在數位系統中，我們通常需要使用高維的矩陣來呈現一張照片。然而，自然影像的統計樣態表現出高度不均勻的分佈，代表影像資訊存在許多冗余，因此找尋更有效的解讀方法至關重要。主成分分析(Principal component analysis, PCA)便是一種能透過找出數據重要特徵，並降低數據維度的方法。在本文中，我們首先將PCA應用於灰階的自然影像，並發現其背後的主成分代表著影像中的邊緣與亮度資訊。接者，將應用拓展至以RGB為基底的自然彩色照片上，發現當我們將照片的紅、綠及藍色通道結合成單一的輸入訊號時，提取出的主特徵中會混雜著灰階及色彩資訊，且灰階資訊的重要程度更高。於是我們推測，若是能將最為關鍵的灰階資訊留下，是否就能用少量的色彩重現原始的彩色自然照片?為了證實這個猜想，我們首先使用HSV模型做實驗，因為其本身就存在將色彩與亮度分離的特性。而實驗的結果與我們的猜想一致，但此模型依然存在著一些限制。為解決這些問題，我們提出一種創新的方法:保留彩色照片最關鍵且為灰階的第一主成分不變，並使用4.7%的比例壓縮剩下的二維 色彩資訊，以此重建的影像幾乎讓人眼無法辨別與原始影像的差異。

In digital systems, we often use high-dimensional arrays to interpret an im- age. However, natural image statistics indicate significant redundancy due to their highly non-uniform distribution. Exploring more effective ways to interpret image data is essential. Principal Component Analysis (PCA) can reduce this redundancy by identifying important patterns and lowering data dimensionality. We apply PCA to grayscale images, revealing that principal components represent edge and intensity information. For color images, treating RGB channels as a single input yields features that mix achromatic and chromatic variations, with achromatic terms dominating. This suggests that by retaining the achromatic (edge) information, a few color components are capable of reproducing natural color images. Testing this with the HSV color model, we find the reconstruction results align with our hypothesis, but still have some limitations. We then propose a novel approach: keeping the achromatic first principal component, which is the most critical feature of the images, then perform complex PCA to compress the remaining two-dimensional color data with a 4.7% compression ratio. The recon- structed images are nearly indistinguishable to the human eye.

Contents i
1 Introduction 1
2 Color Vision 3
2.1 LightandVision ............................... 4
2.2 TheStructureoftheRetina ...................... 5
2.3 TheLight-sensitiveCells....................... 7
3 Color Space 10
3.1 RGBColorSpace............................. 10
3.2 HSVColorSpace ............................ 11
3.3 CIEXYZColorSpace ......................... 12
4 Principal Component Analysis 16
4.1 PrincipalComponentAnalysis.................... 17
4.2 ApplytoGrayscaleImages........................ 20
4.3 ApplytoRGBChannelsSeparately.................. 24
4.4 ApplytoAllRGBChannelsTogether................. 25
5 Complex PCA 28
5.1 ComplexPCA.............................. 29
5.2 TheHS⊗VMethod........................... 29
5.3 Results................................. 30
6 Optimized Color Encoding 33
6.1 OptimizedGrayscaleAxis ....................... 33
6.2 ColorPlane ................................... 35
6.3 Results....................................... 37
7 Discussions and Outlook 41
Appendix 43
A Results for More Grayscale Images .................. 44
B Results for More Color Images - RGB ................ 45
C Results for More ColorImages - OG Color Encoding ... 46
D Comparison of Results from All Methods.............. 47
Bibliography 49

[1] Mark F. Bear, Barry W. Connors, and Michael A. Paradiso. The eye. In Neuroscience: Exploring the Brain, chapter 9, pages 293–330. Wolters Kluwer, Philadelphia, PA, 4 edition, 2016.

[2] Rolf G. Kuehni and Andreas Schwarz. The universe of human color exper- ences. In Color ordered: a survey of color order systems from antiquity to the present, chapter 1, pages 1–26. Oxford University Press, New York, 2008.

[3] John Doe. Vision. In C. Giovanni Galizia and Pierre-Marie Lledo, editors, Neurosciences - From Molecule to Behavior: A University Textbook, chap- ter 18, pages 299–320. Springer Spektrum, Berlin Heidelberg, 2nd edition, 2013.

