1 / 24

Order Structure, Correspondence, and Shape Based Categories

Order Structure, Correspondence, and Shape Based Categories. Stefan Carlsson. Presented by Piotr Dollar. October 24, 2002. Visual Correspondence. No “correct” answer Humans Exceptionally Good Tell & Carlsson (later work) Object Recognition. Basic Idea.

yamin
Download Presentation

Order Structure, Correspondence, and Shape Based Categories

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Order Structure, Correspondence, and Shape Based Categories Stefan Carlsson Presented by Piotr Dollar October 24, 2002

  2. Visual Correspondence • No “correct” answer • Humans Exceptionally Good • Tell & Carlsson (later work) • Object Recognition

  3. Basic Idea • What are the two groups here? • What makes these Similar?

  4. Basic Idea (cont.) • Select Points • Tangent Lines • Structure and location of points and lines • More general than affine and projective transformation that preserve the side from which the image is taken.

  5. Order Types Fundamentally different Structures

  6. Interlude: Homogeneous Coordinates • Points and lines have same representation (3 element vector) • For point: [x y 1] (x and y coordinates) • For line v = [u v –d], the set of points x s.t. vT.x = 0. This line is orthogonal to the standard vector [u v] • Every point can be thought of as a line and vice versa (normalize lines to [u v 1])

  7. Order type: Points • Order structure for points is orientation • Three points may be collinear, or, traversing them in order can give clockwise or counterclockwise rotation • Math: given three points x1, x2 and x3, the determinant of the matrix [ x1 x2 x3 ] is 0 if the points are collinear, negative if they define a clockwise rotation, positive otherwise. • Why: [show informal proof if time] • For >3 points, take all subsets of 3 points and get each subsets ordering. Maintain a consistent ordering of the points.

  8. Order type: lines • Just as we took the orientation of 3 points, can get orientation of 3 lines • Sine each line corresponds to a unique point (scale the line to [ x y 1]), taking the sign of the determinant of the matrix of three lines gives us their orientation • Again, for multiple lines take each ordered subset of 3 lines

  9. Order type: lines (II) • Actually, Carlsson normalizes the middle component of the line vector (i.e. it becomes [ a 1 b ]). • Nice because then it defines the set of points x s.t. vT.x = 0 or in other words ax + y + b =0. Useful for taking combination of points and lines. • However same basic idea.

  10. Order type: Points & Lines • Since the defining vector for a line now always points up ([a 1 b]), we can think of line as having a direction (say always right). • Each point in plane now falls to the right or left of the line (or on the line). • Can use vT.x = 0. Let y2 = -ax – b (y2 is the y coordinate of the point with x coordinate x that falls on the line). Now, if y>y2, (i.e. the point falls to the left of the line) then ax + y +b < 0 is, and likewise y>y2 implies vT.x > 0. • Hence can get orientation of a point relative to a line.

  11. Order type: “0” • The orientation “0” for 3 points (i.e. they are collinear) is an unstable orientation. By this I mean that a miniscule change in the alignment of the three points will cause the orientation to go to either +1 or -1. • Good idea to not use collinear points. What does Carlsson do? Hmm… • One could also try to filter against noise by discarding points that are nearly collinear – i.e. have a determinant that is within some epsilon of 0.

  12. Combining the information • So now for each set of points, lines, and points and lines we can get a series of binary values that contain information about their structure. • What information is useful? • Specifically: How do we filter out dependencies that arose because of how we numbered our lines and points?

  13. Equivalence Classes for Points • Initial ordering of points determines their orientation, in fact all sets of 3 points have same “structure”, even though they can have different orientations. • For sets of >3 points, however, there are different structures. • We can use the information from the orientation of each subset of 3 points to deduce the actual equivalence class for the points.

  14. Equivalence Classes for Points (II) • Two Equivalence classes

  15. Equivalence Classes for Points (III) • Thus we can learn the equivalence class or the order type of a set of points. • Three different order types exist for 5 points

  16. Canonical Orderings • For each equivalence class, we define an arbitrary but fixed numbering scheme. This is the “canonical ordering”. • If cyclic equivalence class, we will call the leftmost point #1. • So, for five points:

  17. Combining the information (II) • Now we have a fixed numbering scheme! • Using this number scheme (both for the points and their corresponding tangent lines), we now get the orientation information for the sets of lines for each pair of a point and line. • This information is now directly useful. No false orientation from the initial order of the points. • Note that we used the point ordering differently from the line and point/line ordering.

  18. Combining the information (III) • For any subset of points, we can get a single number representing their structure. We call this the order structure index. • Two different sets of points with the same order structure index are very similar. We use this as the basis for our algorithm.

  19. The Algorithm • The algorithm is very simple. It is divided into two stages: model building and actual indexing. • The two stages are similar, except in the first we use the first image to build a table, in the second we use the second image and the table to vote on point correspondences.

  20. The Algorithm: Model Building For each subset of 5 points: • Compute the order structure index (OSI) for the points. • The OSI identifies a location in the table, place the 5 point set at that location in the table.

  21. The Algorithm: Indexing For each subset of 5 points (p1,…p5): • Compute the (OSI) for the points, and the index in the table the OSI indicates. • For every set of points (m1,…,m5) at that index, give one vote for p1 corresponding to m1, p2 to m2, and so on. This gives us a table that indicates how strongly points correspond.

  22. Results: (normalized vote counts)

  23. Discussion • Using point structure and tangent lines seemed to give good results. • Just one of many possible schemes to extract feature relations. • If n is the number of points extracted from an image, the running time is at least O(n^5). Thus this algorithm is useful only for small number of points (authors say it is 25-30). • Also used for object recognition (multiple models), classify an image. Choose model with the highest number of point correspondences.

  24. References • Combinatorial Geometry for Shape Representation and Indexing, Carlsson 1996.

More Related