Singular Value Decomposition (SVD)
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1 Singular Value Decomposition (SVD)
2 Matrix Transpose
3 Matrix Multiplication
4 Matrix Inverse If, A B = I,identity matrix Then, B= A -1 Identity matrix:
5 X= UΣV T U and V are Orthonormal matrices rxr diagonal matrix with r non-zero diagonal elements
6 X U = x x V T OPTIONAL 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)
7 Applications of SVD in image processing closest rank-k approximation for a matrix X X k = k u i= 1 Each term in the summation expression above is called principal image i Σ i v T i
8 Original matrix (X) Original size *5=20 bytes X U = x x k=1 0 x 4 x = Compressed size *1+1+1*5=10 bytes V T
9 k=2 0 0 x 4 0 x = Compressed size *2+2+2*5=20 bytes k= x 4 0 0x = Compressed size *3+3+3*5=30 bytes k= = Compressed size *4+4+4*5=40 bytes x x
10 The image compression example in Original size = 384*384 bytes = 147,456 bytes k=1: 384*1+1+1*384=769 bytes k=10: 384* *384=7,690 bytes k=20: 384* *384=15,380 bytes k=50: 384* *384=38,450 bytes k=100: 384* *384=76,900 bytes k=200: 384* *384=153,800 bytes
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