Interpretation of the low resolution image formation using the convolution theorem |

In one of the first examples we saw, the image formed using only the
Fourier coefficients close to the Fourier space origin (low resolution)
lost the details present on the original image and became as blurred:
This characteristic was explained in Fourier space by considering that
the Fourier coefficients of high resolution were set to zero and since
it is in particular these Fourier coefficients that code for details on
the image it results that this latter became blurred.
This effect, linked to the notion of resolution, can also be thought in
direct space using the convolution theorem, which states that the
Fourier transform of a convolution product of two functions (two images
for example) is equal to the simple product between the Fourier
transforms of each of these functions.
In Fourier space, the low resolution FT is obtained as the simple product between the FT of the original image and a mask having pixels equal to 1 inside a given disk and 0 outside:
Then, according to the convolution theorem, in direct space the image
corresponding to the low resolution FT will be obtained by the
convolution of the original image with the inverse FT of the mask used:
The inverse FT of the mask is a function having maxima at the corners, surrounded by concentric rings of decreasing amplitudes:
Then, in order to create the final low resolution image in direct
space, one need to convolute the original image with the inverse FT of
the mask: the strongest contribution arrises from the top-left corner
(which is the origin (0,0) of this inverse FT), and then one have to
add to this original image (non shifted) a copy of it, shifted and
weighted according to the oscillations of the inverse FT of the mask:
The final image is obtained by repeating this addition of shifted and
weighted copies, according to the shape if the inverse FT of the mask,
leading to its blurred appearance.
Remark: the 3 other maxima on the inverse FT of the mask (at top-right and bottom left and right corners) are linked to the fact that the determination of the FT considers the original image as periodically repeated in vertical and horizontal directions which introduces minor artefacts on the image edges (these latter are not visible in the present example but can be heavier in other cases). |