Open the bird cage…

Now the question one is asking is:

How free this bird from its cage ???  (je sais, je dessine assez mal… ;-))

 The bird in its cage

First, one have to study the Fourier transform of this image:

 Fourier transform of the captive bird

If one looks closely, one can see that this FT (here it is its moduli that are displayed) consists in a set of equidistant peaks on the horizontal direction going through the origin (image centre) and of a more complex variation without any particular favoured direction. According to what we have seen concerning the lattices, the set of peaks in the FT should come from a lattice in the image, then from the cage in front of the bird; the more complex moduli variation then comes from the bird itself which has a more complex shape than the grid.
To check this, one can eliminate these peaks in the horizontal axis using an adequate masking tool and then calculate the inverse Fourier transform of this modified FT:

 Fourier transform of the original image where one eliminated the peaks in the horizontal direction using a mask (left) & its inverse FT

The bird is now free!!!

To conclude this example one can notice that the FT of the original image (bird in cage) is the convolution of the FT of the free bird with the empty cage (not the sum):

 FT of the free bird (left) & FT of the empty cage

To see that, one can have a close look around the peaks sitting on the horizontal line of the FT of the bird in cage image: one can notice that these peaks are broader than the peaks of the FT of the empty cage and even display an asymmetry that also exists around the origin of the FT of the free bird.