    Next: Pyramidal Algorithm Up: The discrete wavelet transform Previous: Multiresolution Analysis

## The à trous algorithm

The discrete approach of the wavelet transform can be done with the special version of the so-called à trous algorithm (with holes) [#holschn1<#14462,#shensa<#14463]. One assumes that the sampled data are the scalar products at pixels k of the function f(x) with a scaling function which corresponds to a low pass filter.

The first filtering is then performed by a twice magnified scale leading to the set. The signal difference contains the information between these two scales and is the discrete set associated with the wavelet transform corresponding to . The associated wavelet is therefore . (14.29)

The distance between samples increasing by a factor 2 from the scale (i-1) (i > 0) to the next one, ci(k) is given by: (14.30)

and the discrete wavelet transform wi(k) by:

 wi(k) = ci-1(k) - ci(k) (14.31)

The coefficients derive from the scaling function : (14.32)

The algorithm allowing one to rebuild the data frame is evident: the last smoothed array cnp is added to all the differences wi. (14.33)

If we choose the linear interpolation for the scaling function (see figure 14.5):  we have: (14.34)

c1 is obtained by: (14.35)

and cj+1 is obtained from cj by: (14.36)

The figure 14.6 shows the wavelet associated to the scaling function. The wavelet coefficients at the scale j are: (14.37)

The above à trous algorithm is easily extensible to the two dimensional space. This leads to a convolution with a mask of pixels for the wavelet connected to linear interpolation. The coefficents of the mask are: At each scale j, we obtain a set (we will call it wavelet plane in the following), which has the same number of pixels as the image.

If we choose a B3-spline for the scaling function, the coefficients of the convolution mask in one dimension are ( ), and in two dimensions:     Next: Pyramidal Algorithm Up: The discrete wavelet transform Previous: Multiresolution Analysis
Petra Nass
1999-06-15