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)

α1

α2

α3

• Illustration

1-D Discrete Fourier Transform (DFT)

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

Filtering in The Frequency Domain

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