Multidimensionnal wavelet admissibility criteria

Question

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)

1. Condition for zero-mean: $$\int_{-\infty}^{\infty}\psi(t)dt=0 $$
2. 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:

1. $$\int_{x=-\infty}^{\infty}\int_{y=-\infty}^{\infty}\psi(x,y)dx dy=0 $$
2. $$\int_{x=-\infty}^{\infty}\int_{y=-\infty}^{\infty}|\psi(x,y)|^2dxdy=1,$$

but I get inconsistent results in my computations.

Is my guess right? If not, what are they? Can you provide a reliable source?

Answer 1

It turns out the admissibility criterion does not dictate that the energy of the wavelet must be unity. To be classified as a wavelet, a wavelet must follow the following criteria:

• The wavelet must have finite energy, so $E = \int_{-\infty}^{+\infty} |\psi(t)|^{2} dt < \infty $
• The second condition is that the wavelet must have zero mean. (The intuition behind this is because the wavelet is acting like a matched filter, and we care only about how nicely the shape of the received signal matches the wavelet, and not the energy of the received signal). So if $\hat\psi(f) $ is the fourier transform of your wavelet, then: $$ C_g = \int_{0}^{\infty} \frac{|\hat\psi(f)|^2}{f} df < \infty $$ The $C_g$ here is called the admissibility constant, and the above property is called the admissibility criterion. This implies that $\hat\psi(0)=0$, because if it didn't, the above integral would blow up as you can see.
• Finally, for complex wavelets, the last criterion states that the Fourier transform must be both real, and vanish for negative frequencies.

Now notice, no where does it state that the square norm of the wavelet must be equal to unity. It only states that the $E < \infty$.

In your case, it sounds as though the energy of your signal in the Wavelet domain, is not matching the energy of your signal in the Fourier domain. My best guess without further information is that you not normalize your wavelet to unit energy, since it is not demanded by the admissibility criterion. (First check though, if the energy of your signal in the spatial domain actually matches the energy of your signal in the wavelet domain). I would start there.