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Convolution

Basis beeldverwerking (8D040 ) d r. Andrea Fuster Prof.dr . Bart ter Haar Romeny dr. Anna Vilanova Prof.dr.ir . Marcel Breeuwer. Convolution. Contents. Spatial filtering Correlation Convolution Filters: Smoothing filters Sharpening filters Borders. Spatial filtering.

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Convolution

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  1. Basis beeldverwerking (8D040)dr. Andrea FusterProf.dr. Bart terHaarRomenydr. Anna VilanovaProf.dr.ir. Marcel Breeuwer Convolution

  2. Contents • Spatial filtering • Correlation • Convolution • Filters: • Smoothing filters • Sharpening filters • Borders Basis beeldverwerking 8D040

  3. Spatial filtering • Input image , use a filter to obtain processed image • Filter consists of • Neighbourhood (rectangular) • Mostly odd dimensions • Predefined operation • Create new pixel value in center of neighbourhood Basis beeldverwerking 8D040

  4. Spatial filtering Filter operation More compact notation - filter Basis beeldverwerking 8D040

  5. Intuition to filtering Basis beeldverwerking 8D040

  6. Move filter over image Basis beeldverwerking 8D040

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  11. Correlation • While moving the filter, at each position • Multiply values of overlapping locations • Sum all multiplication results Basis beeldverwerking 8D040

  12. Correlation vs. Convolution - Equivalent to first rotate the filter 180 degrees and correlate- Discrete Correlation 2D Discrete Convolution 2D Basis beeldverwerking 8D040

  13. Correlation vs. Convolution convolution correlation With convolution, first mirror the filter kernel with respect to its origin. With correlation, do not mirror the filter kernel with respect to its origin. Basis beeldverwerking 8D040

  14. Example See blackboard ☺ (or figure 3.30 Gonzalez and Woods) Basis beeldverwerking 8D040

  15. Convolution – 1D cont. case • Imagine a system with • input signal • transfer function • output signal • then Basis beeldverwerking 8D040

  16. input output Definition system transfer function Basis beeldverwerking 8D040

  17. Dirac delta function (unit impulse) • Definition • Constraint • Sifting property • Specifically for t=0 Basis beeldverwerking 8D040

  18. Convolution We saw this already in the discrete case Let Basis beeldverwerking 8D040

  19. Properties of convolution • Commutative • Associative • Distributive Basis beeldverwerking 8D040

  20. Convolution is commutative • Proof • Let • Q.E.D. Basis beeldverwerking 8D040

  21. Convolution is associative - 1 Proof Basis beeldverwerking 8D040

  22. Convolution is associative - 2 Basis beeldverwerking 8D040

  23. Convolution is associative - 3 • Let Basis beeldverwerking 8D040

  24. Convolution is associative - 4 Q.E.D. Basis beeldverwerking 8D040

  25. Convolution is distributive - 1 Proof Basis beeldverwerking 8D040

  26. Convolution is distributive - 2 Q.E.D. Basis beeldverwerking 8D040

  27. Discrete convolution 1D 2D Basis beeldverwerking 8D040

  28. Discrete convolution • Formulas take summation from to • Filters have a limited size, e.g., • 1D 2a + 1 • 2D (2a + 1, 2b + 1) Basis beeldverwerking 8D040

  29. Kernel Filter Kernels Basis beeldverwerking 8D040

  30. Filters Idea: correlate or convolve image with different filters in order to obtain different results, i.e., processed images Basis beeldverwerking 8D040

  31. Smoothing filters … NxN filter • Average intensities – result is blurred image, less details • Response: (z’s image intensities) Basis beeldverwerking 8D040

  32. Smoothing filters • Note that: • Sum of filter coefficients is 1 (normalized filter) • Correlation = convolution (symmetric filter) • Filter size effect? Basis beeldverwerking 8D040

  33. Smoothing filters - example Original 3x3 smoothing filter NxN filter (see figure 3.33 in Gonzalez and Woods!) Basis beeldverwerking 8D040

  34. Effect of normalized smoothing kernel normalized non- normalized Basis beeldverwerking 8D040

  35. Sharpening filters Enhance parts of the image where intensities change rapidly, such as edges Basic derivative filters Measure change of intensity in x or y direction Basis beeldverwerking 8D040

  36. Example Basis beeldverwerking 8D040

  37. Arbitrary angle derivative Given and Basis beeldverwerking 8D040

  38. Arbitrary angle derivative Basis beeldverwerking 8D040

  39. Prewitt gradient kernel • Derivativein one direction, smoothingin the perpendiculardirection Basis beeldverwerking 8D040

  40. Example Prewitt Basic derivative Basis beeldverwerking 8D040

  41. Sobel kernel Basis beeldverwerking 8D040

  42. Example (Thanks to Wikipedia☺)

  43. Derivative filters Note that coefficients in all of the previous filters sum to zero, i.e., zero response in area of constant intensity Also: gradient, Laplacian, … Basis beeldverwerking 8D040

  44. Borders Do you see any problems at image borders? Try position (0,0) Basis beeldverwerking 8D040

  45. Border problems • How to handle? • No border handling • Border is not filtered • Padding • Put values outside image border • Cyclic padding • Use values from the other side of the image Basis beeldverwerking 8D040

  46. Zero padding Basis beeldverwerking 8D040

  47. Cyclic padding Basis beeldverwerking 8D040

  48. Padding Remember: padding is artificial! The values chosen outside the border influence the outcome image Basis beeldverwerking 8D040

  49. End of part 2 Thanks and see you Wednesday 16! Basis beeldverwerking 8D040

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