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# Edges and lines

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### Edges and lines

1. 1. Edges and Lines
2. 2. Edge detection • Goal: Identify sudden changes (discontinuities) in an image – Intuitively, most semantic and shape information from the image can be encoded in the edges – More compact than pixels • Ideal: artist’s line drawing (but artist is also using object-level knowledge) Source: D. Lowe
3. 3. Why do we care about edges?• Extract information, recognize objects Vertical vanishing point (at infinity) Vanishing• Recover geometry and line viewpoint Vanishing Vanishing point point
4. 4. Origin of Edges surface normal discontinuity depth discontinuity surface color discontinuity illumination discontinuity • Edges are caused by a variety of factors Source: Steve Seitz
5. 5. Closeup of edges Source: D. Hoiem
6. 6. Closeup of edges Source: D. Hoiem
7. 7. Closeup of edges Source: D. Hoiem
8. 8. Closeup of edges Source: D. Hoiem
9. 9. Characterizing edges• An edge is a place of rapid change in the image intensity function intensity function image (along horizontal scanline) first derivative edges correspond to extrema of derivative
10. 10. With a little Gaussian noise Gradient Source: D. Hoiem
11. 11. Effects of noise • Consider a single row or column of the image – Plotting intensity as a function of position gives a signal Where is the edge? Source: S. Seitz
12. 12. Effects of noise• Difference filters respond strongly to noise – Image noise results in pixels that look very different from their neighbors – Generally, the larger the noise the stronger the response• What can we do about it? Source: D. Forsyth
13. 13. Solution: smooth first f g f*g d ( f ∗ g) dx d• To find edges, look for peaks in ( f ∗ g) dx Source: S. Seitz
14. 14. Derivative theorem of convolution• Differentiation is convolution, and convolution is d d associative: ( f ∗ g) = f ∗ g dx dx• This saves us one operation: f d g dx d f∗ g dx Source: S. Seitz
15. 15. Derivative of Gaussian filter * [1 -1] =
16. 16. Tradeoff between smoothing and localization 1 pixel 3 pixels 7 pixels • Smoothed derivative removes noise, but blurs edge. Also finds edges at different “scales”. Source: D. Forsyth
17. 17. Designing an edge detector• Criteria for a good edge detector: – Good detection: the optimal detector should find all real edges, ignoring noise or other artifacts – Good localization • the edges detected must be as close as possible to the true edges • the detector must return one point only for each true edge point• Cues of edge detection – Differences in color, intensity, or texture across the boundary – Continuity and closure – High-level knowledge Source: L. Fei-Fei
18. 18. Canny edge detector • This is probably the most widely used edge detector in computer vision • Theoretical model: step-edges corrupted by additive Gaussian noise • Canny has shown that the first derivative of the Gaussian closely approximates the operator that optimizes the product of signal- to-noise ratio and localizationJ. Canny, A Computational Approach To Edge Detection, IEEETrans. Pattern Analysis and Machine Intelligence, 8:679-714, 1986. Source: L. Fei-Fei
19. 19. Example original image (Lena)
20. 20. Derivative of Gaussian filter x-direction y-direction
21. 21. Compute Gradients (DoG)X-Derivative of Gaussian Y-Derivative of Gaussian Gradient Magnitude
22. 22. Get Orientation at Each Pixel• Threshold at minimum level• Get orientation theta = atan2(gy, gx)
23. 23. Non-maximum suppression for each orientation At q, we have a maximum if the value is larger than those at both p and at r. Interpolate to get these values.Source: D. Forsyth
24. 24. Before Non-max Suppression
25. 25. After non-max suppression
26. 26. Hysteresis thresholding• Threshold at low/high levels to get weak/strong edge pixels• Do connected components, starting from strong edge pixels
27. 27. Hysteresis thresholding• Check that maximum value of gradient value is sufficiently large – drop-outs? use hysteresis • use a high threshold to start edge curves and a low threshold to continue them. Source: S. Seitz
28. 28. Final Canny Edges
29. 29. Canny edge detector1. Filter image with x, y derivatives of Gaussian2. Find magnitude and orientation of gradient3. Non-maximum suppression: – Thin multi-pixel wide “ridges” down to single pixel width4. Thresholding and linking (hysteresis): – Define two thresholds: low and high – Use the high threshold to start edge curves and the low threshold to continue them• MATLAB: edge(image, ‘canny’) Source: D. Lowe, L. Fei-Fei
30. 30. Effect of σ (Gaussian kernel spread/size)original Canny with Canny withThe choice of σ depends on desired behavior • large σ detects large scale edges • small σ detects fine features Source: S. Seitz
