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6.3.2.9 Fourier operations in two dimensions ¶

Once all the relations in the previous sections have been clearly understood in one dimension, it is very easy to generalize them to two or even more dimensions since each dimension is by definition independent. Previously we defined \(l\) as the continuous variable in 1D and the inverse of the period in its direction to be \(\omega\). Let’s show the second spatial direction with \(m\) the inverse of the period in the second dimension with \(\nu\). The Fourier transform in 2D (see Fourier transform) can be written as:

$$F(\omega, \nu)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} f(l, m)e^{-i({\omega}l+{\nu}m)}dl$$

$$f(l, m)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} F(\omega, \nu)e^{i({\omega}l+{\nu}m)}dl$$

The 2D Dirac \(\delta(l,m)\) is non-zero only when \(l=m=0\). The 2D Dirac comb (or Dirac brush! See Dirac delta and comb) can be written in units of the 2D Dirac \(\delta\). For most image detectors, the sides of a pixel are equal in both dimensions. So \(P\) remains unchanged, if a specific device is used which has non-square pixels, then for each dimension a different value should be used.

$${\rm III}_P(l, m)\equiv\displaystyle\sum_{j=-\infty}^{\infty} \displaystyle\sum_{k=-\infty}^{\infty} \delta(l-jP, m-kP) $$

The Two dimensional Sampling theorem (see Sampling theorem) is thus very easily derived as before since the frequencies in each dimension are independent. Let’s take \(\nu_m\) as the maximum frequency along the second dimension. Therefore the two dimensional sampling theorem says that a 2D band-limited function can be recovered when the following conditions hold¹⁷⁸:

$${2\pi\over P} > 2\omega_m \quad\quad\quad {\rm and} \quad\quad\quad {2\pi\over P} > 2\nu_m$$

Finally, let’s represent the pixel counter on the second dimension in the spatial and frequency domains with \(y\) and \(v\) respectively. Also let’s assume that the input image has \(Y\) pixels on the second dimension. Then the two dimensional discrete Fourier transform and its inverse (see Discrete Fourier transform) can be written as:

$$F_{u,v}=\displaystyle\sum_{x=0}^{X-1}\displaystyle\sum_{y=0}^{Y-1} f_{x,y}e^{-i({ux\over X}+{vy\over Y})} $$

$$f_{x,y}={1\over XY}\displaystyle\sum_{u=0}^{X-1}\displaystyle\sum_{v=0}^{Y-1} F_{u,v}e^{i({ux\over X}+{vy\over Y})} $$