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Image Processing

Image Processing. Image Enhancement in the Frequency Domain Part III. 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.

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Image Processing

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  1. Image Processing Image Enhancement in the Frequency Domain Part III

  2. 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.

  3. Some Basic Filters

  4. filter in the frequency domain guide for small filter mask DFT IDFT filter in the spatial domain 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

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

  6. Gaussian Filters

  7. Ideal low pass • The transfer function for the ideal low pass filter can be given as: • where D0 is a specified nonnegative quantity, and D(u,v) is the distance from point (u,v) to the origin of the frequency rectangle.

  8. 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

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

  10. 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

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

  12. Ideal Low pass Filters

  13. Butterworth Low pass Filters • Butterworth low pass filter (BLPF) of order n, and with cutoff frequency at a distance D0 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) = D0

  14. Butterworth Low pass Filters • Example No ringing effect

  15. 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 = D0 where D0 is the cutoff frequency • When D(u,v) = D0, the filter is down to 0.607 of its maximum value

  16. 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 Hlp(u, v) is the corresponding lowpass filter

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

  18. Ideal Highpass Filters • Ideal highpass filter :cutoff frequency

  19. Butterworth highpass filter • Butterworth highpass filter

  20. 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

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

  22. 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

  23. The Laplacian in Freq. Domain 2-D image of Laplacian in the frequency domain 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

  24. The Laplacian in Freq. Domain

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

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