Relation Between the Filter Responses and the Optimized Parameters

The results of the optimization of a WN on an image depends largely on choice of the mother-wavelet. An example can be seen in fig. 4. A WN is optimized by increasing the number of wavelets until either a maximal wavelet number $N$ or an energy threshold is reached. Each new wavelet is thus optimized based on the residual between the original function $f$ and the already optimized wavelets:

$$R = f - \sum w_i \psi_{{\bf n}_i}\; .$$

The optimization procedure parameterizes each new wavelet such that

$$\sum_{{\bf x}} (R({\bf x}) - w_{\mbox{new}} \psi_{{\bf n}_{\mbox{new}}}({\bf x}))^2 = \min$$

is minimized which is true, where the correlation between the wavelet and the residual $R$ is maximal:

$$\sum_{{\bf x}} R({\bf x})\psi_{{\bf n}_{\mbox{new}}}({\bf x}) = \max \; .$$

In case of the odd Gabor function, which is an excellent edge detector, the optimized wavelets will end at edges (see figs. 2 and 4). The chosen mother wavelets seems to introduce a model for local image features. While, e.g., the odd Gabor function seems to models edge segments, the anisotropic DOF seems to favor homogeneous image regions (fig. 4).

Figure 4: This figure shows images of a wooden toy block (top, left) on which a WN was trained. The black line segments sketch the positions, sizes and orientations of all the wavelets of the WN (right) The bottom left image shows the residual image $R$ between the original image and the approximation by the wavelets at the top left image. The bottom right image sketches the parameters of the largest optimized anisotropic DOG wavelets.