Introduction to Computer Vision

Transcription

1 Introduction to Computer Vision Michael J. Black Nov 2009 Perspective projection and affine motion

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3 Goals Today Perspective projection 3D motion Wed Projects Friday Regularization and robust statistics

4 Reading Szeliski 2.2.2: 2D tranformations including affine 2.1.3: 3D transformations including rotation matrices (more detailed than we need) 2.1.4: 3D to 2D projections

5 Affine Flow

6 Why Affine? Where does this affine approximation come from? All our models are approximations to the world. What are the assumptions in the affine approximation? For this we need some geometry.

7 Pinhole cameras Abstract camera model - box with a small hole in it. Easy to build but needs a lot of light.

8 The equation of projection Image plane

9 Perspective Projection Y O Move image plane in front of focal point (image not inverted) Camera coordinate frame X (x,y ) (x,y,z) Many to one Optical Axis Z f

10 Perspective Projection Y (x,y,z) X (x,y ) Short focal length means wider field of view. O Z f

11 Perspective Projection Y (x,y,z) X r (x,y ) r O f Z

12 Perspective Projection Y (x,y,z) X r (x,y ) r O f z Z

13 Perspective Projection Y (x,y,z) X r (x,y ) r O f z Z

14 (important slide) Perspective Projection Y = f z (x,y,z) x' = fx z (x,y ) y' = fy z X r r O Z f z

15 Distant objects are smaller

16 Orthographic Projection Y X Z Assume an infinite focal length and that the world is infinitely far away.

17 Weak Perspective (scaled orthography) Y X Assume variation in depth is small relative to the distance from the camera. Approximate scene as a fronto-parallel plane Z

18 Weak Perspective (scaled orthography) Y X Z

19 Claim For small motions, affine flow approximates the motion of a plane viewed under orthographic projection. u = v x = a 1 x + a 2 y + a 3 v = v y = a 4 x + a 5 y + a 6 u v = x y x y 1 a 1 a 2 a 3 a 4 a 5 a 6

20 Recall: 3D motion

21 Motion Models 3D Rigid Motion

22 Review: Ch 2.1.3, Euclidean Geometry Rotation Rotation matrix: orthogonal with determinant =1 R T = R 1 det(r) = 1

23 Review: Ch 2.1.3, Euclidean Geometry Rotation

24 Assumption: Small Motion (If is small)

25 3D Motion

26 Assumption: Planar World + Orthographic Projection

27 Assumption: Planar World Substitute: a 1 = Y b a 3 = Y a + T X u = v x = a 1 x + a 2 y + a 3 v = v y = a 4 x + a 5 y + a 6 a 2 = Y c Z a 4 = Z X b a 5 = X c a 6 = T Y X a

28 Affine Flow Small motion assumption * e.g. at video frame rate Planar surface * look at only a small region of the scene Orthographic projection * surface distant from camera * long focal length u = v x = a 1 x + a 2 y + a 3 v = v y = a 4 x + a 5 y + a 6

29 Assumptions What might be wrong with this? Is there a probabilistic interpretation? Minimize the negative log.

30 Multiple Motions

31 Occlusion occlusion disocclusion shear Multiple motions within a finite region.

32 Coherent Motion Possibly Gaussian.

33 Multiple Motions Definitely not Gaussian.

34 Multiple Motions What is the best fitting translational motion?

35 Multiple Motions X Least squares fit.

36 Simpler problem: fitting a line to data y x

37 y x

38 y x

39 y x

40 Robust Statistics Recover the best fit to the majority of the data. Detect and reject outliers. History.

41 Estimating the mean Gaussian distribution Mean is the optimal solution to: residual

42 Estimating the Mean The mean maximizes this likelihood: The negative log gives (with sigma=1): least squares estimate

43 Estimating the mean

44 Estimating the mean What happens if we change just one measurement? With a single bad data point I can move the mean arbitrarily far.