Computer vision

Discrete Fourier Transform

The Fourier transform, defined for a single channel greyscale image, is defined as

F[X]=F[u,v]=1WHnx=0W1ny=0H1X[nx,ny]ej2π(unxW+unyH)\mathcal{F}[X] = \mathcal{F}[u,v] = \frac{1}{WH}\sum_{n_x=0}^{W-1}\sum_{n_y=0}^{H-1}{X[n_x,n_y]e^{-j2\pi(\frac{un_x}{W}+\frac{un_y}{H})}}

where j=ij=\sqrt{i}. Note that each F[u,v]F[u,v] depends on all pixels the in the original image XX. F can be thought of as “complex-valued image”, having the same size as the original image XX. The transformation gives a representation of the image in the frequency domain. The image produced by the transformation has pixels that each represent a particular frequency in the original spacial domain image.