I've read there are two admissibility criteria for wavelets, both of which are designed to preserve total power of the signal (source: http://en.wikipedia.org/wiki/Wavelet#Mother_wavelet, as well as various scientific papers)
- Condition for zero-mean: $$\int_{-\infty}^{\infty}\psi(t)dt=0 $$
- Condition for square norm one: $$\int_{-\infty}^{\infty}|\psi(t)|^2dt=1,$$
where $\psi(t)$ is the wavelet kernel. This leads to having a normalization factor of $\frac{1}{\sqrt{2\pi}}$ for the Gabor/Morlet wavelet.
Now, my question is: how do these admissibility criteria apply to 2D wavelets? (Let's say $\psi(x,y)$.)
My guess would be:
- $$\int_{x=-\infty}^{\infty}\int_{y=-\infty}^{\infty}\psi(x,y)dx dy=0 $$
- $$\int_{x=-\infty}^{\infty}\int_{y=-\infty}^{\infty}|\psi(x,y)|^2dxdy=1,$$
but I get inconsistent results in my computations.
Is my guess right? If not, what are they? Can you provide a reliable source?