The end of affine cameras

Transcription

1 The end of affine cameras Affine SFM revisited Epipolar geometry Two-view structure from motion Multi-view structure from motion Planches :

2 Weak-Perspective Projection Model r (p and P are in homogeneous coordinates) p = M P (P is in homogeneous coordinates) p = A P + b (neither p nor P is in hom. coordinates)

3 Affine Spaces: (Semi-Formal) Definition

4 Affine projections induce affine transformations from planes onto their images.

5 Affine Structure from Motion Reprinted with permission from Affine Structure from Motion, by J.J. (Koenderink and A.J.Van Doorn, Journal of the Optical Society of America A, 8: (1990) Optical Society of America. Given m pictures of n points, can we recover the three-dimensional configuration of these points? (structure) the camera configurations? (motion)

6 Geometric affine scene reconstruction from two images (Koenderink and Van Doorn, 1991).

7 Epipolar Geometry Epipolar Plane Baseline Epipoles Epipolar Lines

8 The Affine Epipolar Constraint Note: the epipolar lines are parallel.

9 Affine Epipolar Geometry

10 The Affine Fundamental Matrix where

11 Data courtesy of R. Mohr and B. Boufama.

12 Estimation of affine epipolar geometry Mean errors: 3.24 and 3.15pixel

13 The Affine Structure-from-Motion Problem Given m images of n fixed points P we can write j Problem: estimate the m 2x4 matrices M i and the n positions P from the mn correspondences p. j 2mn equations in 8m+3n unknowns ij Overconstrained problem, that can be solved using (non-linear) least squares!

14 The Affine Ambiguity of Affine SFM When the intrinsic and extrinsic parameters are unknown If M and P are solutions, i j So are M and P where i j and Q is an affine transformation.

15 An Affine Trick..

16 The Affine Epipolar Constraint Note: the epipolar lines are parallel.

17 An Affine Trick.. Algebraic Scene Reconstruction Method

18 Affine reconstruction. Mean relative error: 3.2%

19 The Affine Structure of Affine Images Suppose we observe a static scene with m fixed cameras.. The set m-tuples of all image points in a scene is a 3D affine space!

20 When do m+1 points define a p-dimensional subspace Y of an n-dimensional affine space X equipped with some coordinate frame basis? Rank ( D ) = p+1, where Writing that all minors of size (p+2)x(p+2) of D are equal to zero gives the equations of Y. has rank 4!

21 From Affine to Vectorial Structure Idea: pick one of the points (or their center of mass) as the origin.

22 What if we could factorize D? (Tomasi and Kanade, 1992) Affine SFM is solved!

23 Singular Value Decomposition

24 Singular Value Decomposition square roots of

25 Singular Value Decomposition

26 Singular Value Decomposition

27 What if we could factorize D? (Tomasi and Kanade, 1992) Affine SFM is solved! Singular Value Decomposition We can take

28 Affine reconstruction. Mean relative error: 2.8%

29 Projective cameras Motivation Elements of Projective Geometry Projective structure from motion

30 The Projective Structure-from-Motion Problem Given m perspective images of n fixed points P we can write j Problem: estimate the m 3x4 matrices i M and the n positions P from the mn correspondences p. j 2mn equations in 11m+3n unknowns ij Overconstrained problem, that can be solved using (non-linear) least squares!

31 The Projective Ambiguity of Projective SFM When the intrinsic and extrinsic parameters are unknown If M and P are solutions, i j So are M and P where i j and Q is an arbitrary non-singular 4x4 matrix. Q is a projective transformation.

32 Projective Spaces: (Semi-Formal) Definition

33 3 A Model of P( R )

34 Projective Subspaces and Projective Coordinates

35 Projective Subspaces and Projective Coordinates P Projective coordinates

36 Projective Subspaces Given a choice of coordinate frame Line: Plane:

37 Hyperplanes and duality Consider n+1 points P 0,, P n-1, P in a projective space X of dimension n. They lie in the same hyperplane when Det(D)=0. D = x x... x n x x... x n x0 x 1... xn This can be rewritten as u 0 x 0 +u 1 x 1 + +u n x n = 0, or Π T P = 0, where Π = (u 0,u 1,,u n ) T. Hyperplanes form a dual projective space X * of X, and any theorem that holds for points in X holds for hyperplanes in X *. What is the dual of a straight line?

38 Affine and Projective Spaces

39 Affine and Projective Coordinates

40 Affine and Projective Coordinates