Lecture 18 (Apr 12)

Two dimensional generalizations

1) Fourier Transform
$$
f(x,y)=\int \int \hat{f}(u,v) e^{i2\pi(ux+vy)} du dv \\
\hat{f}(u,v)=\int \int f(x,y) e^{-i2\pi(ux+vy)} dx dy \, .
$$
2) Discrete Fourier Transform
$$
f_{m,n}= \frac{1}{MN} \sum_{p=0}^{M-1} \sum_{q=0}^{N-1} \hat{f}_{p,q} e^{i 2\pi \left( \frac{pm}{M}+\frac{qn}{N} \right)} \\
\hat{f}_{p,q}=\sum_{m=0}^{M-1} \sum_{n=0}^{N-1} f_{m,n} e^{-i 2\pi \left( \frac{pm}{M}+\frac{qn}{N} \right)} \, .
$$

Some properties

1) Periodicity:
$$
\hat{f}_{p,q}=\hat{f}_{p+k_1 M,q}=\hat{f}_{p,q+k_2 N}=\hat{f}_{p+k_1 M,q+k_2 N}
$$
for any integer $k_1$ and $k_2$.
2) Translation:
$$
\mathcal{F} \left[
f_{m,n} e^{i 2\pi \left( \frac{p^*m}{M}+\frac{q^*n}{N} \right)} \right] = \hat{f}_{p-p^*,q-q^*}
$$
3) From (2), take $p^*=M/2$ and $q^*=N/2$, we have
$$
\mathcal{F} \left[
f_{m,n} (-1)^{m+n} \right] = \hat{f}_{p-M/2,q-N/2} \, .
$$

Image Enhancement

Design $\hat{h}$ such that $\tilde{f}=\mathcal{F}^{-1}(\hat{h} \cdot \hat{g})$ achieves some properties.

Reading Materials:

Lecture Notes: p.94