1. Basis beeldverwerking (8D040) dr. Andrea Fuster dr. Anna Vilanova Prof
Basis beeldverwerking (8D040) dr. Andrea Fuster dr. Anna Vilanova Prof.dr.ir. Marcel Breeuwer
Convolution

2. Contents Spatial filtering Correlation Convolution Filters:
Smoothing filters
Sharpening filters
Borders
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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
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4. Spatial filtering Filter operation (3x3 filter)
More compact notation filter
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5. Intuition to filtering
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6. Move filter over image
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11. Correlation While moving the filter, at each position
Multiply values of overlapping locations
Sum all multiplication results
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12. Correlation vs. Convolution
Discrete Correlation 2D
Discrete Convolution 2D
- Equivalent to first rotate the filter 180 degrees and correlate-
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13. Example See blackboard ☺ (or figure 3.30 Gonzalez and Woods)
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14. Correlation vs. Convolution
Correlation and convolution give the same result if the filter used is symmetric!
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15. Convolution – 1D cont. case
Imagine a system with
input signal
transfer function
output signal
then
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16. system transfer function
Definition
input
output
system transfer function
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17. Dirac delta function (unit impulse)
Definition
Constraint
Sifting property
Specifically for t=0
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18. Convolution Let We saw this already in the discrete case
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19. Properties of convolution
Commutative
Associative
Distributive
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20. Convolution is commutative
Proof
Let
Q.E.D.
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21. Convolution is associative - 1
Proof
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22. Convolution is associative - 2
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23. Convolution is associative - 3
Let
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24. Convolution is associative - 4
Q.E.D.
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25. Convolution is distributive - 1
Proof
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26. Convolution is distributive - 2
Q.E.D.
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27. Discrete convolution 1D 2D
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28. Discrete convolution Formulas take summation from to
Filters have a limited size, e.g.,
1D a + 1
2D (2a + 1, 2b + 1)
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29. Filter Kernels
Kernel
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30. Filters
Idea: correlate or convolve image with different filters in order to obtain different results, i.e., processed images
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31. Smoothing filters
Average intensities – result is blurred image, less details
Response: (z’s image intensities)
… NxN filter
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32. Smoothing filters Note that:
Sum of filter coefficients is 1 (normalized filter)
Correlation = convolution (symmetric filter)
Filter size effect?
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33. Smoothing filters - example
Original x3 smoothing filter NxN filter
(see figure 3.33 in Gonzalez and Woods!)
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34. Effect of normalized smoothing kernel
non- normalized
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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
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36. Example
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37. Arbitrary angle derivative
Given and
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38. Arbitrary angle derivative
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39. Prewitt gradient kernel
Derivative in one direction, smoothing in the perpendicular direction
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40. Example Prewitt Basic derivative Basis beeldverwerking 8D040

41. Sobel kernel
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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, …
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44. Borders Do you see any problems at image borders? Try position (0,0)
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45. Border problems How to handle? No border handling Padding
Border is not filtered
Padding
Put values outside image border
Cyclic padding
Use values from the other side of the image
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46. Zero padding
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47. Cyclic padding
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48. Padding Remember: padding is artificial!
The values chosen outside the border influence the outcome image
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49. End of part 2 Thanks and see you Tuesday 21!
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