Image processing techniques using Singular Value Decomposition

1. Singular Value Decomposition -
Applications in Image Processing
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By-
Chithaj Mallikarjun Niteesh S Shanbog
PES1201702412 PES1201702421
MTech, EEE MTech, EEE

2. 1. Singular value decomposition
Consider a (real) matrix
A ∈ Rn×m, r = rank (A) ≤ min {n, m} .
A has
m. columns of length n ,
n. rows of length m ,
r is the maximal number of linearly independent columns
(rows)of A .
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3. 3
There exists an S V D decomposition of A in the form
A = U Σ V T ,
where U = [u1, . . . , un] ∈ Rn×n, V = [v1, . . . , vm] ∈ Rm×m are orthogo-
nal matrices, and

4. Singular value decomposition – the matrices:
A U  VT
{ui} i = 1,...,n
{vi} i = 1,...,m
{σi } i=1,...,r
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are left singular vectors (columns of U ),
are right singular vectors (columns of V ),
are singular values of A.

5. 5
T h e S V D gives us:
span (u1, . . . , ur) ≡ range (A ) ⊂ Rn,
span (vr+1, . . . , vm) ≡ ker ( A ) ⊂ Rm,
span (v1, . . . , vr) ≡ range ( A T ) ⊂ Rm,
span (ur+1, . . . , un) ≡ ker ( A T ) ⊂ Rn,

6. Singular value decomposition – the subspaces:
v1
v2
...
vr
u1
u2
...
ur
vr+1
...
vm
ur+1
...
un
1
2
r
}
}0
0
AT
A
range(A)
ker(AT
)
ker(A)
range(A )T
Rn
Rm
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7. 7
T h e outer product (dyadic) form:
We can rewrite A as a sum of rank-one matricesin the dyadic form

8. Matrix A as a sum of rank-one matrices:
A
A1
+ A2
+ ...
+ Ar -1
+ Ar
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r
A  Ai
i = 1
SVD reveals the dominating information encoded in a matrix. The
first terms are the “ m o s t ” important.

10. Application - image compression
Grayscale image =matrix; each entry represents a pixel brightness.
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11. Grayscale image: scale0 , . . . , 255 from black to white
Colored image: 3 matrices for Red, Green and Blue brightness
values
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Memory required to store:
An uncompressed image of size (m × n):mn values
SVD approximation:k(m + n + 1) values

13. Other applications:
• Computer Tomography
( C T ) ;
• Magnetic Resonance;
• Seismology;
• Crystallography;
• Material Sciences;
• Image De blurring
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14. Thank You!