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# Great Pyramid of Giza and Golden Section Transform Preview

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Great Pyramid of Giza and Golden Section Transform Preview

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### Great Pyramid of Giza and Golden Section Transform Preview

1. 1. GGrreeaatt PPyyrraammiidd ooff GGiizzaa aanndd GGoollddeenn SSeeccttiioonn TTrraannssffoorrmm PPrreevviieeww Jun Li http://GoldenSectionTransform.com/ http://en.wikipedia.org/wiki/Great_Pyramid_of_Giza
2. 2. Introduction The Great Pyramid of Giza (also known as the Pyramid of Khufu or the Pyramid of Cheops) is the oldest and largest of the three pyramids in the Giza Necropolis bordering what is now El Giza, Egypt. It is the oldest of the Seven Wonders of the Ancient World, and the only one to remain largely intact. In mathematics, Golden Section Transform is a new class of discrete orthogonal transform. Its theory involves golden ratio, Fibonacci number, Fourier analysis, wavelet theory, stochastic process, integer partition, matrix analysis, group theory, combinatorics on words, its applied fields include digital signal (image/audio/video) processing, denoising, watermarking, compression, coding and so on. see here.
3. 3. Are there any connections http://en.wikipedia.org/wiki/Great_Pyramid_of_Giza 1-level normalized reverse order Low Golden Section Transform(RLGST)
4. 4. between Great Pyramid and RLGST? http://en.wikipedia.org/wiki/Johannes_Kepler
5. 5. http://en.wikipedia.org/wiki/Fibonacci
6. 6. The answer is ... to be contiued on: http://GoldenSectionTransform.org/
7. 7. RLGST Matlab Implementation rlgst2d.m % forward 2d reverse order low golden section transform irlgst2d.m % inverse 2d reverse order low golden section transform testrlgstfn.m % 2d RLGST demo all the codes can be downloaded here: http://goldensectiontransform.org/lifting-scheme-reverse-order-low-golden-section- transform-matlab-part/
8. 8. rlgst2d.m function H = rlgst2d(X,nlevel) % Author: Jun Li, more info@ http://goldensectiontransform.org/ % the lifting scheme of 2d reverse order low golden section transform % here matrix X is of Fn*Fn where Fn is a fibonacci number Fn>=2. % Wythoff sequence is used here, no need to call function_lword. global lj; % only used by function rlgst1d(S) below. global FBL; % only used by function rlgst1d(S) below. [xx,yy] = size(X); ind = floor(log(xx*sqrt(5)+1/2)/log((sqrt(5)+1)/2)); % determine index FBL = filter(1,[1 -1 -1],[1 zeros(1,ind-1)]); % FBL = Fibonacci sequence -> [1 1 2 3 5 8...]; H=X; for lj=1:nlevel for j=1:xx [ss,dd] = rlgst1d(H(j,1:yy)); % row transform H(j,1:yy) = [ss,dd]; end for k=1:yy [ss,dd] = rlgst1d(H(1:xx,k)'); % column transform H(1:xx,k) = [ss,dd]'; end xx = FBL(end-lj); % round((sqrt(5)-1)/2*xx); 8*8 block: xx=8->5->3->2 yy = FBL(end-lj); % round((sqrt(5)-1)/2*yy); 8*8 block: yy=8->5->3->2 end %% 1d reverse order low golden section transform lifting scheme function [ss,dd] = rlgst1d(S) %% Author: Jun Li, more info@ http://goldensectiontransform.org/ global lj; global FBL; for i=1:FBL(end-lj-1) lw = floor((1+sqrt(5))/2*i); % Lower Wythoff sequence->1,3,4,6,8... % uw = floor((3+sqrt(5))/2*i); % Upper Wythoff sequence->2,5,7,10,13... uw = lw + i; % ss(lw) = (sqrt(FBL(lj+1))*S(uw-1)+sqrt(FBL(lj))*S(uw))/sqrt(FBL(lj+2)); % dd(i) = (sqrt(FBL(lj))*S(uw-1)-sqrt(FBL(lj+1))*S(uw))/sqrt(FBL(lj+2)); %% we can use lifting scheme here: ga = sqrt(FBL(lj+1)/FBL(lj)); gb = sqrt(FBL(lj)*FBL(lj+1))/FBL(lj+2); gc = sqrt(FBL(lj+2)/FBL(lj)); d1 = S(uw-1) - ga*S(uw); s1 = S(uw) + gb*d1; ss(lw) = gc*s1; dd(i) = d1/gc; if (FBL(end-lj) ~= 1) & (i <= FBL(end-lj-2)) ss(uw) = S(lw+uw); end end
