1. Robust estimation Problem: we want to determine the displacement (u,v) between pairs of images. We are given 100 points with a correlation score computed for each one. Using the Kalman Filter: –For each matrix we fit a Gaussian –We now know how to combine the information from all of the 100 points. Problem: Even if we have a single outlier, we will find a wrong solution !

2. First try For each point, choose the displacement with the biggest score (Like in optical flow) Compute the median of the horizontal and vertical displacements. Problem: We will get a correct result only if more than 50% of the points are inliers. 33% What will be the median in this case?

3. Second try For each point, choose the displacement with the biggest score Perform a vote: take the winner. Advantages: –we can receive a good solution even with less then 50% good points. –The weight of each point is bounded by 1. Problems: –We need to determine a grid. –We give the same weight to points in smooth regions or in 1D regions.

4. A possible solution For each point, calculate a matrix of “probabilities” for each displacement. Perform a “vote”: sum all the weights of each displacement and take the best one. Points located in more informative regions will have more effect on the resulting displacement. Yet – the weights of each point are bounded by 1 (so outliers have limited effect).

5. RANSAC- randomized algorithm Lotter K points. Solve the problem using the K points. Check the solution on the rest of the points.

6. Finding shapes using the Hough Transform (proposed by Hough in ~1959) Straight Lines Rectangles Circles Others

7. The challenge: –The image contains a lot of irrelevant information. –There are partial occlusions. –The image might be noisy Applications: –Find signs in the images (rectangles, circles). –3D reconstruction of buildings (lines). –Find the eye in a face image. –Tracking –More…

9. Use a voting scheme Find lines of the form y=ax+b (slope-intercept representation) –Apply an edge detector on the image (gradient + threshold, or using Canny’s method) –Prepare a table h(a,b) for a,b in a certain range. –Loop over the image- for each pixel (x,y) on an edge, vote for the cells (a,b) which satisfy the equation y=ax+b. (increase the cell accumulator- h(a,b)++) –Choose the cell with the maximal value (or the cells which pass a given threshold)

10. Some properties of this representation A point corresponds to the set of lines passing through it. Lines are represented by a set of points in the transformed representation. All these points lie on a line (in the transformed representation): b = -ax+y So with have Duality: A point is mapped to a line, and a line is mapped to a point.

11. Example: Slope-Intercept Representation

12. Problems: Vertical lines are not covered by the representation y=ax+b. Selecting the range of values for the table (a and b are not bounded!) Quantization: choosing a grid. Thresholding: Similar lines will get similar scores.

13. Solutions: Representation Use a polar representation instead: - The line normal d - The distance from (0,0) In this representation, each point (x,y) will be mapped to a sinusoidal curve in the (d,  ) space.

14. Solutions: Range of Values and Quantization In the polar representation it is easier to determine the range: –The values of d and of  are now bounded ! –The actual range of image can be application dependent. We have natural units (pixels for d, and degrees for  ) Quantization: –Option1: Vote for the “nearest” cell (f.g – for each  take the nearest d). –Option2: Perform a weighted vote.

15. Solutions: Thresholding How can we find the two most dominant lines in the image ? The problem: if (r,  ) is the cell with the maximal value, then (r+ ,  +  ) will get a high score too. Solution 1: After finding the first line, omit its neighborhood from the search. Solution 2: Search only for local maxima. Solution 3: “Back Mapping”- Perform a second vote, where each point votes only for the best line intersecting it.

16. Detecting Complex Shapes - Circle Naïve solution: –Construct a table h(a,b,r). –For each pixel (x,y) which is edge in the image, vote for all the cells satisfying: The problem: –The size of the table O(N 3 ). –The voting for each pixels takes: O(N 2 ). Solution 1: Use the gradient information - Solve for (a,b) and then for r. Solution 2: Randomized Hough Transform.