CS 1699: Intro to Computer Vision Matching and Fitting

1. CS 1699: Intro to Computer VisionMatching and Fitting Prof. Adriana KovashkaUniversity of Pittsburgh September 29, 2015

2. Today • Fitting models (lines) to points, i.e. find the parameters of a model that best fits the data • Least squares • Hough transform • RANSAC • Matching = finding correspondences between points, i.e. find the parameters of the transformation that best aligns points • Homework 2 is due 10/08

3. Fitting • Want to associate a model with observed features [Fig from Marszalek & Schmid, 2007] For example, the model could be a line, a circle, or an arbitrary shape. Kristen Grauman

4. Example: Line fitting • Why fit lines? Many objects characterized by presence of straight lines • Why aren’t we done just by running edge detection? Kristen Grauman

5. Difficulty of line fitting • Extra edge points (clutter), multiple models: • which points go with which line, if any? • Only some parts of each line detected, and some parts are missing: • how to find a line that bridges missing evidence? • Noise in measured edge points, orientations: • how to detect true underlying parameters? Kristen Grauman

6. Least squares line fitting • Data: (x1, y1), …, (xn, yn) • Line equation: yi = mxi + b • Find (m, b) to minimize y=mx+b (xi, yi) Matlab: p = A \ y; Modified from Svetlana Lazebnik

7. Hypothesize and test • Propose parameters • Try all possible • Each point votes for all consistent parameters • Repeatedly sample enough points to solve for parameters • Score the given parameters • Number of consistent points, possibly weighted by distance • Choose from among the set of parameters • Global or local maximum of scores • Possibly refine parameters using inliers Derek Hoiem

8. Voting • It’s not feasible to check all combinations of features by fitting a model to each possible subset. • Voting is a general technique where we let the features vote for all models that are compatible with it. • Cycle through features, cast votes for model parameters. • Look for model parameters that receive a lot of votes. • Noise & clutter features? • They will cast votes too, but typically their votes should be inconsistent with the majority of “good” features. Kristen Grauman

9. Fitting lines: Hough transform • Given points that belong to a line, what is the line? • How many lines are there? • Which points belong to which lines? • Hough Transform is a voting technique that can be used to answer all of these questions. Main idea: 1. Record vote for each possible line on which each edge point lies. 2. Look for lines that get many votes. Kristen Grauman

10. Finding lines in an image: Hough space Connection between image (x,y) and Hough (m,b) spaces • A line in the image corresponds to a point in Hough space y b b0 m0 x m image space Hough (parameter) space Steve Seitz

11. Answer: the solutions of b = -x0m + y0 • This is a line in Hough space Finding lines in an image: Hough space Connection between image (x,y) and Hough (m,b) spaces • A line in the image corresponds to a point in Hough space • What does a point (x0, y0) in the image space map to? • To go from image space to Hough space: • given a set of points (x,y), find all (m,b) such that y = mx + b y b y0 x0 x m image space Hough (parameter) space Steve Seitz

12. Finding lines in an image: Hough space What are the line parameters for the line that contains both (x0, y0) and (x1, y1)? • It is the intersection of the lines b = –x0m + y0 and b = –x1m + y1 y b (x1, y1) y0 (x0, y0) b = –x1m + y1 x0 x m image space Hough (parameter) space Steve Seitz

13. Finding lines in an image: Hough algorithm How can we use this to find the most likely parameters (m,b) for the most prominent line in the image space? • Let each edge point in image space vote for a set of possible parameters in Hough space • Accumulate votes in discrete set of bins; parameters with the most votes indicate line in image space. y b x m image space Hough (parameter) space Steve Seitz

14. y m 3 5 3 3 2 2 3 7 11 10 4 3 2 3 1 4 5 2 2 1 0 1 3 3 x b Hough transform y m b x Silvio Savarese

15. Parameter space representation • Problems with the (m,b) space: • Unbounded parameter domains • Vertical lines require infinite m Svetlana Lazebnik

