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Image Processing Image Enhancement in the Frequency Domain Part III

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Some Basic Filters Low frequencies in the Fourier transform are responsible for the general gray-level appearance of an image over smooth areas. High frequencies are responsible for detail such as edges and noise. Lowpass Filter : Attenuates high frequency while “passing” low frequency. Less sharpen the details because the high frequencies are attenuated highpass Filter : Attenuates low frequency while “passing” high frequency. Less gray level variation in smooth areas and emphasized transitional.

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Some Basic Filters

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Correspondence between Filtering in the spatial and Frequency domains Relationship It is more computationally efficient to do the filtering in the frequency domain Filtering is more intuitive in the frequency domain It makes more sense to filter in the spatial domain using small filter mask filter in the frequency domain filter in the spatial domain DFT IDFT guide for small filter mask

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Gaussian Filters Example( Filters based on Gaussian functions ) both functions are real two functions behave reciprocally with respect to one another

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Gaussian Filters

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Ideal low pass The transfer function for the ideal low pass filter can be given as: where D 0 is a specified nonnegative quantity, and D(u,v) is the distance from point (u,v) to the origin of the frequency rectangle.

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Ideal Low pass Filters The center of frequency rectangle is (u,v) = (M/2,N/2) In this case, the distance from any point (u,v) to the center (origin) D(u,v) of the Fourier transform is given by The lowpass filters considered here are radially symmetric about the origin

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Ideal Low pass Filters The transition point is called the cutoff frequency Standard cutoff frequency for comparing filters α percent of the power

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Ideal Low pass Filters The severe blurring in (b) is a clear indication that most of the sharp detail information in the picture is contained in the 8% power removed by the filter

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Ideal Low pass Filters (c)~(e) “ringing” behavior The ringing behavior is a characteristic of ideal filters

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Ideal Low pass Filters

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Butterworth Low pass Filters Butterworth low pass filter (BLPF) of order n, and with cutoff frequency at a distance D 0 from the origin A cutoff frequency is defined at points for which H(u,v) is down to a certain fraction of its maximum value In this case, H(u,v) = 0.5 when D(u,v) = D 0

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Butterworth Low pass Filters Example No ringing effect

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Gaussian Low pass Filters Gaussian lowpass filters (GLPFs) of two dimensions are given by σ is a measure of the spread of the Gaussian curve By letting σ = D 0 where D 0 is the cutoff frequency When D(u,v) = D 0, the filter is down to 0.607 of its maximum value

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Sharpening Freq. Domain Filters Image can be blurred by attenuating the high-frequency components of its Fourier transform Edges and other abrupt changes in gray levels are associated with high-frequency components The transfer function of the highpass filters can be obtained using the relation Where H lp (u, v) is the corresponding lowpass filter

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Sharpening Freq. Domain Filters ideal highpass filter Butterworth highpass filter Gaussian highpass filter

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Ideal Highpass Filters Ideal highpass filter :cutoff frequency

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Butterworth highpass filter Butterworth highpass filter

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Gaussian Highpass Filters Gaussian highpass filter (GHPF) The results obtained are smoother than with the previous two Filters. Smaller objects and thin bars is cleaner with the GHPF

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The Laplacian in Freq. Domain It can be shown that It follow that Which result

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The Laplacian in Freq. Domain Laplacian can be implemented in the frequency domain by using the filter The center of the filter function also needs to be shifted

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The Laplacian in Freq. Domain Laplacian in the frequency domain 2-D image of Laplacian in the frequency domain Inverse DFT of Laplacian in the frequency domain Zoomed section of the image on the left compared to spatial filter

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The Laplacian in Freq. Domain

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Assignments Read the following sections from ch.4 1, 2, 3,4

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