Discrete Fourier Transform

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

$$\mathcal{F}[X] = \mathcal{F}[u,v] = \frac{1}{WH}\sum_{n_x=0}^{W-1}\sum_{n_y=0}^{H-1}{X[n_x,n_y]e^{-j2\pi(\frac{un_x}{W}+\frac{un_y}{H})}}$$

where $j=\sqrt{i}$. Note that each $F[u,v]$ depends on all pixels the in the original image $X$. F can be thought of as “complex-valued image”, having the same size as the original image $X$. 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.