[4] DH Sliney. What is light? the visible spectrum and beyond. Eye, 30:222–229, 2016.

[5] Samuel G. Solomon and Peter Lennie. The machinery of colour vision. Nature Reviews Neuroscience, 8:276–286, 2007.

[6] Jonathan N. Tinsley, Maxim I. Molodtsov, Robert Prevedel, David Wart- mann, Jofre Espigule-Pons, Mattias Lauwers, and Alipasha Vaziri. Direct detection of a single photon by humans. Nature Communications, 7:12172, 2016. Published online 19 July 2016, Accepted 7 June 2016, Received 15 January 2016.

[7] Andrew Stockman. Cone fundamentals and cie standards. Current Opinion in Behavioral Sciences, 30:87–93, 2019.

[8] Mark S. Nixon and Alberto S. Aguado. Color images. In Feature Extrac- tion and Image Processing for Computer Vision, chapter 11, pages 511–570. Academic Press, 4th edition, 2020.

[9] John Doe. Cie colorimetry. In Ja ́nos Schanda, editor, Colorimetry: Under- standing the CIE System, chapter 3, pages 45–67. Wiley-Interscience, Hobo- ken, NJ, 2007.

[10] Nina Meinzer. In the eye of the beholder. Nature Physics, 14:320, 2018.

[11] Peter J. B. Hancock, Roland J. Baddeley, and Leslie S. Smith. The principal components of natural images. Network: Computation In Neural Systems, 3:61–70, 1991.

[12] V. Hosu, H. Lin, T. Sziranyi, and D. Saupe. Koniq-10k: An ecologically valid database for deep learning of blind image quality assessment. IEEE Transactions on Image Processing, 29:4041–4056, 2020.

[13] I.T. Jolliffe and J. Cadima. Principal component analysis: a review and recent developments. Philosophical Transactions of the Royal Society A: Mathemat- ical, Physical and Engineering Sciences, 374(2065):20150202, 2016.

[14] Xiangyu Kong, Changhua Hu, and Zhansheng Duan. Introduction. In Prin- cipal Component Analysis Networks and Algorithms, chapter 1, pages 1–16. Springer, Singapore, 2017.

[15] Markus Ringn ́er. What is principal component analysis? Nature Biotechnol- ogy, 26(3):303–304, 2008.

[16] Iain M. Johnstone and Arthur Yu Lu. On consistency and sparsity for prin- cipal components analysis in high dimensions. Journal of the American Sta- tistical Association, 104(486):682–693, 2009.

[17] I.T. Jolliffe. Introduction. In Principal Component Analysis, chapter 1, pages 1–9. Springer-Verlag, New York, 2nd edition, 2002.
[18] Simon Haykin. Principal-components analysis. In Neural Networks and Learn- ing Machines, chapter 8, pages 367–424. Pearson Education, Inc., Upper Sad- dle River, New Jersey, 3rd edition, 2009.

[19] Keith Jack. Color spaces. In Video Demystified: A Handbook for the Digital Engineer, chapter 3, pages 15–36. Elsevier, Burlington, MA, 5th edition, 2007.

[20] Aapo Hyv ̈arinen, Jarmo Hurri, and Patrik O. Hoyer. Introduction. In Natural Image Statistics: A Probabilistic Approach to Early Computational Vision, chapter 1, pages 1–21. Springer, London, 2009.

[21] Bruno A. Olshausen and David J. Field. Emergence of simple-cell receptive field properties by learning a sparse code for natural images. Nature, 381:607– 609, 1996.

[22] Gunther Heidemann. The principal components of natural images revisited. IEEE Transactions on Pattern Analysis and Machine Intelligence, 28:822– 826, 2006.

[23] Gilbert Strang. Eigenvalues and eigenvectors. In Linear Algebra And Its Applications, chapter 5, pages 260–344. Cengage Learning, 4th edition, 2006.

[24] Jean-Jacques Denimal and Sergio Camiz. Complex principal component anal- ysis: Theory and geometrical aspects. Journal of Classification, 39:376–408, 2022.

[25] Alain Hore and Djemel Ziou. Image quality metrics: Psnr vs. ssim. In 2010 20th international conference on pattern recognition, pages 2366–2369. IEEE, 2010.

[26] Signal and Image Processing Institute. USC-SIPI Image Database. https: //sipi.usc.edu/database/database.php.