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:


Original image

FT of the original image


FT of the original image where only the low resolution is kept 

Low resolution image obtained by inverse FT of the previous modified Fourier transform

    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:



In Fourier space one multiplies the FT of the original image with a mask which selects the low resolution Fourier coefficients

    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:


In direct space the low resolution image is obtained by the convolution between the original image and tke inverse FT of the mask


    The inverse FT of the mask is a function having maxima at the corners, surrounded by concentric rings of decreasing amplitudes:


The mask used in order to select the low resolution Fourier coefficients & its inverse FT (together with the profile along the indicated line)

    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:


Image obtained by the addition of the original image (contribution of the peak in (0,0) of the inverse FT of the mask) and of one copy of it shifted and of lower intensity


Blow-up of the original image
 
 

Original image to which a copy of it but shifted and of lower intensity is superposed

    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).