9. 9. irlgst2d.m function X = irlgst2d(H,nlevel) % Author: Jun Li, more info@ http://goldensectiontransform.org/ % the lifting scheme of inverse 2d reverse order low golden section transform % here matrix H is of Fn*Fn where Fn is a fibonacci number Fn>=2. % Wythoff sequence is used here, no need to call function_lword. global lji; % only used by function irlgst1d(S) below. global FBLI; % only used by function irlgst1d(S) below. [xx,yy] = size(H); ind = floor(log(xx*sqrt(5)+1/2)/log((sqrt(5)+1)/2)); % determine index FBLI = filter(1,[1 -1 -1],[1 zeros(1,ind-1)]); % FBLI = Fibonacci sequence -> [1 1 2 3 5 8...]; X=H; for lji=nlevel:-1:1 for j=1:FBLI(end-lji+1) ss = X(1:FBLI(end-lji),j); dd = X(FBLI(end-lji)+1:FBLI(end-lji+1),j); X(1:FBLI(end-lji+1),j) = irlgst1d(ss',dd')'; % inverse column transform end for k=1:FBLI(end-lji+1) ss = X(k,1:FBLI(end-lji)); dd = X(k,FBLI(end-lji)+1:FBLI(end-lji+1)); X(k,1:FBLI(end-lji+1)) = irlgst1d(ss,dd); % inverse row transform end end %% inverse 1d reverse order low golden section transform lifting scheme function S = irlgst1d(ss,dd) %% Author: Jun Li, more info@ http://goldensectiontransform.org/ global lji; global FBLI; for i=1:FBLI(end-lji-1) lw = floor((1+sqrt(5))/2*i); % Lower Wythoff sequence->1,3,4,6,8... % uw = floor((3+sqrt(5))/2*i); % Upper Wythoff sequence->2,5,7,10,13... uw = lw + i; % S(uw-1) = (sqrt(FBLI(lji+1))*ss(lw)+sqrt(FBLI(lji))*dd(i))/sqrt(FBLI(lji+2)); % S(uw) = (sqrt(FBLI(lji))*ss(lw)-sqrt(FBLI(lji+1))*dd(i))/sqrt(FBLI(lji+2)); %% we can use lifting scheme here: ga = sqrt(FBLI(lji+1)/FBLI(lji)); gb = sqrt(FBLI(lji)*FBLI(lji+1))/FBLI(lji+2); gc = sqrt(FBLI(lji+2)/FBLI(lji)); d1 = gc*dd(i); s1 = ss(lw)/gc; S(uw) = s1 - gb*d1; S(uw-1) = d1 + ga*S(uw); if (FBLI(end-lji) ~= 1) & (i <= FBLI(end-lji-2)) S(lw+uw) = ss(uw); end end
10. 10. testrlgstfn.m function testrlgstfn % load 8 bit lena of size 377*377 X = im2double(imread('C:lena377.bmp')); [xx,yy] = size(X); nlevel = 1; H = rlgst2d(X,nlevel); imshow(H); title([num2str(nlevel),'-level normalized reverse order low golden section transform of',' Lena ',num2str(xx),'*',num2str(yy)]); % R = irlgst2d(H,nlevel); % EQ = isequal(round(X*255),round(R*255)) % Perfect Reconstruction imwrite(H,'rlgstlena377.png','png');
11. 11. Example: Lena 377*377
12. 12. Pyramid 377*377
13. 13. Pyramid 377*377
14. 14. Links Jun Li's Golden Section Transform Home Page http://goldensectiontransform.org/ & goldensectiontransform.com facebook: https://www.facebook.com/goldensectiontransform twitter: @JasonLiJun http://en.wikipedia.org/wiki/Great_Pyramid_of_Giza http://en.wikipedia.org/wiki/Kepler_triangle http://blog.world-mysteries.com/science/the-great-pyramid-and-the-speed-of-light/ http://www.wanttoknow.nl/hoofdartikelen/mysterieuze-krachten-van-piramide-structuren/ http://ifdawn.com/esa/tipharet.htm http://jwilson.coe.uga.edu/EMAT6680/Parveen/ancient_egypt.htm http://portal.groupkos.com/index.php?title=Great_Pyramid_Dimensions http://www.cheops-pyramide.ch/khufu-pyramid/pyramid-alignment.html http://www.crystalinks.com/greatpyramid.html http://www.cut-the-knot.org/do_you_know/GoldenRatio.shtml http://www.gizapyramid.com/overview.htm http://www.goldennumber.net/phi-pi-great-pyramid-egypt/ http://www.grahamhancock.com/forum/KollerstromN2.php http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibInArt.html http://www.sacred-geometry.es/en/content/phi-great-pyramid http://www.sriyantraresearch.com/Article/GoldenRatio/golden%20ratio%20triangles.html http://www.world-mysteries.com/mpl_2.htm https://brilliant.org/discussions/thread/the-golden-ratio-kepler-triangle/ https://www.dartmouth.edu/~matc/math5.geometry/unit2/unit2.html http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibrab.html http://en.wikipedia.org/wiki/Fibonacci_word http://mathworld.wolfram.com/RabbitSequence.html The infinite Fibonacci word, https://oeis.org/A005614 Lower Wythoff sequence, http://oeis.org/A000201 Upper Wythoff sequence, http://oeis.org/A001950
