 # Chapter 4 Image Enhancement in the Frequency Domain.

## Presentation on theme: "Chapter 4 Image Enhancement in the Frequency Domain."— Presentation transcript:

Chapter 4 Image Enhancement in the Frequency Domain

Fourier Transform 1-D Fourier Transform 1-D Discrete Fourier Transform (DFT) Magnitude Phase Power spectrum

2D DFT Definition: if f(x,y) is real

Centered Fourier Spectrum It can be shown that:

Example SEM Image

Filtering in the Frequency Domain 1. Multiply the input image by (-1)^x+y to center the transform 2. Compute F(u,v), the DFT of input 3. Multiply F(u,v) by a filter H(u,v) 4. Computer the inverse DFT of 3 5. Obtain the real part of 4 6. Multiply the result in 5 by (-1)^(x+y)

Fourier Domain Filtering

Some Basic Filters Notch filter:

Lowpass and Highpass Filters

Convolution Theorem Definition Theorem Need to define the discrete version of impulse function to prove these results.

Gaussian Filters Difference of Gaussians (DoG)

Illustration

Smoothing Filters Ideal lowpass filters Butterworth lowpass filters Gaussian lowpass filters

Ideal Lowpass Filters

Example

Ringing Effect

Butterworth Lowpass Filters Definition:

Example

Ringing Effect

Gaussian Lowpass Filters Definition:

Example

More example

Sharpening Filters High-pass filters In general, Ideal highpass filter Butterworth highpass filter: Gaussian highpass filters

Relationship between Lowpass and Highpass Filters

Spatial Domain Representation

Ideal Highpass Example

Butterworth Highpass Example

Gaussian Highpass Example

Laplacian in the Frequency Domain It can be shown that: Therefore,

Illustration

Other Filters Unsharp masking High-boost filtering High-frequency emphasis filtering

Homomorphic Filtering

DFT: Implementation Issues Rotation Periodicity and conjugate symmetry Separability Need for padding Circular convolution FFT