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The resolution of an image created from Fourier coefficients is then linked to the highest (h,k) indices of these non-zero coefficients. The following image (256*256 pixels) shows two points (15 pixels in diameter) which centres are separated by 40 pixels:
Now, one is looking to the images formed from the Fourier transform of the original image, keeping only the coefficients inside a circular area centred on the Fourier space origin and of varying radius.
The lowest frequency contribution to draw these two disks is a sine wave of period 40 pixels, having a spatial frequency 256/40=6.4 times larger than the lowest possible frequency for that particluar image (associated with the first Fourier coefficient close to the origin on the horizontal axis):
The Fourier coefficient associated to this sine wave (T=40pixels) is then the 6th (or 7th) from the origin F(0,0)in the horizontal direction. When the reconstructed image contains the Fourier coefficients up to this limit (6 pixels radius) and beyond ((a) & (b)) it is then possible to distinguish the two disks, contrary to situation (c) where only the Fourier coefficients within a circular area of 4 pixels radius are used:
Images reconstructed from the truncated Fourier transform of the original image.Only a central area of (a) 8 (b) 6 & (c) 4 pixels (then Fourier coefficients) in radius is used to build each of these images