16. Parameter space representation • Problems with the (m,b) space: • Unbounded parameter domains • Vertical lines require infinite m • Alternative: polar representation Each point (x,y) will add a sinusoid in the (,) parameter space Svetlana Lazebnik

17. Hough transform P.V.C. Hough, Machine Analysis of Bubble Chamber Pictures, Proc. Int. Conf. High Energy Accelerators and Instrumentation, 1959 Use a polar representation for the parameter space y x Hough space Silvio Savarese

18. Algorithm outline • Initialize accumulator H to all zeros • For each feature point (x,y) in the image For θ = 0 to 180ρ = x cos θ + y sin θ H(θ, ρ) = H(θ, ρ) + 1 endend • Find the value(s) of (θ*, ρ*) where H(θ, ρ) is a local maximum • The detected line in the image is given by ρ* = x cos θ* + y sin θ* ρ θ Svetlana Lazebnik

19. Hough transform example http://ostatic.com/files/images/ss_hough.jpg Derek Hoiem

20. Impact of noise on Hough d y x  Image space edge coordinates Votes Silvio Savarese

21. Impact of noise on Hough d y x  Image space edge coordinates Votes What difficulty does this present for an implementation? Kristen Grauman

22. Impact of noise on Hough Noisy data Need to adjust grid size or smooth features votes Silvio Savarese

23. Impact of noise on Hough Image space edge coordinates Votes Here, everything appears to be “noise”, or random edge points, but we still see peaks in the vote space. Kristen Grauman

24. Algorithm outline • Initialize accumulator H to all zeros • For each feature point (x,y) in the imageFor θ = 0 to 180ρ = x cos θ + y sin θ H(θ, ρ) = H(θ, ρ) + 1 endend • Find the value(s) of (θ, ρ) where H(θ, ρ) is a local maximum • The detected line in the image is given by ρ = x cos θ + y sin θ ρ θ Svetlana Lazebnik

25. Incorporating image gradients • Recall: when we detect an edge point, we also know its gradient direction • But this means that the line is uniquely determined! • Modified Hough transform: • For each edge point (x,y) θ = gradient orientation at (x,y)ρ = x cos θ + y sin θ H(θ, ρ) = H(θ, ρ) + 1end Svetlana Lazebnik

26. Hough transform for circles • Circle: center (a,b) and radius r • For a fixed radius r, unknown gradient direction Hough space Image space Kristen Grauman

27. Hough transform for circles • Circle: center (a,b) and radius r • For a fixed radius r, unknown gradient direction Intersection: most votes for center occur here. Hough space Image space Kristen Grauman

28. Hough transform for circles • Circle: center (a,b) and radius r • For an unknown radius r, unknown gradient direction r ? b a Hough space Image space Kristen Grauman

29. Hough transform for circles • Circle: center (a, b) and radius r • For an unknown radius r, unknown gradient direction r b a Hough space Image space Kristen Grauman

30. Hough transform for circles For every edge pixel (x,y) : For all a: For all b: r = H[a,b,r] += 1 end end end

31. Hough transform for circles • Circle: center (a,b) and radius r • For an unknown radius r, known gradient direction x θ Hough space Image space Kristen Grauman

32. Hough transform for circles • A circle with radius r and center (a, b) can be described as: x = a + r cos(θ) y = b + r sin(θ) (x, y) (a, b)

33. Hough transform for circles x = a + r cos(θ) y = b + r sin(θ) For every edge pixel (x,y) : For each possible radius value r: For each possible gradient direction θ: // or use estimated gradient at (x,y) a = x – rcos(θ) // column b = y – r sin(θ) // row H[a,b,r] += 1 end end end x θ Modified from Kristen Grauman

34. Example: detecting circles with Hough Original Edges Votes: Penny Note: a different Hough transform (with separate accumulators) was used for each circle radius (quarters vs. penny). Kristen Grauman, images from VivekKwatra

35. Example: detecting circles with Hough Original Combined detections Edges Votes: Quarter Note: a different Hough transform (with separate accumulators) was used for each circle radius (quarters vs. penny). Kristen Grauman, images from VivekKwatra

36. Example: iris detection • HemersonPistori and Eduardo Rocha Costa http://rsbweb.nih.gov/ij/plugins/hough-circles.html Gradient+threshold Hough space (fixed radius) Max detections Kristen Grauman

37. Voting: practical tips • Minimize irrelevant tokens first • Choose a good grid / discretization • Too coarse:large votes obtained when too many different lines correspond to a single bucket • Too fine: miss lines because points that are not exactly collinear cast votes for different buckets • Vote for neighbors, also (smoothing in accumulator array) • Use direction of edge to reduce parameters by 1 • To read back which points voted for “winning” peaks, keep tags on the votes Too fine ? Too coarse Kristen Grauman

38. Generalized Hough transform • We want to find a template defined by its reference point (center) and several distinct types of landmark points in stable spatial configuration Template c Svetlana Lazebnik

39. Generalized Hough transform • What if we want to detect arbitrary shapes? Intuition: Displacement vectors x x x Ref. point x x Vote space Model image Now suppose those colors encode gradient directions… Novel image Kristen Grauman

40. x Generalized Hough transform • Define a model shape by its boundary points and a reference point. Offline procedure: a At each boundary point, compute displacement vector: r = a – pi. Store these vectors in a table indexed by gradient orientation θ. θ θ p1 p2 Model shape θ … … θ … [Dana H. Ballard, Generalizing the Hough Transform to Detect Arbitrary Shapes, 1980] Kristen Grauman

41. Generalized Hough transform Detection procedure: x • For each edge point: • Use its gradient orientation θ to index into stored table • Use retrieved r vectors to vote for reference point x x x x p1 Novel image θ θ θ θ θ θ … Assuming translation is the only transformation here, i.e., orientation and scale are fixed. … θ … Kristen Grauman

42. Generalized Hough transform • Template representation: for each type of landmark point, store all possible displacement vectors towards the center Model Template Svetlana Lazebnik

43. Generalized Hough transform • Detecting the template: • For each feature in a new image, look up that feature type in the model and vote for the possible center locations associated with that type in the model Model Test image Svetlana Lazebnik

44. “visual codeword” withdisplacement vectors training image Generalized Hough for object detection • Index displacements by “visual codeword” B. Leibe, A. Leonardis, and B. Schiele, Combined Object Categorization and Segmentation with an Implicit Shape Model, ECCV Workshop on Statistical Learning in Computer Vision 2004 Svetlana Lazebnik

45. Generalized Hough for object detection • Index displacements by “visual codeword” test image B. Leibe, A. Leonardis, and B. Schiele, Combined Object Categorization and Segmentation with an Implicit Shape Model, ECCV Workshop on Statistical Learning in Computer Vision 2004 Svetlana Lazebnik

46. Implicit shape models: Training • Build codebook of patches around extracted interest points using clustering (more on this later in the course) Svetlana Lazebnik

47. Generalized Hough transform • Template representation: for each type of landmark point, store all possible displacement vectors towards the center Model Template Svetlana Lazebnik

48. Implicit shape models: Training • Build codebook of patches around extracted interest points using clustering • Map the patch around each interest point to closest codebook entry Svetlana Lazebnik

49. Implicit shape models: Training • Build codebook of patches around extracted interest points using clustering • Map the patch around each interest point to closest codebook entry • For each codebook entry, store all positions it was found, relative to object center Svetlana Lazebnik

50. Hough transform: pros and cons Pros • All points are processed independently, so can cope with occlusion, gaps • Some robustness to noise: noise points unlikely to contribute consistently to any single bin • Can detect multiple instances of a model in a single pass Cons • Complexity of search time increases exponentially with the number of model parameters • Non-target shapes can produce spurious peaks in parameter space • Quantization: can be tricky to pick a good grid size Kristen Grauman