Chapter Four Image Enhancement in the Frequency Domain

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

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

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

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

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

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

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

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

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)

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

Fourier Series (cont’d) α1α1 α2α2 α3α3 Illustration

1-D Discrete Fourier Transform (DFT)

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.

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.

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

1-D Discrete Fourier Transform (DFT)

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

2-D DFT

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

DC component

Frequency domain basics

Filtering in The Frequency Domain

Some basic filters: 1- Notch filter:

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

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