Image Enhancement in the Frequency Domain

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Basic steps for filtering in the frequency domain

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Basics of filtering in the frequency domain

1. multiply the input image by (-1)x+y to center the transform to u = M/2 and v = N/2 (if M and N are even numbers, then the shifted coordinates will be integers)

2. computer F(u,v), the DFT of the image from (1)3. multiply F(u,v) by a filter function H(u,v)4. compute the inverse DFT of the result in (3)5. obtain the real part of the result in (4)6. multiply the result in (5) by (-1)x+y to cancel the

multiplication of the input image.

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Notch filter

otherwise 1

N/2 (M/2, v)(u, if 0),(

) vuH

• this filter is to force the F(0,0) which is the average value of an image (dc component of the spectrum)• the output has prominent edges• in reality the average of the displayed image can’t be zero as it needs to have negative gray levels. the output image needs to scale the gray level

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Low pass filter

high pass filter

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Add the ½ of filter height to F(0,0) of the high pass filter

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Correspondence between filter in spatial and frequency domains

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Smoothing Frequency-domain filters: Ideal Lowpass filter

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image power circles

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Result of ILPF

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Example

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Butterworth Lowpass Filter: BLPF

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Example

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Spatial representation of BLPFs

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Gaussian Lowpass Filter: GLPF

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Example

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Example

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Example

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Example

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Sharpening Frequency Domain Filter: Ideal highpass filter

Butterworth highpass filter

Gaussian highpass filter

0

0

Dv)D(u, if 1

Dv)D(u, if 0),( vuH

nvuDvuH 2

0 ),(D1

1),(

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2 2/),(1),( DvuDevuH

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Spatial representation of Ideal, Butterworth and Gaussian highpass filters

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Example: result of IHPF

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Example: result of BHPF

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Example: result of GHPF

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Laplacian in the Frequency domain

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Example: Laplacian filtered image

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Example: high-boost filter

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Examples

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Homomorphic Filter

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Result of Homomorphic filter

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