Frequency domain ImageProcessingComp344TutorialKai Zhang
Frequency domain Image Processing Comp344 Tutorial Kai Zhang
OutlineComplexNumber MultiplicationHistogram equalizarionHow to understand digital frequencyHighPassFilters
Outline ◼ Complex Number Multiplication ◼ Histogram equalizarion ◼ How to understand digital frequency ◼ High Pass Filters
Question: given a FFT transform F(u), is thefollowing true?I F P=|F2Proof (exercise)
◼ Question: given a FFT transform F(u), is the following true? ◼ Proof (exercise) 2 2 | F | = F
Histogram EqualizationProblemGiven one random variables, x, with knownprobability distribution px(x) Another variable has the relation y = T(x) What is the pdf of random variable y, py(y) ?Basic theorem in statistics:Let x = T-1(y);Then we have: py(y) = px(T-1(y)ldT-1(y)/dyl
Histogram Equalization ◼ Problem ◼ Given one random variables, x, with known probability distribution px (x). ◼ Another variable has the relation y = T(x) ◼ What is the pdf of random variable y, py (y) ? ◼ Basic theorem in statistics: ◼ Let x = T-1 (y); ◼ Then we have: py (y) = px (T-1 (y))|dT-1 (y)/dy|
ExampleSuppose x is uniformly distributed as p,(x) = 1,0≤ x ≤1Let y = 2x, T(x) = 2xdxThen we have: p,(y) = p,(T-'(y)dy12= p(T-(y)2The domain of the new pdf can be obtained by 0<0.5y<1, i.e., 0<y<2
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. px (x) =1,0 x 1 2 2 1 1 1 ( ) ( ( )) ( ) ( ( )) 1 1 1 = = = = − − − dy d y dy dT y p T y dy dx p y p T y x y x