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From last time …

From last time …. Rodrigues’ formula. Beautiful! Toss Euler angles, Y-P-R, quaternions etc. Check Matlab code in laboratory exercises. Homogeneous representation. Points Vectors Transformation representation. Exponential coordinates for rigid body motions.

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From last time …

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  1. From last time … UCLA Vision Lab

  2. Rodrigues’ formula • Beautiful! • Toss Euler angles, Y-P-R, quaternions etc. • Check Matlab code in laboratory exercises UCLA Vision Lab

  3. Homogeneous representation • Points • Vectors • Transformation • representation UCLA Vision Lab

  4. Exponential coordinates for rigid body motions UCLA Vision Lab

  5. Rodrigues’ formula for rigid body motion UCLA Vision Lab

  6. Summary UCLA Vision Lab

  7. Lecture 4: Image formation UCLA Vision Lab

  8. Image formation (Chapter 3) UCLA Vision Lab

  9. Representation of images UCLA Vision Lab

  10. Lenses UCLA Vision Lab

  11. Pinhole model UCLA Vision Lab

  12. Forward pinhole UCLA Vision Lab

  13. Lambertian reflection • Radiance is independent of viewpoint UCLA Vision Lab

  14. Preview of coming attractions • How to go from measurements of light to estimates of geometric primitives? UCLA Vision Lab

  15. Geometric image formation • Coordinate transformations between camera frame and world frame • Perspective projection from 3D to 2D • Coordinate transformation in image coordinate frame UCLA Vision Lab

  16. Ideal geometric camera UCLA Vision Lab

  17. Ideal geometric camera (contd.) UCLA Vision Lab

  18. Intrinsic parameters UCLA Vision Lab

  19. Intrinsic parameters (contd.) • Skew pixels • Overall intrinsic parameter matrix UCLA Vision Lab

  20. CAMERA PARAMETERS – Radial Distortion Nonlinear transformation along the radial direction Distortion correction: make lines straight UCLA Vision Lab

  21. Radial distortion • Will assume compensated (Tsai ’86) see Intel OpenCV in lab assignment UCLA Vision Lab

  22. Overall projection model UCLA Vision Lab

  23. Overall projection model (contd.) UCLA Vision Lab

  24. Calibrated vs. non-calibrated camera UCLA Vision Lab

  25. IMAGE FORMATION – Image of a Point Homogeneous coordinates of a 3-D point Homogeneous coordinates of its 2-D image Projection of a 3-D point to an image plane UCLA Vision Lab

  26. NOTATION – Image, Coimage, Preimage of a Point Image of a 3-D point Coimage of the point Preimage of the point UCLA Vision Lab

  27. NOTATION – Image, Coimage, Preimage of a Line Coimage of a 3-D line Preimage of the line Image of the line UCLA Vision Lab

  28. IMAGE FORMATION – Coimage of a Line Homogeneous representation of a 3-D line Homogeneous representation of its 2-D coimage Projection of a 3-D line to an image plane UCLA Vision Lab

  29. IMAGE FORMATION – Multiple Images “Preimages” are all incident at the corresponding features. . . . UCLA Vision Lab

  30. SUMMARY UCLA Vision Lab

  31. 3D EUCLIDEAN SPACE - Cartesian Coordinate Frame Coordinates of a point in space: Standard base vectors: UCLA Vision Lab

  32. 3D EUCLIDEAN SPACE - Vectors A “free” vector is defined by a pair of points : Coordinates of the vector : UCLA Vision Lab

  33. 3D EUCLIDEAN SPACE – Inner Product and Cross Product Inner product between two vectors: Cross product between two vectors: UCLA Vision Lab

  34. RIGID-BODY MOTION – Rotation Rotation matrix: Coordinates are related by: UCLA Vision Lab

  35. RIGID-BODY MOTION – Rotation and Translation Coordinates are related by: UCLA Vision Lab

  36. RIGID-BODY MOTION – Homogeneous Coordinates 3D coordinates are related by: Homogeneous coordinates are related by: Homogeneous coordinates of a vector: UCLA Vision Lab

  37. IMAGE FORMATION – Lens, Light, and Surfaces image irradiance surface radiance BRDF Lambertian thin lens small FOV UCLA Vision Lab

  38. IMAGE FORMATION – Pinhole Camera Model Pinhole Frontal pinhole UCLA Vision Lab

  39. IMAGE FORMATION – Pinhole Camera Model 2D coordinates Homogeneous coordinates UCLA Vision Lab

  40. CAMERA PARAMETERS – Pixel Coordinates calibrated coordinates Linear transformation pixel coordinates UCLA Vision Lab

  41. CAMERA PARAMETERS – Calibration Matrix and Camera Model Ideal pinhole Pixel coordinates Calibration matrix (intrinsic parameters) Projection matrix Camera model UCLA Vision Lab

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