# Wavelet transform : how to recover wavelet basis matrix?

In this article, It is said that an image $$x$$ can be sparsely expressed, if the basis is well chosen. It is said something like

"$$x$$ can be expressed as follows : $$x = \psi\theta$$ [...] Many differents basis expansions can achieve sparse approximations of natural image, including wavelets, Gabor frames, and curvelets."

I tried to use a python package, pywavelets, to recover $$\theta$$, but I actually get something like :

cA, cD = pywt.dwt(my_image, kind_of_wavelet) # 'discrete wavelet transform'


I don't understand how to get my $$\theta$$ back from theses coefficients, and how to recover the $$\psi$$ matrix. During my research, I found here that

"The wavefun() method can be used to calculate approximations of scaling function ($$\phi$$) and wavelet function ($$\psi$$) at the given level of refinement."

but I don't know if this is what I need here. One last idea I had, was to apply the dwt() transformation on a one-pixel-images basis (basis of images with one pixel on 1 and the rest on 0), and to get $$\psi$$ from there. But I don't understand wavelet transformation deeply enough to be sure that it will work. Is there maybe a built-in, more orthodox way to get $$\psi$$ and $$\theta$$ ? Can I recover $$\theta$$ through cA and cD for example ?