1. Part II Plane Equation and Epipolar Geometry

2. Duality

3. Grassman Coordinates

4. Points on the Same line and... By Duality,

5. Points and Planes in 3D

6. Points and Planes in 3D(Cont’d) For a Fourth point lying on the plane,

7. Linear Transformation Transformation Group Linear Transformation Classification Euclidean Transformation, Affine Transformation, Projective Transformation

8. Rigid Motions - Euclidean Trans... Using Homogeneous Coordinates, Action of the euclidean group on five points

9. As a Subgroup.. And, Inverse is also possible for a following suitable matrix R -1.. Transformation Group For a Euclidean Group,

10. Affine Transformations Action of the affine group on five points Parallel lines are preseved. R is Replaced by a general non-singular transformation matrix A

11. General Linear Transformations It preserves the collinearity of points. For general non-singular linear transformations T, We get general linear or projective group of transformations. Action of the projective group on five points

12. Projection From 3D to 2D Pin-hole camera model

13. The Projection Equation

14. The Projection Equation(Cont’d)

15. Matrix Decomposition

16. Internal & External parameters Internal orientation parameters External orientation parameters A Internal camera parameters R Camera rotation P 0 Camera position(projection point) :

17. Tsai’s Camera Model Ow yw xw zw z,Z P(x,y,z) =P(xw,yw,zw) O y x Y XO1 Pu(Xu,Yu) Pd(Xd,Yd)

18. Tsai’s Camera Model(Cont’d) Step 1 Step 2 Step 3 Step 4, f : focal length (xw,yw,zw) 3D world coordinate (x,y,z) 3D camera coordinate system (Xu,Yu) Ideal undistorted image coordinate (Xd,Yd) Distorted Image coordinate (Xf,Yf) Computer image coordinate in frame memory

19. Determining the Parameters x 0,y 0 : The Origin of the normalized image system  x,  x : Scaling factor from the normalized image coordinates to the original  : The deviation in the normalized system

20. Summary of Camera Calibration Definition The process of determining internal and external parameters Determine in the following way 1) Find the 11 parameters of the projection matrix P using at least 6 known points in space 2) Find the unique factorization. This gives the internal parameters of the matrix A and the external parameters of the camera rotation and position

21. Mapping from a Planar Surface

22. Epipolar Geometry

23. ※ This constraint relation is known as the epipolar constraints The Fundamental Matrix

24. Epipolar points, lines and planes : Epipolar line U B0B0 BeBe B A AeAe A0A0 평면 UA 0 B 0 : Epipolar plane A e,B e : Epipolar point Since,

25. Computing the Fundamental matrix Given 8 points we can in general compute the F-matrix using just linear equations However, if we use the non-linear singularity constraint, [F]=0, we need only 7 points at the price of a more complicated algorithm

26. Calibrated Cameras If we use the first camera as the reference frame, we have 계속

27. Essential Matrix E is called the essential matrix