Frequency-domain filtering
1 Exercise: Fourier transform
Solve this Fourier transform problem using Excel. We will create a spreadsheet to calculate each term in the DFT and inverse DFT equations. That is, $f(x)e^{-2i\pi ux/M}$ and $F(u)e^{2i\pi ux/M}$.
1.1 Complex numbers in Excel
COMPLEX(a, b): $a+ib$IMREAL(COMPLEX(a, b)): $a$IMAGINARY(COMPLEX(a, b)): $b$IMEXP(COMPLEX(a, b)): $e^{a+ib}$IMEXP(COMPLEX(0, -2*PI())): $e^{-2i\pi}$IMPRODUCT(c, IMEXP(COMPLEX(a, b))): $ce^{a+ib}$IMSUM(...): $\sum\cdots$
1.2 State the problem
First, copy the problem. What is $M$?
| x | 0 | 1 | 2 |
|---|---|---|---|
| f(x) | 5 | 3 | 2 |
| M | ? |
1.3 $f(x)e^{-2i\pi ux/M}$
\[F(u)=\frac{1}{M}\sum_{x=0}^{M-1}f(x)e^{-2i\pi ux/M}\]
For each $u=0,\cdots,M-1$, we need $x=0,\cdots,M-1$.
| u | x | f(x) | f(x)*exp() |
|---|---|---|---|
| 0 | 0 | 5 | =IMPRODUCT($f(x)$, IMEXP(COMPLEX(0, -2*PI()*$u$*$x$/$M$))) |
| 0 | 1 | 3 | |
| 0 | 2 | 2 | |
| 1 | 0 | 5 | |
| 1 | 1 | 3 | |
| 1 | 2 | 2 | |
| 2 | 0 | 5 | |
| 2 | 1 | 3 | |
| 2 | 2 | 2 |
1.4 $F(u)$
Now, for each $u$, we need to sum $f(x)e^{-2i\pi ux/M}$ and divide it by $M$ to get $F(u)$.
| u | x | f(x) | f(x)*exp() | F(u) |
|---|---|---|---|---|
| 0 | 0 | 5 | =IMPRODUCT(IMSUM($f(x)e^{-2i\pi ux/M}$), 1/$M$) | |
| 0 | 1 | 3 | ||
| 0 | 2 | 2 | ||
| 1 | 0 | 5 | =IMPRODUCT(IMSUM($f(x)e^{-2i\pi ux/M}$), 1/$M$) | |
| 1 | 1 | 3 | ||
| 1 | 2 | 2 | ||
| 2 | 0 | 5 | =IMPRODUCT(IMSUM($f(x)e^{-2i\pi ux/M}$), 1/$M$) | |
| 2 | 1 | 3 | ||
| 2 | 2 | 2 |
1.5 $F(u)e^{2i\pi ux/M}$
\[f(x)=\sum_{u=0}^{M-1}F(u)e^{2i\pi ux/M}\]
For each $x=0,\cdots,M-1$, we need $u=0,\cdots,M-1$.
| x | u | F(u) | F(u)*exp() |
|---|---|---|---|
| 0 | 0 | =IMPRODUCT($F(u)$, IMEXP(COMPLEX(0, 2*PI()*$u$*$x$/$M$))) | |
| 0 | 1 | ||
| 0 | 2 | ||
| 1 | 0 | ||
| 1 | 1 | ||
| 1 | 2 | ||
| 2 | 0 | ||
| 2 | 1 | ||
| 2 | 2 |
1.6 $f(x)$
For each $x$, we need to sum $F(u)e^{2i\pi ux/M}$ and take its real part to get $f(x)$.
| x | u | F(u) | F(u)*exp() | f(x) |
|---|---|---|---|---|
| 0 | 0 | =IMREAL(IMSUM($F(u)e^{-2i\pi ux/M}$)) | ||
| 0 | 1 | |||
| 0 | 2 | |||
| 1 | 0 | =IMREAL(IMSUM($F(u)e^{-2i\pi ux/M}$)) | ||
| 1 | 1 | |||
| 1 | 2 | |||
| 2 | 0 | =IMREAL(IMSUM($F(u)e^{-2i\pi ux/M}$)) | ||
| 2 | 1 | |||
| 2 | 2 |
2 Exercise: Ideal low-pass filter
\[H(u,v)= \begin{cases} 1& \text{if }D(u,v)\leq D_0\\ 0& \text{otherwise} \end{cases}\]
filt = ideal_lowpass_filterThis filter smooths the image, but it has a ringing effect because of a sudden drop in the filter surface.
3 Exercise: Gaussian low-pass filter
\[H(u,v)=e^{-D^2(u,v)/(2D_0^2)}\]
filt = gaussian_lowpass_filterThis filter also smooths the image, but it does not have the same ringing effect by the ideal low-pass filter because it has a smooth drop in the filter surface.
4 Exercise: Gaussian high-pass filter
\[H(u,v)=1-e^{-D^2(u,v)/(2D_0^2)}\]
filt = gaussian_highpass_filterThis filter highlights transitions between smooth areas, which in effect, sharpens the image.
5 Homework: Fourier transform
Solve this Fourier transform problem using Excel. Please submit your Excel file and the final answer in the table below (only real parts).
