Back to resolution

    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:

Initial image (256*256 pixels) Fourier transform of the initial image and boundary (dashed line) of the central area used to reconstruct the susequent images

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):

Sine wave of 40 pixels period superposed to the original image (the lowest frequency is displayed on the bottom)

    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:

                                           (a)                                                                                   (b)                                                                                   (c)
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