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MSP1
References
• Jain (a text book?; IP per se; available)
• Castleman (a real text book ; image analysis; less available)
• Lim (unavailable?)
MSP2
Image Transforms – Why?
• Simplicity
• Applications
• Image compression (JPEG
• Image enhancement (e.g., filtering)
• Image analysis (e.g., feature extraction)
MSP3
Image Transforms
• Preliminary definitions Orthogonal matrix
Unitary matrix
IAAAA
AA
TT
T
or
1
IAAAA
AA
TT
T
**
*1
or
MSP4
Preliminary Definitions (cont’)
• Real orthogonal matrix is unitary
• Unitary matrix need not be orthogonal
• Columns (rows) of an NxN unitary matrix are orthogonal and form a complete set of basis vectors in an N-dimensional vector space
MSP5
Preliminary Definitions (cont’)
• Examples (Jain, 1989)
1
1
2
1
2
2
11
11
2
1321 j
jA
j
jAA
orthogonal & unitary not unitary unitary
MSP6
Image Transforms (cont’)
• ... are a class of unitary matrices used to facilitate image representation
• Representation using a discrete set of basis images (similar to orthogonal series expansion of a continuous function)
MSP7
Image Transforms (cont’)
• For a 1D sequence , a unitary transformation is written as
where (unitary). This gives
10 ),( Nnnu
TAA *1
10 ),(),()( 1
0
NknunkakvN
n
Auv
10 ),,()()( 1
0
*
NnnkakvnuN
k
vAu T*
MSP8
Basic Vectors of 8x8 Orthogonal TransformsJain, 1989Jain, 1989
MSP9
2D Orthogonal & Unitary Transformations
• A general orthogonal series expansion for an NxN image u(m,n) is a pair of transformations
where is called an image transform, the elements v(k,l) are called the transform coefficients and is the transformed image.
),(, nma lk
1
0
1
0, 1,0 ),,(),(),(
N
m
N
nlk Nlknmanmulkv
1
0
1
0
*, 1,0 ),,(),(),(
N
k
N
llk Nnmnmalkvnmu
),( lkvV
MSP10
2D Orthogonal & Unitary Transformations (cont’)
is a set of complete orthonormal discrete basis functions satisfying ),(, nma lk
)','(),(),( :lityOrthonorma1
0
1
0
*',', llkknmanma
N
m
N
nlklk
)','()','(),( :ssCompletene1
0
1
0
*,, nnmmnmanma
N
k
N
llklk
MSP11
2D Orthogonal & Unitary Transformations (cont’)
• The orthonormality property assures that any truncated series expansion of the form
will minimise the sum-square-error
for v(k,l) as above, and the completeness property guarantees that this error will be zero for P=Q=N.
NQNPnmalkvnmuP
k
Q
llkQP
1
0
1
0
*,, , ),,(),(),(
21
0
1
0,
2 ),(),(
N
m
N
nQPe nmunmu
MSP12
Basis Images
• Define the matrices , where is the kth column of , and the matrix inner product of two NxN matrices F and G as
• Then Equations 2 & 1 provide series representation for the image as
Tlklk***
, aaA *ka
*TA
.),(),(,1
0
1
0
*
N
m
N
n
nmgnmfGF
*,
*,
1
0
1
0
,),(
),(
lk
lk
N
k
N
l
lkv
lkv
AU
AU
MSP13
Basic Images of the 8x8 2D TransformsJain, 1989Jain, 1989
MSP14
The Continuous 1D Fourier Transform
• The Fourier transform pair
dsesFtf
dtetfsF
stj
stj
2
2
)()(
)()(