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© 2002-2003 by Yu Hen Hu 1 ECE533 Digital Image Processing Image Enhancement in Frequency Domain

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© 2002-2003 by Yu Hen Hu 2 ECE533 Digital Image Processing Image and Its Fourier Spectrum

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© 2002-2003 by Yu Hen Hu 3 ECE533 Digital Image Processing Filtering in Frequency Domain: Basic Steps Basic Steps 1. Multiply pixel f(x,y) of the input image by (-1) x+y. 2. Compute F(u,v), the DFT 3. G(u,v)=F(u,v)H(u,v) 4. g1(x,y)=F -1 {G(u,v)} 5. g(x,y) = g1(x,y)*(-1) x+y

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© 2002-2003 by Yu Hen Hu 4 ECE533 Digital Image Processing Notch Filter l The frequency response F(u,v) has a notch at origin (u = v = 0). l Effect: reduce mean value. l After post-processing where gray level is scaled, the mean value of the displayed image is no longer 0.

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© 2002-2003 by Yu Hen Hu 5 ECE533 Digital Image Processing Low-pass & High-pass Filtering

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© 2002-2003 by Yu Hen Hu 6 ECE533 Digital Image Processing Gaussian Filters l Fourier Transform pair of Gaussian function l Depicted in figures are low- pass and high-pass Gaussian filters, and their spatial response, as well as FIR masking filter approximation. l High pass Gaussian filter can be constructed from the difference of two Gaussian low pass filters.

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© 2002-2003 by Yu Hen Hu 7 ECE533 Digital Image Processing Gaussian Low Pass Filters D(u,v): distance from the origin of Fourier transform

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© 2002-2003 by Yu Hen Hu 8 ECE533 Digital Image Processing Ideal Low Pass Filters l The cut-off frequency D o determines % power are filtered out. l Image power as a function of distance from the origin of DFT (5, 15, 30, 80, 230)

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© 2002-2003 by Yu Hen Hu 9 ECE533 Digital Image Processing Effects of Ideal Low Pass Filters l Blurring can be modeled as the convolution of a high resolution (original) image with a low pass filter.

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© 2002-2003 by Yu Hen Hu 10 ECE533 Digital Image Processing Ringing and Blurring

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© 2002-2003 by Yu Hen Hu 11 ECE533 Digital Image Processing Butterworth Low Pass Filters

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© 2002-2003 by Yu Hen Hu 12 ECE533 Digital Image Processing l Ideal high pass filter l Butterworth high pass filter l Gaussian high pass filter High Pass Filters

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© 2002-2003 by Yu Hen Hu 13 ECE533 Digital Image Processing Applications of HPFs l Ideal HPF »D o = 15, 30, 80 l Butterworth HPF »n = 2, »D o = 15, 30, 80 l Gaussian HPF »D o = 15, 30, 80

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© 2002-2003 by Yu Hen Hu 14 ECE533 Digital Image Processing Laplacian HPF l 3D plots of the Laplacian operator, l its 2D images, l spatial domain response with center magnified, and l Compared to the FIR mask approximation

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Filters in the Frequency Domain. Image Smoothing Using Frequency Domain Filters: ◦ Ideal Lowpass Filters ◦ Butterworth Lowpass Filters ◦ Gaussian Lowpass.

Filters in the Frequency Domain. Image Smoothing Using Frequency Domain Filters: ◦ Ideal Lowpass Filters ◦ Butterworth Lowpass Filters ◦ Gaussian Lowpass.

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