1. Chapter Four Image Enhancement in the Frequency Domain

2. Mathematical Background: Complex Numbers A complex number x has the form: a: real part, b: imaginary part Addition Multiplication

3. Mathematical Background: Complex Numbers (cont’d) Magnitude-Phase (i.e.,vector) representation Magnitude: Phase : φ

4. Mathematical Background: Complex Numbers (cont’d) Multiplication using magnitude-phase representation Complex conjugate Properties

5. Mathematical Background: Complex Numbers (cont’d) Euler’s formula Properties j

6. Mathematical Background: Sine and Cosine Functions Periodic functions General form of sine and cosine functions:

7. Mathematical Background: Sine and Cosine Functions Special case: A=1, b=0, α=1 π

8. Mathematical Background: Sine and Cosine Functions (cont’d) Shifting or translating the sine function by a const b

9. Mathematical Background: Sine and Cosine Functions (cont’d) Changing the amplitude A

10. Mathematical Background: Sine and Cosine Functions (cont’d) Changing the period T=2π/|α| e.g., y=cos(αt) period 2π/4=π/2 shorter period higher frequency (i.e., oscillates faster) α =4 Frequency is defined as f=1/T Different notation: sin(αt)=sin(2πt/T)=sin(2πft)

11. Any periodic function can be represented by the sum of sines/cosines of different frequencies, multiplied by a different coefficient (Fourier series). Non-periodic functions can also be represented as the integral of sines/cosines multiplied by weighing function (Fourier transform). Important characterestic: a function can be reconstructed completely via inverse transform with no loss of information. Fourier Series Theorem

12. Fourier Series (cont’d) α1α1 α2α2 α3α3 Illustration

13. 1-D Discrete Fourier Transform (DFT)

14. The domain (values of u) over which F(u) range is called the frequency domain Each of th M terms of F(u) is called frequency compnent of the transform.

15. 1-D Discrete Fourier Transform (DFT) |F(u)| is called magnitude or spectrum of the DFT. Φ(u) is called the phase angle of the spectrum. In terms of image enhancement we are interested in the properties of the spectrum.

16. 1-D DFT: Example Example: Let f (x) = {1, − 1, 2, 3}. (Note that M=4.)

17. 1-D Discrete Fourier Transform (DFT)

18. 2-D DFT The Two-Dimensional Fourier Transform and its Inverse

19. 2-D DFT

20. Conjugate symmetry The Fourier transform of a real function is conjugate symmetric This means Which says that the spectrum of the DFT is symmetric.

21. DC component

22. Frequency domain basics

23. Filtering in The Frequency Domain

25. Some basic filters: 1- Notch filter:

26. 2- Lowpass filter: Attenuates a high frequencies, while passing a low frequencies (average gray level). Filtering in The Frequency Domain

27. 3- Highpass filter: Attenuates a low frequencies, while passing a high frequencies (details). Filtering in The Frequency Domain