1. Class 51 Multi-linear Systems and Invariant Theory in the Context of Computer Vision and Graphics Class 5: Self Calibration CS329 Stanford University Amnon Shashua

2. Class 52 Material We Will Cover Today The basic equations and counting arguments The “ absolute conic ” and its image. Kruppa ’ s equations Recovering internal parameters.

3. Class 53 The Basic Equations and Counting Arguments Recall, 3D->2D from Euclidean world frame to image Let K,K ’ be the internal parameters of camera 1,2 and choose canonical frame in which R=I and T=0 for first camera. world frame to first camera frame

4. Class 54 The Basic Equations and Counting Arguments where maps from the projective frame to Euclidean (8 unknown parameters)

5. Class 55 are the points on the plane at infinity (in Euc frame) is the plane at infinity is the plane at infinity in Proj frame (recall: if W maps points to points (Euc -> Proj), then the dual maps planes to planes) The Basic Equations and Counting Arguments

6. Class 56 The Basic Equations and Counting Arguments

7. Class 57 The Basic Equations and Counting Arguments Projective frame

8. Class 58 The Basic Equations and Counting Arguments sincethen, butprovides 5 (non-linear) constraints!

9. Class 59 Since the right-hand side is symmetric and up to scale, we have 5 constraints. The Basic Equations and Counting Arguments

10. Class 510 The Basic Equations and Counting Arguments Lets do some counting: Letbe the number of internal parameters be the number of views

11. Class 511 The Basic Equations and Counting Arguments not enough measurements (!) (fixed internal params)

12. Class 512 The remainder of this lecture is about a geometric insight of

13. Class 513 The Absolute Conic whererepresents a conic in 2D are the points on the plane at infinity (in Euc frame) is the plane at infinity is conic on the plane at infinity when is the “ absolute ” conic (imaginary circle)

14. Class 514 Plane at infinity is preserved under affine transformations: because is preserved under similarity transformation (R,t up to scale) if and then butso in order thatwe must have: is orthogonal The Absolute Conic

15. Class 515 The Image of the Absolute Conic Image of points at infinity: let ifis a conic on the plane at infinity thenis the projected conic onto the image since then the image ofis

16. Class 516 The Image of the Dual Absolute Conic is tangent to the conic at p is the image of the dual absolute conic The basic equation: Becomes: Why 8 parameters? 5 for the conic, 3 for the plane

17. Class 517 Geometric Interpretation of p direction of optical ray The angle between two optical rays given one can measure angles

18. Class 518 Kruppa’s Equations General idea: eliminate n from the basic equation. are degenerate (rank 2) conics

19. Class 519 Kruppa’s Equations Note: is a degenerate conic iffor Letbe the homography induced by the plane of the conic (slide 14)

20. Class 520 Kruppa’s Equations Recall: In our case and the conic is Likewise:

21. Class 521 Determining K given Recall: the location of the plane at inifinity in the projective coordinate frame. We wish to represent the homographyinduced by Let be a point on the plane at infinity.

22. Class 522 Determining K given Recall: (slide 16) Note: this could be derived from “ first principles ” as well: tangents lines to the image of the absolute conic

23. Class 523 Determining K given Assume fixed internal parameters Provides 4 independent linear constraints on Why 4 and not 5? we need 3 views (since has 5 unknowns) Note:

24. Class 524 Why 4 Constraints? andare “ similar ” matrices, i.e., have the same eigenvalues Let be the axis of rotation, i.e., has an eigenvalue = 1, with eigenvector

25. Class 525 Why 4 Constraints? if then is a solution to is also a solution We need one more camera motion (with a different axis of rotation).

26. Class 526 Kruppa’s Equations (revisited) Kruppa ’ s equations: Start with the basic equation: Multiply the terms byon both sides: