2D DFT is given by,

$$ F(u, v)=\sum_{x=0}^{M-1} \sum_{y=0}^{N-1} f(x, y) e^{-j 2 \pi\left(\frac{u x}{M}+\frac{v y}{N}\right)} .\\ $$

There are many types of 2D DFT properties:

• Translation
• Distributive and scaling
• Rotation
• Periodicity and Conjugate Symmetry
• Separability (kernel separating)
• Linearity
• Convolution and Correlation

i) Translation and Rotation:

The Fourier transform pair satisfies the following translation properties.

$ \begin{aligned} &f(x, y) e^{j 2 \pi\left(\frac{u_0 x}{M}+\frac{v_0^{\prime} y}{N}\right)} \Leftrightarrow F\left(u-u_0, v-v_0\right) \\ &\text { and } \\ &f\left(x-x_0, y-y_0\right) \Leftrightarrow F(u, v) e^{-j 2 \pi\left(\frac{x_0 u}{M}+\frac{y_0 v}{N}\right)} \end{aligned} $

Multiplying $f(x, y)$ by the exponential shown shifts the origin of DFT to $\left(u, v, v_0\right)$ and multiplying $F(u, v)$ by the negative of that exponential shifts the origin of $f(x, y)$ to $\left(x_0, y_0\right)$

ii) Periodicity:-

The 2D Fourier transform m and its transform inverse are infinitely periodic in the u and u directions.

$ \begin{aligned} F(u, v) &=F\left(u+k_1 M, v\right)=F\left(u, v+k_2 N\right) \\ &=F\left(u+k_1 M, v+k_2 N\right) \\ \text { and } \\ f(x, y) &=f\left(x+k_1 M, y\right)=f\left(x, y+k_2 N\right) \\ &=f\left(x+k_1 M, y+k_2 N\right) \end{aligned} $

where $k_1$ s $k_2$ are integers.