Expression for DFT of linear 2D ramp

Question

One dimensional solution is in “Expression for discrete fourier transform of linear ramp“

I need two-dimensional for image processing. We have function f(x,y) = $a_1 \cdot x + a_2 \cdot y$,

My experimentatal results:

• only is one row and one col contains data, all others are empty
• except [0,0] all real parts in one row are equal
• similar in one column real parts are equal
• imaginary parts are descending geometric series for k<N/2

for k = N/2 imaginary is zero, for k > N/2 is mirrored with sign change for k = N/4 imaginary part is is equal real part but sign real part

coefficient geometric series for 8x8 is tangent 22.5 degree but for another sizes are other(unknown way) example 8x8 for f(x,y) = x only first col because all other equal zero, only half rows because lower rows are mirrored

(224,0) 
(-32,77.2548) 
(-32,32) 
(-32,13.2548) 
(-32,0) 

for 12x12
 (792,0) 
(-72,268.708) 
(-72,124.708) 
(-72,72) 
(-72,41.5692) 
(-72,19.2923) 
(-72,0) 

for 8x12
(336,0) 
(-48,115.882) 
(-48,48) 
(-48,19.8823) 
(-48,0) 

How can we compute this coefficients?

Answer 1

Solution: for $N$x$1$ is one dimensional formula, for $N$x$M$ near all coefficients are $M$ times formula, only $[0,0]$ element is another, but this element has their simple formula as average.