introduction …………………………………………………………………35
4.2 The Fourier Transform …………………………………………………….37
4.3 The Quaternion Fourier Transform ………………………………………38
4.4 Efficient algorithm of Quaternion Fourier Transform …………….…….41
4.5 Time-frequency analysis and quaternion time-frequency analysis ……...47
4.5.1 Introduction ………………………………………………………….47
4.5.2 Time-frequency analysis ………………………………………...….48
4.5.3 Short-term Fourier transform (STFT) and Gabor transform …...53
4.5.4 Quaternion STFT (Gabor transform) …………………………..…56
4.6 Discrete reduced biquaternion Fourier transform………………. ……...58
4.7 Efficient implementation of the DRBQFT………………..………. ……...59
4.7.1 Implementation of type two DRBQFT …………………………….59
4.7.2 Implementation of type one DRBQFT …………………………….60
4.8 Conclusion …………………………………………………………………..61
Chapter 5 GLCHF for Color Image Processing Applications……...…………….62
5.1 Introduction …………………………………………………………………62
5.2 HGF and GLCHF…………….……………………………………………66
5.3 Eigenfunctions and Eigenvalues of DQFT and DRBQFT ……...………..67
5.3.1 The definitions of DQFT and the derivation of their eigenvalues...67
5.3.2 The definitions of DRBQFT and their eigenvalues………..………71
5.4 Experimental results………………………………………….…...………..77
5.4.1 Verification of derived eigenvalue …………………………..….…..77
5.4.2 Color image expansion & partial reconstruction using GLCHFs...78
5.4.3 Coefficients h_(a,b)^Q & h_(a,b)^RB as rotation invariant feature.................83
5.4.4 Color shape matching by using GLCHFs…………...……………...86
5.5 Conclusion ……………………………………………………………………92
Chapter 6 Edge Detection, Color Quantization, Segmentation, Texture Removal
& Noise Reduction of Color Image Using Quaternion Iterative Filtering……...93
6.1 Introduction ………………………………………………………………….93
6.2 EMD and iterative filtering …………………..……………………………..98
6.3 Proposed algorithms and experimental results ……………..……………102
6.4 Conclusions ………………………………………………………………….122
Chapter 7 Spatial Affine Transformations of 2D Color Images and 3D Objects By Using Quaternion Fractional Shift Fourier Transform………………………...123
7.1 Introduction …………………………………………………………………123
7.2 Fractional shift of quaternion signal …………………...….………………127
7.3 Affine transformation by using fractional shift fourier transform, shearing matrices, and modified inverse discrete fourier transform…….……………..129
7.4 Experimental results…………………………………………...….......…..133
7.5 Conclusion …………………………………………………………………140
Chapter 8 Demosaicing of Color Filter Array Patterns Using Quaternion Fourier Transform and Low Pass Filter……………………………………………….…...141
8.1 Introduction …………………………………………………………………141
8.2 The proposed algorithm for CFA demosaicing …………………...……..……142
8.3 The proposed method…………………………………………...….......…..143
8.4 Experimental results…………………………………………...….......…..145
8.5 Conclusion …………………………………………………………………146
Chapter 9 Simpe Effective Image and Video Color Correction Using Quaternion Distance Metric……………………………………………………………..….…...152
9.1 Introduction …………………………………………………………………152
9.2 Review of quaternion distance metric ………………………...……..……154
9.3 Proposed color correction method for image and vide……………...155
9.4 Experimental results…………………………………………...….......…..157
9.5 Conclusion …………………………………………………………………158
Chapter 10 Luminance-invariant Image and Video Color Correction Using Quaternion Rotation ………………………………..…...........................................162
10.1 Introduction ………………………………………………………...………162
10.2 The proposed algorithm for color correction ………………..…………165
10.2.1 Color correction algorithm for image and video…….…….…...165
10.2.2 The calculation of
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[C6-31] J. Zhou, Y. Xu, and X. Yang, “Quaternion Wavelet Phase Based Stereo Matching for Uncalibrated Images,” Pattern Recognition Letters, vol.28, pp-1509-1522, 2007.
[C6-32] G. Metikas and S. Olhede, “Multiple Multidimensional Morse Wavelets,” IEEE Transactions on Signal Processing, vol.51, no.7, pp.1941-1953, Jul. 2003.
[C6-33] W. Chan, H. Choi, and R. Baraniuk, “Coherent Multiscale Image Processing Using Dual-tree Quaternion Wavelets,” IEEE Transcations on Image Processing, vol.17, no.7, pp.1069-1082, Jul. 2008.
[C6-34] W. Chan, H. Choi, and R. Baraniuk, “Directional Hypercomplex Wavelets for Multidimensional Signal Analysis and Processing,” Proc. IEEE int. Conf. Acoust, Speech, Signal Processing, vol.3, pp.III-996-III-999, May. 2004.
