1. Frequency domain Image Processing Comp344 Tutorial Kai Zhang

2. Outline Complex Number Multiplication Histogram equalizarion How to understand digital frequency High Pass Filters

3. Question: given a FFT transform F(u), is the following true? Proof (exercise)

4. Histogram Equalization Problem Given one random variables, x, with known probability distribution p x (x). Another variable has the relation y = T(x) What is the pdf of random variable y, p y (y) ? Basic theorem in statistics: Let x = T -1 (y); Then we have: p y (y) = p x (T -1 (y))|dT -1 (y)/dy|.

5. Example Suppose x is uniformly distributed as Let y = 2x, T(x) = 2x Then we have: The domain of the new pdf can be obtained by 0<0.5y<1, i.e., 0<y<2.

6. Example(histeq) Now, suppose the transform function is Then what will be the resultant pdf for y? First, note that Then we have

7. Digital frequency

8. Signal 1 [1 1 -1 -1 1 1 -1 -1 …..] 1024 points Signal 2 [1 -1 1 -1 1 -1…..] 1024 points

9. Observations Given a signal of length M The highest possible frequency is the signal that ``vibrates’’ M/2 times. So such signal has the highest digital frequency M/2 This is why for a signal with length M, the frequency domain has two symmetric part each with upper bound M/2. In case the signal is associated with its own time stamp, the the frequency component by FFT can also be transferred properly to reflect the actual frequency in iterms of Hz.

10. High Parss Filters Examples High pass filter: H_hp(u,v) = 1 – H_lp(u,v) Ideal high pass filter Bartworth high pass filter Gaussian high pass filter