21/11/15
21/11/15 Duong Anh Duc - Digital Image Processing
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Digital Image Processing
Unitary Transforms
Unitary Transforms
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21/11/15 Duong Anh Duc - Digital Image Processing
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Unitary Transforms
Unitary Transforms
Sort samples
Sort samples f(x,y)
f(x,y) in an
in an MxN
MxN image (or a rectangular
image (or a rectangular
block in the image) into colunm vector of length
block in the image) into colunm vector of length MN
MN
Compute transform coefficients
Compute transform coefficients
where
where A
A
is a matrix of size
is a matrix of size MNxMN
MNxMN
The transform
The transform A
A
is unitary, iff
is unitary, iff
If
If A
A
is real-valued, i.e.,
is real-valued, i.e., A
A-1
-1=A*,
=A*,
transform is
transform is
„orthonormal“
„orthonormal“
fAc
conjugateHermitian
*1 HT
AAA
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Energy conservation with unitary
Energy conservation with unitary
transforms
transforms
For any unitary transform
For any unitary transform we obtain
we obtain
Interpretation: every unitary transform is simply
Interpretation: every unitary transform is simply
a rotation of the coordinate system.
a rotation of the coordinate system.
Vector lengths („energies“) are conserved.
Vector lengths („energies“) are conserved.
fAc
2
2
ffAAfccc
HHH
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Energy distribution for unitary
Energy distribution for unitary
transforms
transforms
Energy is conserved, but often will be
Energy is conserved, but often will be
unevenly distributed among coefficients.
unevenly distributed among coefficients.
Autocorrelation matrix
Autocorrelation matrix
Mean squared values („average energies“) of
Mean squared values („average energies“) of
the coefficients
the coefficients c
ci
i
are on the diagonal of
are on the diagonal of R
Rcc
cc
H
ff
HHH
cc
AARAffAEccER
ii
H
ff
ii
cci
AARRcE
,
,
2
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Eigenmatrix of the autocorrelation
Eigenmatrix of the autocorrelation
matrix
matrix
Definition: eigenmatrix
Definition: eigenmatrix of autocorrelation matrix
of autocorrelation matrix R
Rff
ff
is unitary
is unitary
The columns of
The columns of form an orthonormalized set of eigenvectors
form an orthonormalized set of eigenvectors
of
of R
Rff
ff, i.e.,
, i.e.,
R
Rff
ff
is a diagonal matrix of eigenvalues.
is a diagonal matrix of eigenvalues.
R
Rff
ff
is symmetric nonnegative definite, hence
is symmetric nonnegative definite, hence
i
i
0
0
for all
for all i
i
R
Rff
ff
is normal matrix, i.e.,
is normal matrix, i.e., , hence unitary
, hence unitary
eigenmatrix exists
eigenmatrix exists
1
1
0
0
0
MN
H
ffffff
H
ff
RRRR