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Chapter Four Image Enhancement in the Frequency Domain

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

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

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

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

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

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

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

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

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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)

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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

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Fourier Series (cont’d) α1α1 α2α2 α3α3 Illustration

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1-D Discrete Fourier Transform (DFT)

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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.

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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.

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

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1-D Discrete Fourier Transform (DFT)

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

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2-D DFT

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

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DC component

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Frequency domain basics

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Filtering in The Frequency Domain

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Some basic filters: 1- Notch filter:

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

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

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Fourier Transform.

Fourier Transform.

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