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Basic ideas of Image Transforms are derived from those showed earlier

Basic ideas of Image Transforms are derived from those showed earlier. Image Transforms. Fast Fourier 2-D Discrete Fourier Transform Fast Cosine 2-D Discrete Cosine Transform Radon Transform Slant Walsh, Hadamard, Paley, Karczmarz Haar Chrestenson Reed-Muller.

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Basic ideas of Image Transforms are derived from those showed earlier

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  1. Basic ideas of Image Transforms are derived from those showed earlier

  2. Image Transforms • Fast Fourier • 2-D Discrete Fourier Transform • Fast Cosine • 2-D Discrete Cosine Transform • Radon Transform • Slant • Walsh, Hadamard, Paley, Karczmarz • Haar • Chrestenson • Reed-Muller

  3. Methods for Digital Image Processing

  4. Spatial FrequencyorFourier Transform Fourier face in Fourier Transform Domain Jean Baptiste Joseph Fourier

  5. Examples of Fourier 2D Image Transform

  6. Fourier 2D Image Transform

  7. Another formula for Two-Dimensional Fourier Image is function of x and y A cos(x2i/N) B cos(y2j/M) fx = u = i/N, fy = v =j/M Lines in the figure correspond to real value 1 Now we need two cosinusoids for each point, one for x and one for y Now we have waves in two directions and they have frequencies and amplitudes

  8. Fourier Transform of a spot Original image Fourier Transform

  9. Transform Results image transform spectrum

  10. Two Dimensional Fast Fourier in Matlab

  11. Filtering in Frequency Domain … will be covered in a separate lecture on spectral approaches…..

  12. H(u,v) for various values of u and v • These are standard trivial functions to compose the image from

  13. < < image ..and its spectrum

  14. Image and its spectrum

  15. Image and its spectrum

  16. Image and its spectrum

  17. Convolution Theorem Let g(u,v) be the kernel Let h(u,v) be the image G(k,l) = DFT[g(u,v)] H(k,l) = DFT[h(u,v)] Then This is a very important result where means multiplication and means convolution. This means that an image can be filtered in the Spatial Domain or the Frequency Domain.

  18. Convolution Theorem Let g(u,v) be the kernel Let h(u,v) be the image G(k,l) = DFT[g(u,v)] H(k,l) = DFT[h(u,v)] Then Instead of doing convolution in spatial domain we can do multiplication In frequency domain Multiplication in spectral domain Convolution in spatial domain where means multiplication and means convolution.

  19. v Image u Spectrum Noise and its spectrum Noise filtering

  20. Spectrum Image v u

  21. Image x(u,v) v u Spectrum log(X(k,l)) l k

  22. Image x(u,v) v u Spectrum log(X(k,l)) l k Image of cow with noise

  23. white noise white noise spectrum kernel spectrum (low pass filter) red noise red noise spectrum

  24. Filtering is done in spectral domain. Can be very complicated

  25. Discrete Cosine Transform (DCT) • Used in JPEG and MPEG • Another Frequency Transform, with Different Set of Basis Functions

  26. Discrete Cosine Transform in Matlab trucks Two-dimensional Discrete Cosine Transform Two dimensional spectrum of tracks. Nearly all information in left top corner absolute

  27. “Statistical” Filters • Median Filter also eliminates noise • preserves edges better than blurring • Sorts values in a region and finds the median • region size and shape • how define the median for color values?

  28. “Statistical” Filters Continued • Minimum Filter (Thinning) • Maximum Filter (Growing) • “Pixellate” Functions Now we can do this quickly in spectral domain

  29. thinning growing • Thinning • Growing

  30. Pixellate Examples Original image After pixellate Noise added

  31. DCT used in compression and recognition 1 2 3 4 5 1 2 3 4 5 Can be used for face recognition, tell my story from Japan. Fringe Pattern DCT Coefficients Zonal Mask DCT (1,1) (1,2) (2,1) (2,2) . . . Artificial Neural Network Feature Vector

  32. Noise Removal Transforms for Noise Removal Image with Noise Transform been removed Image reconstructed as the noise has been removed

  33. Image Segmentation Recall: Edge Detection Gradient Mask -1 -1 -2 0 -1 1 f(x,y) fe(x,y) 0 0 0 2 0 -2 0 2 1 1 1 -1 Now we do this in spectral domain!!

  34. Image Moments 2-D continuous function f(x,y), the moment of order (p+q) is: Moments were found by convolutions Central moment of order (p+q) is:

  35. Image Moments (contd.) Normalized central moment of order (p+q) is: convolutions are now done in spectral domain A set of seven invariant moments can be derived from gpq Now we do this in spectral domain!!

  36. Image Textures Grass Sand Brick wall Now we do texture analysis like this in spectral domain!! The USC-SIPI Image Database http://sipi.usc.edu/

  37. Problems • There is a lot of Fourier and Cosine Transform software on the web, find one and apply it to remove some kind of noise from robot images from FAB building. • Read about Walsh transform and think what kind of advantages it may have over Fourier • Read about Haar and Reed-Muller transform and implement them. Experiment

  38. Sources • Howard Schultz, Umass • Herculano De Biasi • Shreekanth Mandayam • ECE Department, Rowan University • http://engineering.rowan.edu/~shreek/fall01/dip/ http://engineering.rowan.edu/~shreek/fall01/dip/lab4.html

  39. Image Compression Please visit the website http://www.cs.sfu.ca/CourseCentral/365/li/material/notes/Chap4/Chap4.html

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