15. 15. Reverse Order High Golden Section Transform (RHGST) Jun Li
16. 16. RHGST of Pyramid
17. 17. References [1] Jun. Li (2007). "Golden Section Method Used in Digital Image Multi-resolution Analysis". Application Research of Computers (in zh-cn) 24 (Suppl.): 1880–1882. ISSN 1001- 3695;CN 51-1196/TP [2] I. Daubechies and W. Sweldens, "Factoring wavelet transforms into lifting schemes," J. Fourier Anal. Appl., vol. 4, no. 3, pp. 247-269, 1998. [3] Ingrid Daubechies. 1992. Ten Lectures on Wavelets. Soc. for Industrial and Applied Math., Philadelphia, PA, USA. [4] Sun, Y.K.(2005). Wavelet Analysis and Its Applications. China Machine Press. ISBN 7111158768 [5] Jin, J.F.(2004). Visual C++ Wavelet Transform Technology and Engineering Practice. Posts & Telecommunications Press. ISBN 7115119597 [6] He, B.;Ma, T.Y.(2002). Visual C++ Digital Image Processing. Posts & Telecommunications Press. ISBN 7115109559 [7] V. Britanak, P. Yip, K. R. Rao, 2007 Discrete Cosine and Sine Transform, General properties, Fast Algorithm and Integer Approximations" Academic Press [8] K. R. Rao and P. Yip, Discrete Cosine Transform: Algorithms, Advantages, Applications, Academic Press, Boston, 1990. [9] J. Nilsson, On the entropy of random Fibonacci words, arXiv:1001.3513. [10] J. Nilsson, On the entropy of a family of random substitutions, Monatsh. Math. 168 (2012) 563–577. arXiv:1103.4777.
18. 18. [11] J. Nilsson, On the entropy of a two step random Fibonacci substitution, Entropy 15 (2013) 3312—3324; arXiv:1303.2526. [12] Wai-Fong Chuan, Fang-Yi Liao, Hui-Ling Ho, and Fei Yu. 2012. Fibonacci word patterns in two-way infinite Fibonacci words. Theor. Comput. Sci. 437 (June 2012), 69-81. DOI=10.1016/j.tcs.2012.02.020 http://dx.doi.org/10.1016/j.tcs.2012.02.020 [13] Julien Cassaigne (2008). On extremal properties of the Fibonacci word. RAIRO - Theoretical Informatics and Applications, 42, pp 701-715. doi:10.1051/ita:2008003. [14] Wojciech Rytter, The structure of subword graphs and suffix trees of Fibonacci words, Theoretical Computer Science, Volume 363, Issue 2, 28 October 2006, Pages 211-223, ISSN 0304-3975, http://dx.doi.org/10.1016/j.tcs.2006.07.025. [15] Alex Vinokur, Fibonacci connection between Huffman codes and Wythoff array, arXiv:cs/0410013v2 [16] Vinokur A.B., Huffman trees and Fibonacci numbers. Kibernetika Issue 6 (1986) 9-12 (in Russian), English translation in Cybernetics 22 Issue 6 (1986) 692–696; http://springerlink.metapress.com/link.asp?ID=W32X70520K8JJ617 [17] L. Colussi, Fastest Pattern Matching in Strings, Journal of Algorithms, Volume 16, Issue 2, March 1994, Pages 163-189, ISSN 0196-6774, http://dx.doi.org/10.1006/jagm.1994.1008. [18] Ron Lifshitz, The square Fibonacci tiling, Journal of Alloys and Compounds, Volume 342, Issues 1–2, 14 August 2002, Pages 186-190, ISSN 0925-8388, http://dx.doi.org/10.1016/S0925-8388(02)00169-X. [19] C. Godr`eche, J. M. Luck, Quasiperiodicity and randomness in tilings of the plane, Journal of Statistical Physics, Volume 55, Issue 1–2, pp. 1–28. [20] S. Even-Dar Mandel and R. Lifshitz. Electronic energy spectra and wave functions on the square Fibonacci tiling. Philosophical Magazine, 86:759-764, 2006.