[C6-35] T. Tsui, X.-P. Zhang, and D. Androutsons, “Color Image Watermarking Using Multidimensional Fourier Transforms,” IEEE Transactions on Inf. Forensics Security, vol.3, no.1, pp.16-28, Mar.2008.
[C6-36] W. K. Pratt, “Digital Image Processing,” Wiley, N.Y., 1991.
Chapter 7
[C7-1] K. Nomizu and S. Sasaki, “Affine Differential Geometry,” Cambridge University Press, 1994.
[C7-2] P. Thevenaz, et.al, “A Pyramid Approach to Subpixel Registration Based on Intensity,” IEEE Transactions on Image Processing, vol.7, issue 1, Jan 1988.
[C7-3] B. Zitová and J. Flusser, “Image Registration Methods: A Survey,” vol.21, issue 11, pp. 977-1000, Oct 2003.
[C7-4] B. S. Reddy, “An FFT-based Technique for Translation, Rotation, and Scale-invariant Image Registration,” IEEE Transactions on Image Processing, vol.5, issue 8, pp.1266-1271, Aug 1996.
[C7-5] Alan W. Paeth, “A Fast Algorithm for General Raster Rotation,” Grapics Interface 1986.
[C7-6] Robert W. Cox and Raoqiong Tong, “Two- and Three-Dimensional Image Rotation Using the FFT, ” IEEE Transactions on Image Processing, vol.8, no.9, pp.1297-1299, Aug 1999.
[C7-7] Michael Unser et.al, “Convolution-Based Interpolation for Fast High-Quality Rotation of Images, ” IEEE Transactions on Image Processing, vol.4, no.10, pp.1371-1381, Oct. 1995.
[C7-8] P. Hanrahan, “Three-pass Affine Transforms for Volume Rendering,” VVS ''90 Proceedings of the 1990 workshop on Volume visualization, pp. 71-78, 1990.
[C7-9] Xilin Chen,et.al, “Automatic Detection of Signs with Affine Transformation,” Applications of Computer Vision, 2002. (WACV 2002)., pp. 32-36, 2002.
[C7-10] R.E. Barnhill,G. Birkhoff, and W.J. Gordon, “ Smooth Interpolation in Triangles,” Journal of Approximation Theory, vol. 8, issue 2, pp.114-128, June 1973.
[C7-11] W. R. Hamilton, “Elements of Quaternion,” Longmans, Green and Co., London, 1866.
[C7-12] C. Moxey, S. Sangwine, T. Ell, “Hypercomplex Correlation Techniques for Vector Images,” IEEE Transactions on Signal Processing, vol.51, no.7, pp.1941-1953, Jul. 2003.
[C7-13] S. C. Pei, J. J. Ding, and J. H. Chang, “Efficient Implementation of Quaternion Fourier Transform, Convolution, and Correlation by 2-D complex FFT,” IEEE Transactions on Signal Processing, vol.49, no.11, pp.2783-2797, 2001.
[C7-14] E. Bayro-Corrochano, “The Theory and Use of the Quaternion Wavelet Transform,” Journal of Mathematical Imaging and Vision, vol.24, pp.19-35, 2006.
[C7-15] J. Zhou, Y. Xu, and X. Yang, “Quaternion Wavelet Phase Based Stereo Matching for Uncalibrated Images,” Pattern Recognition Letters, vol.28, pp-1509-1522, 2007.
[C7-16] G. Metikas and S. Olhede, “Multiple Multidimensional Morse Wavelets,” IEEE Transactions on Signal Processing, vol.51, no.7, pp.1941-1953, Jul. 2003.
[C7-17] W. Chan, H. Choi, and R. Baraniuk, “Coherent Multiscale Image Processing Using Dual-tree Quaternion Wavelets,” IEEE Transcations on Image Processing, vol.17, no.7, pp.1069-1082, Jul. 2008.
[C7-18] W. Chan, H. Choi, and R. Baraniuk, “Directional Hypercomplex Wavelets for Multidimensional Signal Analysis and Processing,” Proc. IEEE int. Conf. Acoust, Speech, Signal Processing, vol.3, pp.III-996-III-999, May. 2004.
[C7-19] T. Tsui, X.-P. Zhang, and D. Androutsons, “Color Image Watermarking Using Multidimensional Fourier Transforms,” IEEE Transactions on Inf. Forensics Security, vol.3, no.1, pp.16-28, Mar.2008.
[C7-20] S. C. Pei and Y. C. Lai, “ Closed Form Variable Fractional Time Delay Using FFT,” IEEE Signal Processing Letters, vol.19, issue 5, pp.299-302, 2012.
