Introduction to Computer Vision

1. Lecture 05 Roger S. Gaborski Introduction to Computer Vision Roger S. Gaborski

2. Quiz Review • In Class Exercise • Correlation – Convolution • Filtering Roger S. Gaborski

3. Example • Histogram PDF  CDF Equalized Image • Filtering using Correlation continued

4. Histogram Equalization • Consider an image with the following gray level values: • Construct the pdf • Construct the cdf • Equalize the image using the cdf (not histeq)

5. Histogram Equalization • Consider an image with the following gray level values: • Construct the pdf 1/9 2/9 3/9 4/9 5/9 .1 .2 .3 .4 .5

6. pdf cdf 1/9 2/9 3/9 4/9 5/9 .1 .2 .3 .4 .5 Histogram Equalization Look Up Table 1/9 2/9 3/9 4/9 5/9 6/9 7/9 8/9 1 cdf probability .1 .2 .3 .4 .5 Gray level value Gray level value

7. Histogram Equalization • Consider an image with the following gray level values: • Construct the pdf • Construct the cdf • Equalize the image using the cdf (not histeq)

8. Spatial Filtering • Neighborhood processing • Define center point (x, y) • Perform operations involving only pixels in the neighborhood • Result of operation is response to process at that point • Moving the pixel results in a new neighborhood • Repeat process for every point in the image Roger S. Gaborski

9. Linear and Nonlinear Spatial Filtering • Linear operation • Multiply each pixel in the neighborhood by the corresponding coefficient and sum the results to get the response for each point (x, y) • If neighborhood is m x n , then mn coefficients are required • Coefficients are arranged in a matrix, called • filter / filter mask / kernel / template • Mask sizes are typically odd numbers (3x3, 5x5, etc.) Roger S. Gaborski

10. Image origin y Kernel coefficients mask x Image region under mask Roger S. Gaborski

11. Correlation and Convolution • Correlation • Place mask w on the image array f as previously described • Convolution • First rotate mask w by 180 degrees • Place rotated mask on image as described previously • Convolution = 180 degree rotation + correlation Roger S. Gaborski

12. Example: 1D Correlation • Assume w and f are one dimensional • Origin of f is its left most point • Place w so that its right most point coincides with the origin of f • Pad f with 0s so that there are corresponding f points for each w point (also pad end with 0s) • Multiply corresponding points and sum • In this case (example on next page) result is 0 • Move w to the right one value, repeat process • Continue process for whole length of f Roger S. Gaborski

13. Reminder • ‘full’ is the result we obtain from the operations on the previous slide. If instead of aligning the left most element of f with the right most element of w we aligned the center element of w with the left most value of f we would obtain the ‘same’ result, same indicating the result is the same length of the original w Roger S. Gaborski

14. Chapter 3 www.prenhall.com/gonzalezwoodseddins Roger S. Gaborski

15. ‘Full’ correlation Roger S. Gaborski

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18. ‘Same’ correlation etc. Roger S. Gaborski

19. Example - Convolution • Convolution is the same procedure, but the filter is first rotated 180 degrees. • Convolution = 180 degree rotation + correlation • If the filter is symmetric, correlation and convolution results are the same Roger S. Gaborski

20. Chapter 3 www.prenhall.com/gonzalezwoodseddins This can be simply extend to images Roger S. Gaborski

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26. Linear Filtering in MATLAB g = imfilter(f, w, filtering mode, boundary, size) • filters the imput image f with the filter mask w. • f is input image. It can be of any class (logical/numeric) and dimension. • g is output image • filter mode: - 'corr' : correlation, and default mode - 'conv' : convolution Roger S. Gaborski

27. Parameters • g = imfilter(f, w, filtering mode, boundary, size) Boundary options - X pad boundary with value X. Default X = 0. - 'symmetric' symmetric padding - 'replicate' replicate padding - 'circular' circular padding Size options - 'same' g is the same size of f (default mode) - 'full' g is full filtered by w, so size of g is increased Roger S. Gaborski

28. MATLAB function for filtering: imfilter • g = imfilter(f, w, ‘replicate’) • Correlation is the default filtering mode. • If filters are pre-rotated 180 degrees, can simply use default(corr) for convolution • If filter is symmetric, doesn’t matter Roger S. Gaborski

29. Chapter 3 www.prenhall.com/gonzalezwoodseddins Roger S. Gaborski

30. Example:Smoothing • w = ones(31); (31x31 filter) • % Normally the coefficients (w) are scaled to sum to one • % In this example only coefficients are not scaled by 312 • % Convolution should result in a blurred result • gd = imfilter(f, w); • % Default mode: correlation filtering • imshow(gd, [ ]); Roger S. Gaborski

31. Chapter 3 www.prenhall.com/gonzalezwoodseddins Input Default padding ‘replicate’ ‘symmetric’ ‘circular’ ‘replicate’, uint8 Roger S. Gaborski