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Chapter 4 Image Enhancement in the Frequency Domain

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

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

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Centered Fourier Spectrum It can be shown that:

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Example SEM Image

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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)

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Fourier Domain Filtering

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Some Basic Filters Notch filter:

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Lowpass and Highpass Filters

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

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Gaussian Filters Difference of Gaussians (DoG)

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Illustration

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Smoothing Filters Ideal lowpass filters Butterworth lowpass filters Gaussian lowpass filters

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Ideal Lowpass Filters

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Example

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Ringing Effect

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Butterworth Lowpass Filters Definition:

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Example

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Ringing Effect

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Gaussian Lowpass Filters Definition:

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Example

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More example

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

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Relationship between Lowpass and Highpass Filters

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Spatial Domain Representation

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Ideal Highpass Example

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Butterworth Highpass Example

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Gaussian Highpass Example

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

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Illustration

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

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Homomorphic Filtering

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

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