1. October 3, 2013Computer Vision Lecture 10: Contour Fitting 1 Edge Relaxation Typically, this technique works on crack edges: pixelpixelpixel pixelpixelpixelebg i h a c d f

2. October 3, 2013Computer Vision Lecture 10: Contour Fitting 2 Edge Relaxation Edges h and i can be used for non-maxima suppression similar to the Canny detector. We will not discuss this here. The edge pattern at an edge e can be described as a pair of integers: the number of connecting edges on the left (edges a, b, c) and the number of connecting edges on the right (d, f, g). Since the left and right sides are exchangeable, we simply write the smaller number first, e.g. 0-1, 1-3, 2-2. These edge patterns determine whether we should increase or decrease the current confidence value for e or leave it unchanged.

3. October 3, 2013Computer Vision Lecture 10: Contour Fitting 3 Edge Relaxation These are the actions we should take: 0-0: isolated edge – decrease edge confidence 0-1: uncertain – increase slightly or leave unchanged 0-2, 0-3: dead end – decrease 1-1: continuation: increase strongly 1-2, 1-3: continuation to border intersection: increase 2-2, 2-3, 3-3: bridge between borders – no change

4. October 3, 2013Computer Vision Lecture 10: Contour Fitting 4 Edge Relaxation Now we can describe the edge relaxation algorithm: (1)Evaluate a confidence c (1) (e) for all crack edges e in the image. (2)Find the edge type of each edge based on edge confidences c (k) (e) in its neighborhood. (3)Update the confidence c (k+1) (e) of each edge e according to its type and its previous confidence c (k) (e). (4)Stop if all edge confidences have converged either to 0 or 1. Repeat steps (2) and (3) otherwise.

5. October 3, 2013Computer Vision Lecture 10: Contour Fitting 5 Edge Relaxation At the start of the algorithm, we can determine c (1) (e) for all crack edges e as the normalized (between 0 and 1) edge strength yielded by the edge detector. Often local normalization works better, because it limits the influence of individual, extremely high edge strengths. Since during the iterations we often have edge confidence values between 0 and 1, how can we determine edge types? In order to determine the edge pattern and the edge type, we can simply use confidence thresholds. Another method is shown on the next slide.

6. October 3, 2013Computer Vision Lecture 10: Contour Fitting 6 Edge Relaxation Consider the three adjacent edges on one side of e. Let’s call them a, b, and c so that a  b  c. Let’s further define a value q (something like a threshold); usually q = 0.1. Finally, let’s define m = max(a, b, c, q) Then the edge type on that side is defined as k, where type(k) is the maximum of the following numbers: type(0) = (m – a)(m – b)(m – c) type(1) = a(m – b)(m – c) type(2) = ab(m – c) type(3) = abc

7. October 3, 2013Computer Vision Lecture 10: Contour Fitting 7 Edge Relaxation Based on the edge type we decide whether we want to increase or decrease the confidence for e (or leave it unchanged). This can be done as follows: Increase: c (k+1) (e) = min(1, c (k) (e) +  ) Decrease: c (k+1) (e) = max(0, c (k) (e) -  ) Appropriate values for  are typically in the range between 0.1 and 0.3.

8. October 3, 2013Computer Vision Lecture 10: Contour Fitting 8 Edge Relaxation After a large number of iterations, it is possible that the results of edge relaxation deteriorate. A possible solution is to use an upper threshold T 1 and a lower threshold T 2 and use them as follows: If c (k+1) (e) > T 1 then assign c (k+1) (e) = 1, If c (k+1) (e) < T 2 then assign c (k+1) (e) = 0.

9. October 3, 2013Computer Vision Lecture 10: Contour Fitting 9Contours Once we have a good idea of where our contours are, what is the best way to represent them? A good contour representation should meet the following criteria: Efficiency: The contour should be a simple, compact representation. Efficiency: The contour should be a simple, compact representation. Accuracy: The contour should accurately fit the image features. Accuracy: The contour should accurately fit the image features. Effectiveness: The contour should be suitable for the operations to be performed in later stages of the application. Effectiveness: The contour should be suitable for the operations to be performed in later stages of the application.

10. October 3, 2013Computer Vision Lecture 10: Contour Fitting 10Contours The accuracy of a contour representation is determined by the form of the curve used to model the contour, the form of the curve used to model the contour, the performance of the curve-fitting algorithm, and the performance of the curve-fitting algorithm, and the accuracy of the estimation of edge locations. the accuracy of the estimation of edge locations. Using an ordered list of edges as the representation is simple and as precise as the edge information itself. It is not a compact representation and may be difficult to process in subsequent stages.

11. October 3, 2013Computer Vision Lecture 10: Contour Fitting 11Contours Using an appropriate curve model increases the accuracy of the representation. This is because errors in the location of individual edges can be eliminated (if the correct model is used!). The result is a more compact representation that is well-suited for subsequent analysis. We will talk about the elementary differential geometry of 2D curves, the elementary differential geometry of 2D curves, techniques for calculating contour properties, and techniques for calculating contour properties, and curve models and how to fit them to contours. curve models and how to fit them to contours.

12. October 3, 2013Computer Vision Lecture 10: Contour Fitting 12Definitions We will use the term edge to refer to edge points regardless of the edge orientation. Most standard algorithms do not consider the orientation of edges, but just their location. Definitions: An edge list is an ordered set of edge points or fragments. A contour is an edge list or the curve that has been used to represent the edge list. A boundary is the closed contour that surrounds a region.

13. October 3, 2013Computer Vision Lecture 10: Contour Fitting 13 Geometry of Curves We usually avoid the description of curves in the x-y plane by functions of the kind y = f(x). This is because this notation only allows one y-value for a given x-value. So, for example, it cannot be used to describe a circle, rectangle, or any other closed contour. Instead, we will use the parametric form (x(u), y(u)). It uses two functions x(u) and y(u) of a parameter u to specify the points along the curve from a starting point p 1 = (x(u 1 ), y(u 1 )) to an end point p 2 = (x(u 2 ), y(u 2 )).

14. October 3, 2013Computer Vision Lecture 10: Contour Fitting 14 Geometry of Curves The length of the curve is given by the arc length: The unit tangent vector is given by: where p(u) = (x(u), y(u)) and p’(u) = (x’(u), y’(u)).

15. October 3, 2013Computer Vision Lecture 10: Contour Fitting 15 Geometry of Curves Imagine three points along our curve: p(u +  ), p(u), and p(u -  ). If these three points all differ from each other, then there is exactly one circle that passes through all three points. In the limit   0, this circle is the osculating (touching) circle of the curve at the point u. The center of this circle lies along the line containing the normal to the curve at point u. The curvature is the inverse of the radius of the osculating circle.

16. October 3, 2013Computer Vision Lecture 10: Contour Fitting 16 Digital Curves When we are dealing with actual curves in digital images, the situation is slightly different. There are only eight possible angles between neighboring pixels, which makes the computation of slope and curvature difficult. The idea to overcome this problem is to also account for non-adjacent edge points. Let p n = (i n, j n ) be the coordinates of edge n in our edge list. Then the k-slope is the (angle) direction vector between points that are k edges apart.

17. October 3, 2013Computer Vision Lecture 10: Contour Fitting 17 Geometry of Curves The left k-slope is the direction from p n-k to p n, and the right k-slope is the direction from p n to p n+k. The k-curvature is the difference between the left and right k-slopes. If we have N edge points (i 1, j 1 ) to (i N, j N ) in the edge list, then we can approximate the length S of the digital curve as follows: