Foundations of Computer Vision

1. Foundationsof Computer Vision Lectures 09 Roger S. Gaborski

2. Exam 1 – Raw Scores

3. Exam 1- Percentage Scores

4. Assignment • How do you calculate the determinant of a 3 x 3 matrix? • What are eigenvalues and how are they calculated? • What are eigenvectors and how are they calculated?

5. Features • Match corresponding feature points in images • In video, two temporally adjacent frames • In still images, two different images of the same scene • Or two different images of similar scene

6. Issues • Lighting variations • View point • Scale • Rotation

7. NEED TO FIND CORRESPONDENCE BETWEEN FEATURE POINTS IN TWO DIFFERENT IMAGES • Cannot match individual pixels • Need to use a window containing many pixels (5x5, 7x7, 21x21, etc)

8. Matching on a continuum like texture and edges not very robust • Many edges (and parts of edges) will match • At the very least: • Need to find interest points • Extract patches around interest points • Match patches

9. Feature Points / Correspondence • Points should be extracted consistently over different views • Points should be invariant to scaling, rotation, changes in illumination • Information in the neighborhood of the point should be unique so that it can be matched

10. Select Window RegionMatch Region in Second Image Calculate difference between the two patches WindowMatching_ACV.m

11. Convert to Grayscale

12. Randomly Select Patch patch = I(80:110,200:230); For Demonstration Use Only Strip Containing Patch

13. Error=Sum of Absolute Difference Between Patch and Strip

14. This simple approach works because the patch was extracted from the image • We need a method that will work with ‘similar’ patches

15. Normalized Cross Correlation (Refer to equation on page p313)Also MATLAB docs • w is template • w is average value of elements in template • f is the image • f is the average of the image where f and w overlap • Denominator normalizes resulting in an output range of -1, +1 • High value for absolute value of output is a good match

16. MATLAB Cross Correlation Function g = normxcorr2(template, f)

17. Find Max Value in |g| d = abs(g); [ypeak, xpeak] = find(d == max(d(:))); %Adjust location by size of template %since g is larger than original image %by size of template ypeak = ypeak-(size(patch,1)-1)/2; xpeak = xpeak-(size(patch,2)-1)/2; fprintf('\n Center of Patch: ypeak is %d and xpeak is %d \n\n\n', ypeak, xpeak); figure, imshow(Igray) hold on plot(xpeak, ypeak, 'ro')

19. Red – strongest response Green – second strongest response (wheel rotation slightly different)

20. One Possible Approach • Read in Purple Flower Image and Select a 31x31 patch • Extract image strip that contains patch • Calculate Error=Absolute Difference Between Patch and Strip and plot • Repeat using Normalized Cross Correlation on whole image • Plot max location on image • CONTINUED NEXT SLIDE

21. One Possible Approach, continued • Repeat Normalized Cross Correlation using several different templates – choose templates from uniform areas and compare with templates containing strong edges, try different size templates • What are the characteristics of ‘good’ templates

22. Selection of Patches • We don’t want to just randomly select patches from an image • We want to select patches that have high information content • We want to do this automatically, not manually chosen

23. But How Do We Select Points? • Junctions or Corners • Stable over changes in viewpoint

24. Feature Points?

25. Subtract First Window from Second Window Sightly Offset from First

26. Moravec’s Corner Detector • Overview: • Select window size • Shift window over image region • If window over uniform region, shifts in all directions will result in small changes • If window over edge, shifts along edge will results in small changes, but shifts across edge will result in large changes • along edge – no change • Perpendicular to edge – large change • If window over corner, than shifts in all directions will result will result in large changes • Detect corner by finding regions that have large changes in all directions

27. Subtract First Window from Second Window Window moved vertically, no change Window moved horizontally, no change Window moved in either direction, large change

28. Corner Detection: Basic Idea We should easily recognize the point by looking through a small window Shifting a window in anydirection should give a large change in intensity “flat” region:no change in all directions “edge”:no change along the edge direction “corner”:significant change in all directions Roger S. Gaborski Source: A. Efros

29. Wikipedia: Corner Detection-No shortage of corner detectors • Moravec • Harris • Laplacian of gaussians, Difference of Gaussian and Determinant of the Hessian • SIFT • SUSAN Corner Detector • Trajkovic – Hedley Corner Detector Roger S. Gaborski

