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# 15 cv mil_models_for_transformations

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• 1. Computer vision: models, learning and inference Chapter 15 Models for transformations Please send errata to s.prince@cs.ucl.ac.uk
• 2. Structure• Transformation models• Learning and inference in transformation models• Three problems – Exterior orientation – Calibration – Reconstruction• Properties of the homography• Robust estimation of transformations• Applications Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 2
• 3. Transformation models• Consider viewing a planar scene• There is now a 1 to 1 mapping between points on the plane and points in the image• We will investigate models for this 1 to 1 mapping – Euclidean – Similarity – Affine transform – Homography Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 3
• 4. Motivation: augmented reality tracking Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 4
• 5. Euclidean Transformation• Consider viewing a fronto-parallel plane at a known distance D.• In homogeneous coordinates the imaging equations are: 3D rotation matrix Plane at known Point is on plane becomes 2D (in plane) distance D (w=0) Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 5
• 6. Euclidean transformation• Simplifying• Rearranging the last equation Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 6
• 7. Euclidean transformation• Simplifying• Rearranging the last equation Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 7
• 8. Euclidean transformation• Pre-multiplying by inverse of (modified) intrinsic matrix Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 8
• 9. Euclidean transformation Homogeneous: Cartesian:or for short: For short: Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 9
• 10. Similarity Transformation• Consider viewing fronto-parallel plane at unknown distance D• By same logic as before we have• Premultiplying by inverse of intrinsic matrix Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 10
• 11. Similarity TransformationSimplifying:Multiply each equation by : Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 11
• 12. Similarity TransformationSimplifying:Incorporate the constants by defining: Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 12
• 13. Similarity Transformation Homogeneous: Cartesian: For short: Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 13
• 14. Affine Transformation Homogeneous: Cartesian: For short:Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 14
• 15. Affine TransformAffine transform describes mapping well When variation in depth is comparable to when the depth variation within the distance to object then the affineplanar object is small and the camera is transformation is not a good model. Here far away we need the homography Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 15
• 16. Projective transformation / collinearity / homographyStart with basic projection equation:Combining these two matrices we get: Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 16
• 17. Homography Homogeneous: Cartesian: For short:Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 17
• 18. Adding noise• In the real world, the measured image positions are uncertain.• We model this with a normal distribution• e.g. Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 18
• 19. Structure• Transformation models• Learning and inference in transformation models• Three problems – Exterior orientation – Calibration – Reconstruction• Properties of the homography• Robust estimation of transformations• Applications Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 19
• 20. Learning and inference problems• Learning – take points on plane and their projections into the image and learn transformation parameters• Inference – take the projection of a point in the image and establish point on plane Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 20
• 21. Learning and inference problems Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 21
• 22. Learning transformation models• Maximum likelihood approach• Becomes a least squares problem Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 22
• 23. Learning Euclidean parametersSolve for transformation:Remaining problem: Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 23
• 24. Learning Euclidean parametersHas the general form:• This is an orthogonal Procrustes problem. To solve: • Compute SVD • And then set Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 24
• 25. Learning similarity parameters• Solve for rotation matrix as before• Solve for translation and scaling factor Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 25
• 26. Learning affine parameters• Affine transform is linear• Solve using least-squares solution Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 26
• 27. Learning homography parameters• Homography is not linear – cannot be solved in closed form. Convert to other homogeneous coordinates• Both sides are 3x1 vectors; should be parallel and so cross product will be zero Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 27
• 28. Learning homography parameters• Write out these equations in full• There are only 2 independent equations here – use a minimum of four points to build up a set of equations Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 28
• 29. Learning homography parameters• These equations have the form , which we need to solve with the constraint• This is a minimum direction problem – Compute SVD – Take last column of• Then use non-linear optimization Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 29
• 30. Inference problems• Given point x* in image, find position w* on object Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 30
• 31. InferenceIn the absence of noise, we have the relation ,or in homogeneous coordinatesTo solve for the points, we simply invert this relation Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 31