Chapter 8
[C8-1] D. Alleysson, S. Susstrunk, and J. Herault, “Linear demosaicing inspired by the human visual system,” IEEE Trans. Image Process., 14(4):439-449, 2005.
[C8-2] E. Dubois, “Frequency-domain methods for demosaicking of Bayersampled color images,” IEEE Signal Process. Lett., 12:847-850, 2005.
[C8-3] J.W. Glotzbach, R.W. Schafer, and K. Illgner, “A method of color filter array interpolation with alias cancellation properties,” Proc. IEEE Int. Conf. Image Processing, vol. 1, 2001, pp. 141-144.
[C8-4] B.K. Gunturk, Y. Altunbasak, and R.M. Mersereau, “Color plane interpolation using alternating projections,” IEEE Trans. Image Processing, vol. 11, no. 9, pp. 997-1013, Sept. 2002.
[C8-5] B.K. Gunturk, J. Glotzbach, Y. Altunbask, R.W. Schafer, and R.M., “Mersereau. Demosaicing: Color filter array interpolation,” IEEE Signal Process. Mag., 22(1):44-54, 2005.
[C8-6] P. Longere, X. Zhang, P.B. Delahunt, and D.H., “Brainard. Perceptual assessment of demosaicing algorithm performance,” Proc. IEEE, vol. 90, no.1, pp. 123-132, Jan. 2002.
[C8-7] R. Ramanath, W.E. Snyder, G.L. Bilbro, and W.A. Sander III, “Demosaicking methods for Bayer color arrays,” J. Electron. Imaging, vol. 11, no. 3, pp.306-315, July 2002.
[C8-8] T. Sakamoto, C. Nakanishi, and T. Hase, “Software pixel interpolation for digital still cameras suitable for a 32-bit MCU,” IEEE Trans. Consum. Electron., vol. 44, no. 4, pp. 1342-1352, Nov. 1998.
[C8-9] Eric Dubois, “Frequency-Domain Methods for Demosaicking of Bayer- Sampled Color Images,” IEEE Signal Processing Letters, vol.12, no.12 dec. 2005.
Chapter 9
[C9-1] P. E. Trahanias and A. N. Venetsanopoulos, “Color Image Enhancement Through 3-D Histogram Equalization,” in Proc. 15th IAPR Int. Conf. Pattern Recognition, vol. 1, pp. 545–548, Aug.-Sep., 1992.
[C9-2] N. Bassiou and C. Kotropoulos, "Color Image Histogram Equalization by Absolute Discounting Back-off," Computer Vision and Image Understanding, vol. 107, no. 1-2, pp.108-122, Jul.-Aug., 2007.
[C9-3] Ji-Hee Han, Sejung Yang, Byung-Uk Lee, "A Novel 3-D Color Histogram Equalization Method with Uniform 1-D Gray Scale Histogram", IEEE Trans. on Image Processing, vol. 20, No. 2, pp. 506-512, Feb., 2011.
[C9-4] K. Barnard et al., “A Comparison of Computational Color Constancy Algorithms Part II: Experiments with Image Data,” IEEE Trans. Imag. Process., pp. 985-996, 2002.
[C9-5] G. Buchsbaum, “A Spatial Processor Model for Object Color Perception,”, J. Frank. Inst., vol. 310, 1980.
[C9-6] E. H. Land, “The Retinex Theory of Color Vision,” Sci. Am. vol. 237, no.6, pp.108-128, 1977.
[C9-7] G. D. Finlayson and E. Trezzi, “Shades of Gray and Color Constancy,” in Proc. IS&T/SID 12th Color Imaging Conf., pp. 37-41, 2004.
[C9-8] D. Forsth, “A Novel Algorithm for Color Constancy,” Int. J. Comput. Vis. vol.5, no.1, pp.5-36, 1990.
[C9-9]. A. Gijsenij, T. Gevers, and J. van de Weijer, “Generalized Gamut Mapping using Image Derivative Structures for Color Constancy,” Int. J. Comput. Vis., 2008.
[C9-10] Javier Vazquez-Corral and Marcelo Bertalmfo, “Color Stabilization Along Time and Across Shots of the Same Scene, for One or Serveral Cameras of Unknown Specifications,” IEEE Trans. on Image Processing, vol.23, no.10, Oct., 2014.
[C9-11] T. W. Huang and H. T. Chen, “Landmark-based Sparse Color Representations for Color Transfer,” in Proc. IEEE 12th Int. Conf. Comput. Vis., pp.199-204, 2009.
[C9-12] Lianghai Jin, Hong Liu, Xiangyang Xu, and Enmin Song, “Quaternion-Based Implulse Noise Removal from Color Video Sequences,” IEEE Transactions on Circuit and System for Video Technology, vol.23, no.5, May. 2013.
