1. October 19, 19981 STEREO IMAGING (continued)

2. October 19, 19982 RIGHT IMAGE PLANE LEFT IMAGE PLANE RIGHT FOCAL POINT LEFT FOCAL POINT BASELINE d FOCAL LENGTH f BINOCULAR STEREO SYSTEM GOAL: Passive 2-camera system for triangulating 3D position of points in space to generate a depth map of a world scene. Example of a depth- map: z=f(x,y) where x,y coordinatizes one of the image planes and z is the height above the respective image plane. (2D topdown view)

3. October 19, 19983 RIGHT IMAGE PLANE LEFT IMAGE PLANE RIGHT FOCAL POINT LEFT FOCAL POINT BASELINE d FOCAL LENGTH f BINOCULAR STEREO SYSTEM Correspondence Problem is a key issue for binocular stereo -- namely identify image features in respective images that correspond to exactly the same world object point. Clearly localization of image features (e.g., edges) is of critical importance to 3D measurement accuracy. (2D topdown view)

4. October 19, 19984 SOME OTHER MAJOR PROBLEMS WITH CORRESPONDENCE (2D VIEW) OCCLUSION LIMITED FIELD OF VIEW

5. October 19, 19985 BINOCULAR STEREO SYSTEM (2D VIEW Nonverged Stereo System Verged Stereo System

6. October 19, 19986 BINOCULAR STEREO SYSTEM Z X (0,0) (d,0) Z=f XL XR DISPARITY (XL - XR) Z = (f/XL) X Z= (f/XR) (X-d) (f/XL) X = (f/XR) (X-d) X = (XR d) / (XL - XR) Z = df (XL - XR)

7. October 19, 19987 BINOCULAR STEREO SYSTEM (3D VIEW) Rays of projection in 3D may not intersect at all. RIGHT IMAGE LEFT IMAGE Points of closest approach

8. October 19, 19988 BINOCULAR STEREO SYSTEM X Y Z (XL,YL) (XR,YR) (0,0,0) (d,0,0) Points of closest approach

9. October 19, 19989 BINOCULAR STEREO SYSTEM (XL,YL) (XR,YR) (0,0,0) (d,0,0) Points of closest approach = (d,0,0) - (0,0,0)   u u = ^ u u u = (XL,YL,f) - (0,0,0) v = ^ v v = (XR+d,YL,f) - (d,0,0) v v FIND a and b that minimizes | au - (  + bv) | where norm | | is Euclidean Norm. ^ ^

10. October 19, 199810 BINOCULAR STEREO SYSTEM (XL,YL) (XR,YR) (0,0,0) (d,0,0) Points of closest approach  u v FIND a and b that minimizes | au - (  + bv) | where norm | | is Euclidean Norm. ^ ^ a = b = (v.u)(u.  ) - v.  u.  - (u.v)(v.  ) 1 - (u.v) 2 ^ 1 - (u.v) 2 ^ ^ ^ o o Best Geometric Computation is: a u + (  + b v) 2 oo ^

11. October 19, 199811 EPIPOLAR GEOMETRY Points of closest approach For an image point C in the left image plane consider the plane determined by the left image focal point A the right image focal point B and the point C. Call this the epipolar plane for image point C with respect to a particular binocular stereo configuration. C A B

12. October 19, 199812 EPIPOLAR GEOMETRY Points of closest approach In order to guarantee intersection of projective rays produced from the left and right image planes, the point in the right image plane corresponding to C must lie on the intersection of the epipolar plane with the right image plane. C A B EPIPOLAR PLANE EPIPOLAR LINE

13. October 19, 199813 EPIPOLAR GEOMETRY (‘SCANLINE COHERENT’ STEREO SYSTEM)

14. October 19, 199814 EPIPOLAR GEOMETRY (VERGED IN)