1. MSU CSE 240 Fall 2003 Stockman CV: 3D to 2D mathematics Perspective transformation; camera calibration; stereo computation; and more

2. MSU CSE 240 Fall 2003 Stockman Roadmap of topics Review perspective transformation Camera calibration Stereo methods Structured light methods Depth-from-focus Shape-from-shading

3. MSU CSE 240 Fall 2003 Stockman Review coordinate systems World or global W Object or model M Camera or sensor C Camera or sensor D

4. MSU CSE 240 Fall 2003 Stockman Convenient notation for points and transformations This point P has 3 real world coordinates in coordinate system W This point P has 2 real coordinates in the image This transformation maps each point in the real world W to a point in the image I

5. MSU CSE 240 Fall 2003 Stockman Current goal Develop the theory in terms of modules (components) so that concepts are understood and can be put into practical application

6. MSU CSE 240 Fall 2003 Stockman Perspective transformation X and Y are scaled by the ratio of focal length to depth Z Camera origin is center of projection, not lens

7. MSU CSE 240 Fall 2003 Stockman In next homework & project fit camera model to image with jig jig has known precise 3D coordinates examine accuracy of camera model use camera model to do graphics use two camera models to compute depth from stereo

8. MSU CSE 240 Fall 2003 Stockman Notes on perspective trans. 3D world scaled according to ratio of depth to focal length scaling formulas are in terms of real numbers with the same units e.g. mm in the 3D world and mm in the image plane real image coordinates must be further scaled to pixel row and column entire 3D ray images to the same 2D point

9. MSU CSE 240 Fall 2003 Stockman Goal: General perspective trans to be developed (accept for now) Camera matrix C transforms 3D real world point into image row and column using 11 parameters

10. MSU CSE 240 Fall 2003 Stockman The 11 parameters Cij model internal camera parameters: focal length f ratio of pixel height and width any shear due to sensor chip alignment external orientation parameters: rotation of camera frame relative to world frame translation of camera frame relative to world The 11 parameters of this model are NOT independent. Radial distortion is not linear and is not modeled.

11. MSU CSE 240 Fall 2003 Stockman Camera matrix via least squares Minimize the residuals in the image plane. Get 2 equations for each pair ((r, c), (x, y, z))

12. MSU CSE 240 Fall 2003 Stockman 2 equations for each pair Here, (u, v) is the point in the image where 3D point (x,y,z) is projected. The 11 unknowns d jk form the camera matrix. Known image points Known 3D points Camera parameters

13. MSU CSE 240 Fall 2003 Stockman 2n linear equations from n pairs ((u,v) (x,y,z)) Standard linear algebra problem; easily solved in Matlab or by using a linear algebra package. Often, package replaces b’s with the residuals.

14. MSU CSE 240 Fall 2003 Stockman Use a jig for calibration Jig has known set of points Measure points in world system W or use the jig to define W Take image with camera and determine 2D points Get pairings ((r, c) (x, y, z))

15. MSU CSE 240 Fall 2003 Stockman Example calibration data # # IMAGE: g1view1.ras # # INPUT DATA | OUTPUT DATA # | Point Image 2-D (U,V) 3-D Coordinates (X,Y,Z) | 2-D Fit Data Residuals X Y | A 95.00 336.00 0.00 0.00 0.00 | 94.53 337.89 | 0.47 -1.89 B 0.00 6.00 0.00 | | C 11.00 6.00 0.00 | | D 592.00 368.00 11.00 0.00 0.00 | 592.21 368.36 | -0.21 -0.36 N 501.00 363.00 9.00 0.00 0.00 | 501.16 362.78 | -0.16 0.22 O 467.00 279.00 8.25 0.00 -1.81 | 468.35 281.09 | -1.35 -2.09 P 224.00 266.00 2.75 0.00 -1.81 | 224.06 266.43 | -0.06 -0.43 # CALIBRATION MATRIX 44.84 29.80 -5.504 94.53 2.518 42.24 40.79 337.9 -0.0006832 0.06489 -0.01027 1.000

16. MSU CSE 240 Fall 2003 Stockman 3D points on jig Dimensions in inches

17. MSU CSE 240 Fall 2003 Stockman Jig set in workspace Mapping is established between 3D points (x, y, z) and image points (u, v)

18. MSU CSE 240 Fall 2003 Stockman Other jigs used at MSU frame with wires and beads placed in car instead of the driver seat (to do stereo measurements of car driver) frame with wires and beads as big as a harp to calibrate space for people walking (up to 6 cameras, persons wear tight body suit with reflecting disks, cameras compute 3D motion trajectory)

19. MSU CSE 240 Fall 2003 Stockman Least squares set up 2n x 11 11 x 1 2n x 1 = AX = B

20. MSU CSE 240 Fall 2003 Stockman Least squares abstraction

21. MSU CSE 240 Fall 2003 Stockman Justify the form of camera matrix Another sequence of slides Rotation, scaling, shear in 3D real world as a 3x3 (or 4x4) matrix Projection to real 2D image as 4x4 matrix Scaling real image coordinates to [r, c] coordinates as 4x4 matrix Combine them all into one 4x4 matrix

22. MSU CSE 240 Fall 2003 Stockman Other mathematical models Two camera stereo

23. MSU CSE 240 Fall 2003 Stockman Baseline stereo: carefully aligned cameras

24. MSU CSE 240 Fall 2003 Stockman Computing (x, y, z) in 3D from corresponding 2D image points

25. MSU CSE 240 Fall 2003 Stockman 2 calibrated cameras view the same 3D point at (r1,c1)(r2,c2)

26. MSU CSE 240 Fall 2003 Stockman Compute closest approach of the two rays: use center point V Shortest line segment between rays

27. MSU CSE 240 Fall 2003 Stockman Connector is perpendicular to both imaging rays

28. MSU CSE 240 Fall 2003 Stockman Solve for the endpoints of the connector Scaler mult. Fix book

29. MSU CSE 240 Fall 2003 Stockman Correspondence problem: more difficult aspect

30. MSU CSE 240 Fall 2003 Stockman Correspondence problem is difficult Can use interest points and cross correlation Can limit search to epipolar line Can use symbolic matching (Ch 11) to determine corresponding points (called structural stereopsis) apparently humans don’t need it

31. MSU CSE 240 Fall 2003 Stockman Epipolar constraint With aligned cameras, search for corresponding point is 1D along corresponding row of other camera.

32. MSU CSE 240 Fall 2003 Stockman Epipolar constraint for non baseline stereo computation If cameras are not aligned, a 1D search can still be determined for the corresponding point. P1, C1, C2 determine a plane that cuts image I2 in a line: P2 will be on that line. Need to know relative orientation of cameras C1 and C2

33. MSU CSE 240 Fall 2003 Stockman Measuring driver body position 4 cameras were used to measure driver position and posture while driving: 2mm accuracy achieved