Chapter 10
[C10-1] P. E. Trahanias and A. N. Venetsanopoulos, “Color Image Enhancement Through 3-D Histogram Equalization,” in Proc. 15th IAPR Int. Conf. Pattern Recognition, vol. 1, pp. 545–548, Aug.-Sep., 1992.
[C10-2] N. Bassiou and C. Kotropoulos, "Color Image Histogram Equalization by Absolute Discounting Back-off," Computer Vision and Image Understanding, vol. 107, no. 1-2, pp.108-122, Jul.-Aug., 2007.
[C10-3] Ji-Hee Han, Sejung Yang, Byung-Uk Lee, "A Novel 3-D Color Histogram Equalization Method with Uniform 1-D Gray Scale Histogram", IEEE Trans. on Image Processing, vol. 20, No. 2, pp. 506-512, Feb., 2011.
[C10-4] K. Barnard et al., “A Comparison of Computational Color Constancy Algorithms Part II: Experiments with Image Data,” IEEE Trans. Imag. Process., pp. 985-996, 2002.
[C10-5] G. Buchsbaum, “A Spatial Processor Model for Object Color Perception,”, J. Frank. Inst., vol. 310, 1980.
[C10-6] E. H. Land, “The Retinex Theory of Color Vision,” Sci. Am. vol. 237, no.6, pp.108-128, 1977.
[C10-7] G. D. Finlayson and E. Trezzi, “Shades of Gray and Color Constancy,” in Proc. IS&T/SID 12th Color Imaging Conf., pp. 37-41, 2004.
[C10-8] D. Forsth, “A Novel Algorithm for Color Constancy,” Int. J. Comput. Vis. vol.5, no.1, pp.5-36, 1990.
[C10-9]. A. Gijsenij, T. Gevers, and J. van de Weijer, “Generalized Gamut Mapping using Image Derivative Structures for Color Constancy,” Int. J. Comput. Vis., 2008.
[C10-10] Javier Vazquez-Corral and Marcelo Bertalmfo, “Color Stabilization Along Time and Across Shots of the Same Scene, for One or Serveral Cameras of Unknown Specifications,” IEEE Trans. on Image Processing, vol.23, no.10, Oct., 2014.
[C10-11] T. W. Huang and H. T. Chen, “Landmark-based Sparse Color Representations for Color Transfer,” in Proc. IEEE 12th Int. Conf. Comput. Vis., pp.199-204, 2009.
[C10-12] Y. Fan et.al, “Bottom-Up Saliency Detection Model Based on Human Visual Sensitivity and Amplitude Spectrum,” IEEE Trans. Multimedia, vol. 14, issue.1, pp.187-198, Jul. 2012.
[C10-13] C. Moxey, S. Sangwine, T. A. Ell, “Hypercomplex correlation techniques for vector images,” IEEE Trans. Signal Processing, vol. 51, no.7, pp.1941-1953, Jul. 2003.
[C10-14] S. C. Pei, J. J. Ding, and J. H. Chang, “Efficient implementation of quaternion Fourier transform, convolution, and correlation by 2-D complex FFT,” IEEE Trans. on Signal Processing, vol. 49, no.11, pp.2783-2797, 2001.
[C10-15] E. Bayro-Corrochano, “The theory and use of the quaternion wavelet transform,” Journal of Mathematical Imaging and Vision, vol. 24, pp.19-35, 2006.
[C10-16] J. Zhou, Y. Xu, and X. Yang, “Quaternion wavelet phase based stereo matching for uncalibrated Images,” Pattern Recognition Letters, vol. 28, pp-1509-1522, 2007.
[C10-17] G. Metikas and S. Olhede, “Multiple multidimensional morse wavelets,” IEEE Trans. on Signal Processing, vol. 51, no.7, pp.1941-1953, Jul. 2003.
[C10-18] W. Chan, H. Choi, and R. Baraniuk, “Coherent multiscale image processing using dual-tree quaternion wavelets,” IEEE Trans. Image Processing, vol. 17, no.7, pp.1069-1082, Jul. 2008.
[C10-19] W. Chan, H. Choi, and R. Baraniuk, “Directional hypercomplex wavelets for multidimensional signal analysis and processing,” Proc. IEEE int. Conf. Acoust, Speech, Signal Processing, vol. 3, pp.III-996-III-999, May. 2004.
[C10-20] T. Tsui, X. P. Zhang, and D. Androutsons, “Color image watermarking using multidimensional Fourier transforms,” IEEE Trans. Inf. Forensics Security, vol. 3, no.1, pp.16-28, Mar.2008.



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