The Reconstruction

If the wavelet is the difference between two resolutions, an evident reconstruction for a wavelet transform is:

But this is a particular case and other wavelet functions can be chosen. The reconstruction can be done step by step, starting from the lowest resolution. At each scale, we have the relations:

we look for knowing , , h and g. We restore with a least mean square estimator:

is minimum. and are weight functions which permit a general solution to the restoration of . By derivation we get:

where the conjugate filters have the expression:

It is easy to see that these filters satisfy the exact reconstruction equation . In fact, equations and give the general solution to this equation. In this analysis, the Shannon sampling condition is always respected. No aliasing exists, so that the dealiasing condition is not necessary.

The denominator is reduced if we choose:

This corresponds to the case where the wavelet is the difference between the square of two resolutions:

: Left, the interpolation function and right, the wavelet .

: On left, the filter , and on right the filter .

We plot in figure the chosen scaling function derived from a B-spline of degree 3 in the frequency space and its resulting wavelet function. Their conjugate functions are plotted in figure .

The reconstruction algorithm is:

1. We compute the FFT of the image at the low resolution.
2. We set j to . We iterate:
3. We compute the FFT of the wavelet coefficients at the scale j.
4. We multiply the wavelet coefficients by .
5. We multiply the image at the lower resolution by .
6. The inverse Fourier Transform of the addition of and gives the image .
7. j = j - 1 and we go back to 3.

The use of a scaling function with a cut-off frequency allows a reduction of sampling at each scale, and limits the computing time and the memory size.