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 ?


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