Euclidean Distance in Wavelet Sub-space

An interesting question is, how to compute the distance between two vectors of wavelet coefficients. Let us consider two vectors ${\bf v}$, ${\bf w}$ of some wavelet subspace, define w.r.t. WN ${\bf\Psi }$. Computing the Euclidean distance between the two vectors, $\Vert{\bf v}-{\bf w}\Vert _2$, fails to reflect the different influences (e.g. due to different scales) of the wavelets in the sum (4). Instead, we suggest to compute the Euclidean distance between the WNs of ${\bf v}$ and ${\bf w}$ as follows: Starting out from the Euclidean distance in the (image) subspace $<{\bf\Psi}>$

$$\left\Vert\sum_{i=1}^N v_i \psi_{{\bf n}_i} - \sum_{i=1}^N w_i \psi_{{\bf n}_i} \right\Vert _2\; ,$$ (8)

algebraic transformations lead to

$$\Vert{\bf v}-{\bf w}\Vert _{{\bf\Psi}} := \left[ \sum_{i,j} (v_i-w_i)(v_j-w_j)\langle \psi_{{\bf n}_i}, \psi_{{\bf n}_j} \rangle \right]^{\frac{1}{2}}$$

$$= ({\bf v}-{\bf w})^t\; ({\bf\Psi})_{i,j}\; ({\bf v}-{\bf w})\;.$$ (9)

$\Vert\cdot\Vert _{{\bf\Psi}}$ compute the Euclidean distance between the two appropriate points in $<{\bf\Psi}>$ and considers thus the different parameters of the wavelets. For orthogonal wavelets, the matrix $({\bf\Psi})_{i,j} = \langle\psi_{{\bf n}_i},\psi_{{\bf n}_j}\rangle$ is the unity matrix and no weighting is needed.

The same techniques can be used to derive further distance or similarity measures, such as,e.g., the normalized cross correlation.