Morlet's wavelet reconstruction formula

Question

The CWT (continuous wavelet transform) of a signal $x(t)$ is $$X_w(a,b)=\frac{1}{\sqrt{|a|}} \int_{-\infty}^{\infty} x(t)\psi^{\ast}\left(\frac{t-b}{a}\right)\, dt$$

In order to reconstruct the signal, we can use this inverse formula : $$x(t)=\frac{1}{B_\psi}\int_{0}^{\infty}\int_{-\infty}^{\infty} \frac{1}{a^2}X_w(a,b)\frac{1}{\sqrt{|a|}} \tilde\psi\left(\frac{t-b}{a}\right)\, db\ da$$ where $$ B_\psi := \int_0^\infty \frac{\left|\hat{\psi}(u)\right|^2}{u} du.$$

But I read in some places (http://wavelets.ens.fr/ENSEIGNEMENT/COURS/CNRS/3_Marie_CWT_1D.pdf) that it is possible to do reconstruction with a different function than the original analysis wavelet (and it should give the same result ! I tried with numerical signals in Python, and it is true, it works!).

For example using $\psi(x)=\delta(x)$ leads to the Morlet reconstruction formula :

$$x(t)= \dfrac{1}{C_\psi}\int_{0}^{\infty} \frac{1}{a^2}X_w(a,t)\frac{1}{\sqrt{|a|}} \,da$$

where

$$ C_\psi := \int_0^\infty \frac{\overline{\psi(u)}}{u} du $$

Do you know where we can find a proof for this ?

Answer 1

I have seen the answer of this question at a book, but I didn't understand so much. I don't know if its useful or no, the notation and mathematics of it is so much hard, hope you understand, explain me if you understand too, also please explain how did you write inverse CWT code for delta function.
the reference is : WAVELET TRANSFORMS AND THEIR APPLICATIONS TO TURBULENCE,Marie Farye

Answer 2

I'm not sure if this answer your question but you can reconstruct your signal from the wavelet transform with use of another function called the scaling function. I can show you the case for one dimensioal functions. I'm not sure how it generalizes to higher dimensions. The theory is taken from Mallat - A wavelet tour of signal processing, Chapter 4.3, where the proof is given as an exercise.

If $ \psi$ is your wavelet function, define $ \phi $ via its Fourier transform $$ \widehat{ \phi }( \omega) = \int_{ \omega} ^{\infty} \frac{ | \widehat{ \psi }(\xi)|^2 }{\xi} d\xi ,$$

And let $ \phi_{a}(t) = a^{-1/2}\phi(a^{-1}t)$ be the dilated function. Define $ Lx(a,b) =\langle x(t) , \phi_{a}(t-b) \rangle $. Then for any $ a_0 > 0 $ we have

$$ x(t) = \frac{1}{C_{\psi}} \int_{0}^{a_0} X_{w}(a,\cdot) \ast \psi_a(t) \frac{da}{a^2} + \frac{1}{C_{\psi}a_0} Lx( \cdot, a_0) \ast \phi_{ a_0}(t), $$

where $C_{\psi} = \int_{ 0} ^{\infty} \frac{ | \widehat{ \psi }(\xi)|^2 }{\xi} d\xi $. However you do need the original analysis wavelet to do full reconstruction.