Direct Calculation of Weights

Wavelet functions are not necessarily orthogonal. For a given family ${\bf\Psi }$ of wavelets it is therefore not generally possible to calculate a wavelet coefficient $w_i$ directly by a simple projection of the wavelet $\psi_{{\bf n}_i}$ onto the considered function. In [Daubechies, 1990,Daugman, 1988] it was therefore proposed to use eq. (2) to find the optimal coefficients $w_i$ for each fixed wavelet. Because optimization is a slow process, we want to suggest a direct calculation for the case of a finite wavelet family. The correct coeffs. $w_i$ are computed by projecting the dual wavelets $\tilde{\psi}_{{\bf n}_i}$. The wavelet $\tilde{\psi}_{{\bf n}_i}$ is the dual wavelet to the wavelet $\psi_{{\bf n}_i}$ if

$$\langle\psi_{{\bf n}_i},\tilde{\psi}_{{\bf n}_j}\rangle =\delta_{i,j} \; .$$ (4)

With $\tilde{{\bf\Psi}} =(\tilde{\psi}_{{\bf n}_1},\ldots,\tilde{\psi}_{{\bf n}_N})^T$, we can write

$$\left[\left<{\bf\Psi},\tilde{{\bf\Psi}}\right>\right]={\rm 1\negthickspace I}$$ (5)

and we find $\tilde{\psi}_{{\bf n}_i}$ to be

$$\tilde{\psi}_{{\bf n}_i} = \sum_j\left(\Psi\right)_{i,j}^{-1}\psi_{{\bf n}_j}\; ,$$ (6)

where $({\bf\Psi})_{i,j}=\langle\psi_i,\psi_j\rangle$. Given a family ${\bf\Psi }$ of optimized wavelets of a WN for the function $f$, we can compute the orthogonal projection of a function $g$ into the subspace $<{\bf\Psi}> \subseteq {\Bbb{L}}^2({\Bbb{R}}^2)$ (see (4)), i.e.

$$\hat{g}= \sum^N_{i=1}w_i \psi_{{\bf n}_i}\; \;{\bf w}= \tilde{{\bf\Psi}}g\;.$$ (7)

The method to compute the orthogonal projection of a function $g$ into the subspace $<{\bf\Psi}>$ is mathematically equivalent to using the pseudo-inverse of ${\bf\Psi }$ directly. However, using the dual wavelets of eq. (7) will proof to be computationally more efficient: For our tracking experiment we will have to deform the entire WN affinely, which means that the pseudo-inverse has to be recomputed. The matrix $({\bf\Psi})_{i,j}$, on the other hand, is invariant, except for a factor, to affine deformations of the WN, and only the projections $\langle g, \psi_{{\bf n}_i}\rangle$ need to be recomputed.