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Digital Image Processing
Unitary Transforms
21/11/15 Duong Anh Duc - Digital Image Processing 1
Unitary Transforms
Sort samples f(x,y) in an MxN image (or a rectangular block in the image) into colunm vector of length MN
Compute transform coefficients
c Af
where A is a matrix of size MNxMN The transform A is unitary, iff A 1
T H Hermitianconjugate
If A is real-valued, i.e., A1=A*, transform is „orthonormal“
21/11/15 Duong Anh Duc - Digital Image Processing 2
Energy conservation with unitary
transforms
For any unitary transform c Af
c 2 cHc f H AH Af
we obtain
f 2
Interpretation: every unitary transform is simply a rotation of the coordinate system.
Vector lengths („energies“) are conserved.
21/11/15 Duong Anh Duc - Digital Image Processing 3
Energy distribution for unitary transforms
Energy is conserved, but often will be unevenly distributed among coefficients.
Autocorrelation matrix
Rcc E ccH E Af f H AH ARff AH
Mean squared values („average energies“) of the coefficients ci are on the diagonal of Rcc
E c2 Rcc i,i ARff AH i,i
21/11/15 Duong Anh Duc - Digital Image Processing 4
Eigenmatrix of the autocorrelation matrix
Definition: eigenmatrix of autocorrelation matrix Rff is unitary
The columns of form an orthonormalized set of eigenvectors of Rff, i.e.,
Rff 0 0
1
0 MN 1
is a diagonal matrix of eigenvalues.
R is symmetric nonnegative definite, hence 0 for all i R is normal matrix, i.e., Rff Rff Rff Rff , hence unitary
eigenmatrix exists
21/11/15 Duong Anh Duc - Digital Image Processing 5
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