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Geometric Model Acquisition

Geometric Model Acquisition. Steve Maybank School of Computer Science and Information Systems Birkbeck College London, WC1E 7HX Edited version of the slides for the VVG Summer School, held at the University of Bath 21 September 2007. Geometric Model Acquisition.

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Geometric Model Acquisition

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  1. Geometric Model Acquisition Steve Maybank School of Computer Science and Information Systems Birkbeck College London, WC1E 7HX Edited version of the slides for the VVG Summer School, held at the University of Bath 21 September 2007 Birkbeck College, U. London

  2. Geometric Model Acquisition • Aim: make a 3D model of a scene from two or more images taken from different viewpoints. • Why is it possible: the image differences depend in part on the shapes of the objects in the scene. Birkbeck College, U. London

  3. Two Images of the Same Scene http://vasc.ri.cmu.edu/idb/images/stereo/fruit SOURCE "University of Illinois, Bill Hoff“ DESCRIPTION "Fruit on table, digitized from 35mm." Birkbeck College, U. London

  4. Two Images of a Point in R3 Image 1 c1 • optical centre q1 • object point • p q2 • optical centre Epipolar plane: <c1,c2,x> c2• Image 2 Birkbeck College, U. London

  5. Corresponding Points • Points in different images correspond, if they are projections of the same scene point p. In projective coordinates, projection is a matrix application, Birkbeck College, U. London

  6. Method for Finding Corresponding Points Birkbeck College, U. London

  7. Example 1 of Correlation Based Matching Points in lh image =(150,100), (250,150), (350,250), (450,350), (250,450) Correlations (ρ) = 0.750, 0.685, 0.912, 0.644, 0.691 Search area = (2d+1)x(2d+1) box, d=20. Birkbeck College, U. London

  8. What Do We Need for GAM? • Description of image formation in the camera. • Description of the relative positions of the cameras. • Equations involving the measurements, the scene points and the relative positions of the cameras. • Statistical description of the errors in the measurements. Birkbeck College, U. London

  9. Pinhole Camera Small hole (optical centre) Viewing screen (image) Object Light rays Light tight box Central perspective projection model for image formation (Brunelleschi, 15th C.). Birkbeck College, U. London

  10. Camera Coordinate Frame y Y x X Z • • (0,0,-f) (0,0,0) • (X,Y,Z) Origin (0,0,0) at the pin hole. Focal length of the camera = f. Axes of image coordinate frame are parallel to X, Y axes of the CCF. Image point = (-Xf/Z, -Yf/Z) Birkbeck College, U. London

  11. Mathematical Version of the Camera Coordinate Frame Y Image plane y X x (0,0,f) • Z • (0,0,0) • • (X,Y,Z) Origin (0,0,0) at the pin hole. Focal length of the camera = f. The image is in front of the pin hole! Image point = (Xf/Z, Yf/Z). The minus signs have gone. Birkbeck College, U. London

  12. Relative Position of the Cameras R, t The relative position of the cameras is described by an orthogonal matrix R and a translation vector t. Birkbeck College, U. London

  13. Transformation of Coordinates ● p R, t If a point p has coordinates (X,Y,Z)T in the first CCF, then in the second CCF the same point p has coordinates Birkbeck College, U. London

  14. Properties of Orthogonal Matrices Birkbeck College, U. London

  15. Projection Ray Y ● X Z CCF Any scene point projecting to (x, y, f)T is on the projection ray. Birkbeck College, U. London

  16. Projection Rays of Corresponding Points 1 ● The projection rays of corresponding points intersect at a scene point. Geometric model acquisition is based on this single constraint. For an extreme example, see http://www.wisdom.weizmann.ac.il/~vision/VideoAnalysis/Demos/Traj2Traj/hall.htm Birkbeck College, U. London

  17. Projection Rays of Corresponding Points 2 ● The equations of the projection rays are known, but they hold in different coordinate systems. Birkbeck College, U. London

  18. Transformation of Coordinates Birkbeck College, U. London

  19. The Essential Matrix Birkbeck College, U. London

  20. Model Acquisition Birkbeck College, U. London

  21. Naïve Estimates of E Birkbeck College, U. London

  22. Better Way of Estimating E Birkbeck College, U. London

  23. Geometric Picture ● ● First image Second image Birkbeck College, U. London

  24. Camera Calibration Measured pixel coordinates Ideal pixel coordinates Ideal CCF Camera calibration is a transformation from measured pixel coordinates to ideal pixel coordinates. Birkbeck College, U. London

  25. Calibration Matrix Birkbeck College, U. London

  26. Fundamental Matrix The fundamental matrix F is defined by Birkbeck College, U. London

  27. Properties of E and F • det(E)=det(Tt)det(R)=0 • The matrix E is essential iff SingularValues(E) = (σ,σ,0) • det(F)=det(K~)det(E)det(K)=0 • The matrix F is fundamental iff det(F)=0. Birkbeck College, U. London

  28. Minimal Data Birkbeck College, U. London

  29. Books • D.A. Forsyth and J. Ponce. Computer Vision: a modern approach. Prentice Hall, 2003. • R.C. Gonzalez and R.E. Woods. Digital Image Processing. Second edition, Prentice Hall, 2002. Birkbeck College, U. London

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