Pyramidal Algorithm with one Wavelet

To modify the previous algorithm in order to have an isotropic wavelet transform, we compute the difference signal by:

$$w_{j+1}(k) = c_j(k) - \tilde c_j (k)$$ (14.43)

but $\tilde c_j$ is computed without reducing the number of samples:

$$\tilde{c}_j(k) = \sum_l h(k-l) c_j(k)$$ (14.44)

and c_j+1 is obtained by:

$$c_{j+1}(k) = \sum_l h(l-2k) c_j(l)$$ (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 P₀ represents the wavelet coefficients pyramid, we look for an image such that the wavelet transform of this image gives P₀. Van Cittert's iterative algorithm gives:

P_n+1 = P₀ + P_n - R(P_n) (14.46)

where

• P₀ 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 cut-off frequency.