Next: Multiresolution with scaling functions
Up: Pyramidal Algorithm
Previous: The Laplacian Pyramid
Pyramidal Algorithm with one Wavelet
To modify the previous algorithm in order to have an isotropic
wavelet transform, we compute the difference signal by:



(14.43) 
but
is computed without reducing the number of samples:



(14.44) 
and c_{j+1} is obtained by:



(14.45) 
The reconstruction method is the same as with the laplacian pyramid,
but the reconstruction is not exact. However, the exact reconstruction
can be performed by an iterative algorithm. If P_{0} represents
the wavelet coefficients pyramid, we look for an image such that the wavelet
transform of this image gives P_{0}. Van Cittert's iterative algorithm gives:
P_{n+1} = P_{0} + P_{n}  R(P_{n}) 


(14.46) 
where
 P_{0} is the pyramid to be reconstructed
 P_{n} is the pyramid after n iterations
 R is an operator which consists in doing a reconstruction followed
by a wavelet transform.
The solution is obtained by reconstructing the pyramid P_{n}.
We need no more than 7 or 8 iterations to converge. Another way to
have a pyramidal wavelet transform with an isotropic wavelet is
to use a scaling function with a cutoff frequency.
Next: Multiresolution with scaling functions
Up: Pyramidal Algorithm
Previous: The Laplacian Pyramid
Petra Nass
19990615