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- 5. X= UΣVT rxr diagonal matrix with r non-zero diagonal elements U and V are Orthonormal matrices
- 6. OPTIONALOPTIONALOPTIONALOPTIONAL R Code > M=matrix(c(1,0,0,0,0,0,0,4,0,3,0,0,0,0,0,0,2,0,0,0),nrow=4,ncol=5) > X=svd(M) > X$u > X$d > X$v > X$u%*%diag(X$d)%*%t(X$v) 1 0 0 0 2 0 0 1 0 4 0 0 0 0 1 0 0 0 0 0 3 0 0 0 1 0 0 0 3 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -1 0 0 2.24 0 0.45 0 0 0 0.89 0 4 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 V T = x x X U ∑
- 7. Applications of SVD in image processing – closest rank-k approximation for a matrix X – Each term in the summation expression above is called principal image vuX T ii k i i k Σ∑= = 1
- 8. Original matrix (X) Original size 1 0 0 0 2 4*5=20 bytes 0 0 3 0 0 0 0 0 0 0 0 4 0 0 0 1 0 0 0 2 0 0 1 0 4 0 0 0 0 1 0 0 0 0 0 3 0 0 0 1 0 0 0 3 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -1 0 0 2.24 0 0.45 0 0 0 0.89 0 4 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 V T = x x X U ∑ k=1 0 x 4 x 0 1 0 0 0 = 0 0 0 0 0 Compressed size 0 0 0 0 0 0 4*1+1+1*5=10 bytes 0 0 0 0 0 0 1 0 4 0 0 0
- 9. k=2 0 0 x 4 0 x 0 1 0 0 0 = 0 0 0 0 0 Compressed size 0 1 0 3 0 0 1 0 0 0 0 3 0 0 4*2+2+2*5=20 bytes 0 0 0 0 0 0 0 1 0 0 4 0 0 0 k=3 0 0 1 x 4 0 0 x 0 1 0 0 0 = 1 0 0 0 2 Compressedsize 0 1 0 0 3 0 0 0 1 0 0 0 0 3 0 0 4*3+3+3*5=30bytes 0 0 0 0 0 2.24 0.45 0 0 0 0.89 0 0 0 0 0 1 0 0 0 4 0 0 0 k=4 0 0 1 0 4 0 0 0 0 1 0 0 0 = 1 0 0 0 2 Compressedsize 0 1 0 0 0 3 0 0 0 0 1 0 0 0 0 3 0 0 4*4+4+4*5=40bytes 0 0 0 -1 0 0 2.24 0 0.45 0 0 0 0.89 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 4 0 0 0 x x
- 10. The image compression example in http://journal.batard.info/post/2009/04/08/svd- fun-profit • Original size = 384*384 bytes = 147,456 bytes • k=1: 384*1+1+1*384=769 bytes • k=10: 384*10+10+10*384=7,690 bytes • k=20: 384*20+20+20*384=15,380 bytes • k=50: 384*50+50+50*384=38,450 bytes • k=100: 384*100+100+100*384=76,900 bytes • k=200: 384*200+200+200*384=153,800 bytes