• 32. Structure• Transformation models• Learning and inference in transformation models• Three problems – Exterior orientation – Calibration – Reconstruction• Properties of the homography• Robust estimation of transformations• Applications Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 32
• 33. Problem 1: exterior orientation Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 33
• 34. Problem 1: exterior orientation• Writing out the camera equations in full• Estimate the homography from matched points• Factor out the intrinsic parameters Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 34
• 35. Problem 1: exterior orientation• To estimate the first two columns of rotation matrix, we compute this singular value decomposition• Then we set Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 35
• 36. Problem 1: exterior orientation• Find the last column using the cross product of first two columns• Make sure the determinant is 1 – if it is -1 then multiply last column by -1.• Find translation scaling factor between old and new values• Finally, set Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 36
• 37. Structure• Transformation models• Learning and inference in transformation models• Three problems – Exterior orientation – Calibration – Reconstruction• Properties of the homography• Robust estimation of transformations• Applications Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 37
• 38. Problem 2: calibrationComputer vision: models, learning and inference. ©2011 Simon J.D. Prince 38
• 39. StructureComputer vision: models, learning and inference. ©2011 Simon J.D. Prince 39
• 40. CalibrationOne approach (not very efficient) is to alternately• Optimize extrinsic parameters for fixed intrinsic• Optimize intrinsic parameters for fixed extrinsic Then use non-linear optimization. Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 40
• 41. Structure• Transformation models• Learning and inference in transformation models• Three problems – Exterior orientation – Calibration – Reconstruction• Properties of the homography• Robust estimation of transformations• Applications Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 41
• 42. Problem 3 - reconstructionTransformation between plane and image:Point in frame of reference of plane:Point in frame of reference of camera Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 42
• 43. Structure• Transformation models• Learning and inference in transformation models• Three problems – Exterior orientation – Calibration – Reconstruction• Properties of the homography• Robust estimation of transformations• Applications Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 43
• 44. Transformations between images• So far we have considered transformations between the image and a plane in the world• Now consider two cameras viewing the same plane• There is a homography between camera 1 and the plane and a second homography between camera 2 and the plane• It follows that the relation between the two images is also a homography Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 44
• 45. Properties of the homographyHomography is a linear transformation of a rayEquivalently, leave rays and linearly transform image plane – all images formed byall planes that cut the same ray bundle are related by homographies. Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 45
• 46. Camera under pure rotationSpecial case is camera under pure rotation.Homography can be showed to be Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 46
• 47. Structure• Transformation models• Learning and inference in transformation models• Three problems – Exterior orientation – Calibration – Reconstruction• Properties of the homography• Robust estimation of transformations• Applications Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 47
• 48. Robust estimation Least squares criterion is not robust to outliers For example, the two outliers here cause the fitted line to be quite wrong. One approach to fitting under these circumstances is to use RANSAC – “Random sampling by consensus”Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 48
• 49. RANSACComputer vision: models, learning and inference. ©2011 Simon J.D. Prince 49
• 50. RANSACComputer vision: models, learning and inference. ©2011 Simon J.D. Prince 50
• 51. Fitting a homography with RANSAC Original images Initial matches Inliers from RANSAC Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 51
• 52. Piecewise planarityMany scenes are not planar, but are nonetheless piecewise planar Can we match all of the planes to one another? Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 52
• 53. Approach 1 – Sequential RANSAC Problems: greedy algorithm and no need to be spatially coherent Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 53
• 54. Approach 2 – PEaRL (propose, estimate and re-learn)• Associate label l which indicates which plane we are in• Relation between points xi in image1 and yi in image 2• Prior on labels is a Markov random field that encourages nearby labels to be similar• Model solved with variation of alpha expansion algorithm Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 54
• 55. Approach 2 – PEaRLComputer vision: models, learning and inference. ©2011 Simon J.D. Prince 55
• 56. Structure• Transformation models• Learning and inference in transformation models• Three problems – Exterior orientation – Calibration – Reconstruction• Properties of the homography• Robust estimation of transformations• Applications Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 56
• 57. Augmented reality tracking Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 57
• 58. Fast matching of keypoints Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 58
• 59. Visual panoramasComputer vision: models, learning and inference. ©2011 Simon J.D. Prince 59
• 60. Conclusions• Mapping between plane in world and camera is one-to-one• Takes various forms, but most general is the homography• Revisited exterior orientation, calibration, reconstruction problems from planes• Use robust methods to estimate in the presence of outliers Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 60