30. Interest Point Detection • http://www.youtube.com/watch?v=_qgKQGsuKeQ

31. Corner Detection: Weighted Sum of Differences Window function Shifted intensity Intensity Window function w(u,v) = Area of patch: (u,v) or 1 in window, 0 outside Gaussian Change in appearance for the shift [x,y]: E(x,y) = ΣΣ w(u,v) (I(u+x, v+y) – I(u,v))2 u v Roger S. Gaborski Source: Wikipedia

32. Corner Detection: Mathematics Change in appearance for the shift [x,y]: I(u,v) E(x,y) E(3,2) E(0,0) Roger S. Gaborski Source: R. Szeliski

33. Corner Detection Change in appearance for the shift [x,y]: • E(x,y) = ΣΣ w(u,v) (I(u+x, v+y) – I(u,v))2 We want to find out how this function behaves for small shifts Roger S. Gaborski

34. Taylor Expansion Change in appearance for the shift [x,y]: • E(x,y) = ΣΣ w(u,v) (I(u+x, v+y) – I(u,v))2 Taylor expansion of I(u+x, v+y)= I(u,v) + Ix(u,v)*x + Iy(u,v)*y Ix and Iy are partial derivatives of x and y So, (I(u,v) + Ix(u,v)*x + Iy(u,v)*y – I(u,v))2 Results in E(x,y) = ΣΣ w(u,v) (Ix(u,v)*x+Iy(u,v)*y)2 Roger S. Gaborski

35. Corner Detection: Mathematics The approximation simplifies to E(x,y) = (x,y) A (x,y)T Where Ais a second moment matrixcomputed from image derivatives: A = ΣΣ w(u,v) Ix2IxIy IxIy Iy2 Roger S. Gaborski

36. Interpreting the second moment matrix Analyze eigenvalues of A to determine if a corner exists If λ1and λ2= 0, then pixel (x,y) is not of interest If one eigenvalue λ is close to 0, and the other is large, (x,y) is on an edge If both eigenvalues λ are large, (x,y) is a corner Roger S. Gaborski

37. Interpreting the eigenvalues Classification of image points using eigenvalues of M: 2 “Edge”2 >> 1 “Corner”1 and 2 are large,1 ~ 2;E increases in all directions 1 and 2 are small;E is almost constant in all directions “Edge”1 >> 2 “Flat” region 1 Roger S. Gaborski

38. Defining Corner Response Function, R Instead of calculating eigenvalues, Harris defined the following function: R = det(A)- α trace(A)2 = λ1λ2 - α (λ1 + λ2) Where α = .04 to .06 Recall: A = [ a b; c d] Det(A) = ad – bc Roger S. Gaborski

39. Corner Response Function R < 0 R > 0 Edge Corners |R| small R < 0 “Flat” region Edge Roger S. Gaborski

40. Harris Detector Algorithm • Compute Derivatives at each point • Compute Second Moment Matrix A • Compute Corner Response Function • Threshold R • Find Local Maxima Roger S. Gaborski

41. Harris Corner Detector • Reference: C.G. Harris and M.J. Stephens “A Combined Corner and Edge Detector” • Code inspired by Peter Kovesi • Derivative Masks: dx = [-1, 0, 1;-1, 0, 1;-1, 0, 1] • dy = dx’ • Image Derivatives; • Ix = imfilter(im, dx, 'conv',‘same’); • Iy = imfilter(im, dy, 'conv',‘same’); • Gaussian Filter • g = fspecial(‘gaussian’, 6*sigma, sigma); Roger S. Gaborski

42. Smooth squared image derivative • Ix2 = imfilter (Ix.^2, g, 'conv', ‘same’); • Iy2 = imfilter (Iy.^2, g, 'conv', ‘same’); • IxIy = imfilter (Ix .* Iy, g, 'conv', ‘same’); c = (Ix2.*Iy2 – IxIy.^2)./(Ix2+Iy2); Roger S. Gaborski

43. Non-maximal Suppression and Threshold • Extract local maxima – gray scale morphological dilation • size = 2*radius+1; %radius is parameter • mx = imdilate(c,ones(size)); %gray scale dilate • cc = (c==mx)&(c>thresh); %find maxima • [r,c] = find(cc) %find row, col coordinates • figure, imagesc( im), colormap(gray) • hold on • plot(c,r, ‘rs’), title(‘Corners) Roger S. Gaborski

44. 100x100 Grid background =0, lines = 1 Roger S. Gaborski

45. Roger S. Gaborski

46. Image rotated 45 Degrees Roger S. Gaborski

47. Image rotated 45 Degreessame parameters Roger S. Gaborski

48. Roger S. Gaborski

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50. Porsche Image Roger S. Gaborski