About using cwt function

Hi guys! I'm a little bit confused about using cwt function. I know, that some wavelets have a scaling function (phi), but some have only wavelet function (psi). But, as far as I know, cwt uses only psi. So, we have only detail coefficients. What about approximation coefficients, obtained with phi?

Here is the video to clarify what I mean http://slideplayer.com/slide/4998644/. Slide on 14:40.

Answer 1

[Wayne King]

Hi Alexander, For the CWT, the scaling functions are not commonly used. For discrete analysis wavelets are typically defined in terms of a multiresolution analysis. A multiresolution analysis is a nested series of subspaces and their orthogonal complements. The bases for the subspaces are the scaling functions (dilated versions of the scaling function for $V_0$) and the wavelets are derived from the scaling functions.

In the CWT, that is not the case. One usually starts with the wavelet, typically some rapidly-decreasing oscillatory function like a modulated Gaussian. You can however obtain the scaling function as the integral, $\int_{s'}^{\infty} \dfrac{\hat{\psi}(s \omega)}{s} ds$ where $s'$ is the maximum scale you have in the CWT. $\hat{\psi}$ here is the Fourier transform of the wavelet.

[Alexander Voznesensky]

Thank you. I see, that we introduce the concept of scaling function in DWT. How the scaling func is connected with the wavelet func, is there any formula of connection? What is primary: scaling or wavelet? By the way, we use 2 filters in DWT: low-pass h[n] and high-pass g[n]. We can obtain these filters by dwt. But how these filters are connected with scaling and wavelet? I know 2 formulas (see the attachment), but i want to understand deeper.