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Image Enhancement Frequency Domain Filter

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Image Enhancement Frequency Domain Filter

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    1. Image Enhancement Frequency Domain Filter School of Electronics & Information Engineering Soochow University

    2. 2 Image Enhancement - 3 Frequency vs. spatial domain Different approaches

    3. 3 1-D Discrete Fourier Transform f(x), x=0,1,,M-1 . discrete function F(u), u=0,1,,M-1. DFT of f(x)

    4. 4 2-D DFT 2-D: x-axis then y-axis

    5. 5 Complex Quantities to Real Quantities Useful representation

    6. 6 Some notes about 2-D Fourier transform Frequency axis

    7. 7 DFT: example

    8. 8 Fourier Transform FTTransform.mFTTransform.m

    9. 9 Fourier Transform FTTransform.mFTTransform.m

    10. 10 Fourier Transform FTTransform.mFTTransform.m

    11. 11 Properties in the frequency domain Fourier transform works globally No direct relationship between a specific components in an image and frequencies Intuition about frequency Frequency content Rate of change of gray levels in an image

    12. 12

    13. 13 Image Enhancement - 3 Enhancement in frequency domain in principle is straightforward. However, it makes more sense to filter in the spatial domain using small filter masks. The relationship between frequency and spatial domain is the convolution theorem Select H(u,v) so that the desired image g(x,y) exhibits some highlighted features of f(x,y)

    14. 14 Image Enhancement - 3

    15. 15 Image Enhancement - 3

    16. 16 Image Enhancement - 3 Lowpass filters Ideal Lowpass filters Butterworth Lowpass filters Gaussian Lowpass filters Highpass filters Homomorphic filters

    17. 17 Image Enhancement - 3 Ideal filters : D(u,v) : distance from point (u,v) to the original D0 : cutoff frequency Ideal filter is nonphysical Radially symmetric about the original Special case : notch filter Power ratio of enhanced and original image If the image in question is of size MxN, its transform is of the same size. The center of the frequency rectangle is at (u,v) = (M/2, N/2) due to the fact that the transform has been centered. In this case, the distance from any point (u,v) to the center (original) of the Fourier transform is given by D(u,v) = [(u-M/2)2+(v-N/2)2]1/2. It is the Euclidean distance.If the image in question is of size MxN, its transform is of the same size. The center of the frequency rectangle is at (u,v) = (M/2, N/2) due to the fact that the transform has been centered. In this case, the distance from any point (u,v) to the center (original) of the Fourier transform is given by D(u,v) = [(u-M/2)2+(v-N/2)2]1/2. It is the Euclidean distance.

    18. 18 Image Enhancement - 3

    19. 19 Image Enhancement - 3

    20. 20 Image Enhancement - 3

    21. 21 Image Enhancement - 3

    22. 22 Image Enhancement - 3

    23. 23 Image Enhancement - 3 Butterworth lowpass filters A Butterworth filter of order 1 has no ringing. Ringing generally is imperceptible in filters of order 2, but can become a significant factor in filters of higher order.A Butterworth filter of order 1 has no ringing. Ringing generally is imperceptible in filters of order 2, but can become a significant factor in filters of higher order.

    24. 24 Image Enhancement - 3 A Butterworth filter of order 2 with cutoff frequencies at radii of 5, 15, 30, 80, and 230.A Butterworth filter of order 2 with cutoff frequencies at radii of 5, 15, 30, 80, and 230.

    25. 25 Image Enhancement - 3 A Butterworth filter of order 20 already exhibits the characteristics of the ILPF. A Butterworth filter of order 20 already exhibits the characteristics of the ILPF.

    26. 26 Image Enhancement - 3 Guassian lowpass filters No ringing for all orders. Does not achieve as much smoothing as the Butterworth filter of order 2. This is an important characteristic in practice, especially in situations where any type of artifact is not acceptable.No ringing for all orders. Does not achieve as much smoothing as the Butterworth filter of order 2. This is an important characteristic in practice, especially in situations where any type of artifact is not acceptable.

    27. 27 Image Enhancement - 3 No ringing for all orders. Does not achieve as much smoothing as the Butterworth filter of order 2. This is an important characteristic in practice, especially in situations where any type of artifact is not acceptable.No ringing for all orders. Does not achieve as much smoothing as the Butterworth filter of order 2. This is an important characteristic in practice, especially in situations where any type of artifact is not acceptable.

    28. 28 Image Enhancement - 3 Ideal filters : Butterworth highpass : Gaussian lowpass :

    29. 29 Image Enhancement - 3

    30. 30 Spatial-domain HPF

    31. 31 Ideal high-pass filters

    32. 32 Butterworth high-pass filters

    33. 33 Gaussian high-pass filters

    34. 34 Laplacian frequency-domain filters Spatial-domain Laplacian (2nd derivative) Fourier transform

    35. 35 Laplacian frequency-domain filters

    36. 36

    37. 37

    38. 38 Image Enhancement - 3 A simple image model: illuminationreflection model f(x,y) : the intensity is called gray level for monochrome image f(x,y)=i(x,y)*r(x,y) 0<i(x,y)<inf, the illumination 0<r(x,y)<1, the reflection

    39. 39 Image Enhancement - 3 The illumination component Slow spatial variations Low frequency The reflectance component Vary abruptly, particularly at the junctions of dissimilar objects High frequency Homomorphic filters Effect low and high frequency differently Compress the low frequency dynamic range Enhance the contrast in high frequency The illumination component of an image generally is characterized by slow spatial variations, while the reflectance component tends to vary abruptly, particularly at the junctions of dissimilar objects. The net results of homomophic filtering is simultaneous dynamic range compression and contrast enhancement.The illumination component of an image generally is characterized by slow spatial variations, while the reflectance component tends to vary abruptly, particularly at the junctions of dissimilar objects. The net results of homomophic filtering is simultaneous dynamic range compression and contrast enhancement.

    40. 40 Image Enhancement - 3

    41. 41 Image Enhancement - 3 f(x,y)=i(x,y)*r(x,y) z(x,y)=ln f(x,y) = ln i(x,y) + ln r(x,y) F{z(x,y)} = F{ln i(x,y)} + F{ln r(x,y)} S(u,v) = H(u,v) I(u,v) + H(u,v) R(u,v) s(x,y) = i(x,y) + r(x,y) g(x,y) = exp[s(x,y)] = exp[i(x,y)]exp[r(x,y)]

    42. 42 Image Enhancement - 3 The paramenters gamma_L and gamma_H are chosen so that gamma_L<1 and gamma_H>1. The curve shape can be approximated using the basic form of any of the ideal highpass filters. The paramenters gamma_L and gamma_H are chosen so that gamma_L<1 and gamma_H>1. The curve shape can be approximated using the basic form of any of the ideal highpass filters.

    43. 43 Image Enhancement - 3 Gamma_L = 0.5, Gamma_H =2.0. A reduction of dynamic range in the brightness, together with an increase in contrast, brought out the details of objects inside the shelter and balanced the gray levels of the outside wall. The enhanced image is also sharper.Gamma_L = 0.5, Gamma_H =2.0. A reduction of dynamic range in the brightness, together with an increase in contrast, brought out the details of objects inside the shelter and balanced the gray levels of the outside wall. The enhanced image is also sharper.

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