31. 31. Learning to detect boundaries image human segmentation gradient magnitude • Berkeley segmentation database: http://www.eecs.berkeley.edu/Research/Projects/CS/vision/grouping/segbench/
32. 32. Finding straight lines• One solution: try many possible lines and see how many points each line passes through• Hough transform provides a fast way to do this
33. 33. Hough transform• An early type of voting scheme• General outline: – Discretize parameter space into bins – For each feature point in the image, put a vote in every bin in the parameter space that could have generated this point – Find bins that have the most votes Image space Hough parameter space P.V.C. Hough, Machine Analysis of Bubble Chamber Pictures, Proc. Int. Conf. High Energy Accelerators and Instrumentation, 1959
34. 34. Parameter space representation• A line in the image corresponds to a point in Hough space Image space Hough parameter space Source: S. Seitz
35. 35. Parameter space representation • What does a point (x0, y0) in the image space map to in the Hough space? Image space Hough parameter space
36. 36. Parameter space representation • What does a point (x0, y0) in the image space map to in the Hough space? – Answer: the solutions of b = –x0m + y0 – This is a line in Hough space Image space Hough parameter space
37. 37. Parameter space representation • Where is the line that contains both (x0, y0) and (x1, y1)? Image space Hough parameter space (x1, y1) (x0, y0) b = –x1m + y1
38. 38. Parameter space representation • Where is the line that contains both (x0, y0) and (x1, y1)? – It is the intersection of the lines b = –x0m + y0 and Image –x m + b = space y1 Hough parameter space 1 (x1, y1) (x0, y0) b = –x1m + y1
39. 39. Parameter space representation• Problems with the (m,b) space: – Unbounded parameter domain – Vertical lines require infinite m
40. 40. Parameter space representation• Problems with the (m,b) space: – Unbounded parameter domain – Vertical lines require infinite m• Alternative: polar representation x cosθ + y sinθ = ρ Each point will add a sinusoid in the (θ,ρ) parameter space
41. 41. Algorithm outline • Initialize accumulator H to all zeros • For each edge point (x,y) ρ in the image For θ = 0 to 180 ρ = x cos θ + y sin θ H(θ, ρ) = H(θ, ρ) + 1 θ end end • 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
42. 42. Basic illustration features votes
43. 43. Other shapes Square Circle
44. 44. Several lines
45. 45. A more complicated image http://ostatic.com/files/images/ss_hough.jpg
46. 46. Effect of noise features votes
47. 47. Effect of noise features votes • Peak gets fuzzy and hard to locate
48. 48. Effect of noise• Number of votes for a line of 20 points with increasing noise:
49. 49. Random points features votes• Uniform noise can lead to spurious peaks in the array
50. 50. Random points• As the level of uniform noise increases, the maximum number of votes increases too:
51. 51. Dealing with noise• 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 some points that are not exactly collinear cast votes for different buckets• Increment neighboring bins (smoothing in accumulator array)• Try to get rid of irrelevant features – Take only edge points with significant gradient magnitude
52. 52. 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) + 1 end
53. 53. Hough transform for circles• How many dimensions will the parameter space have?• Given an oriented edge point, what are all possible bins that it can vote for?
54. 54. Hough transform for circles image space Hough parameter spacey r ( x , y ) + r∇ I ( x , y ) (x,y) x ( x , y ) − r∇ I ( x , y ) x y
55. 55. Hough transform for circles • Conceptually equivalent procedure: for each (x,y,r), draw the corresponding circle in the image and compute its “support” r x yIs this more or less efficient than voting with features?
56. 56. Finding straight lines• Another solution: get connected components of pixels and check for straightness
57. 57. Finding line segments using connected components1. Compute canny edges – Compute: gx, gy (DoG in x,y directions) – Compute: theta = atan(gy / gx)2. Assign each edge to one of 8 directions3. For each direction d, get edgelets: – find connected components for edge pixels with directions in {d-1, d, d+1}• Compute straightness and theta of edgelets using eig of x,y 2nd moment matrix of their points ∑ ( x − µ x ) ( y − µ y ) Larger eigenvector  ∑( x − µx )2M=  [ v, λ] = eig(Μ ) ∑ ( x − µ x ) ( y − µ y ) ∑( y − µ ) θ = atan 2( v(2,2), v(1,2)) 2  y   conf = λ2 / λ1• Threshold on straightness, store segment Slides from Derek Hoiem
58. 58. 1. Image  Canny
59. 59. 2. Canny lines  … straight edges
60. 60. Comparison Hough Transform Method Connected Components Method