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Microsoft PowerPoint - 2-SVD
Singular Value
Decomposition (SVD)
Matrix Transpose
Matrix Multiplication
Matrix Inverse
=
=×
1 0 ... 0 0
0 ...1 0 0
. ... . . . . . .
0 0 ... 1 0
0 0 ... 0 1
:matrixIdentity
A 1-B Then,
matrixidentity , If, IBA
X= UΣVT
rxr diagonal
matrix with r
non-zero
diagonal
elements
U and V are
Orthonormal
matrices
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)
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 ∑
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 Tii
k
i
i
k Σ∑
=
=
1
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
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 Compressed size
0 1 0 0 3 0 0 0 1 0 0 0 0 3 0 0 4*3+3+3*5=30 bytes
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 Compressed size
0 1 0 0 0 3 0 0 0 0 1 0 0 0 0 3 0 0 4*4+4+4*5=40 bytes
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
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
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