Math Review: parameter estimation. Emma

Transcription

1 Math Review: parameter estimation Emma

2 Fitting lines to dots: We will cover how Slides provided by HyunSoo Park

3 1809, Carl Friedrich Gauss

4 What about fitting line on a curved surface?

5 Least squares methods -fitting a line - Data: (x 1, y 1 ),, (x n, y n ) y=mx+b Line equation: y i = m x i + b Find (m, b) to minimize (x i, y i ) E n i 1 ( y i m x i b ) 2 Silvio Savarese

6 0 2 2 Y X XB X db de T T 2 2 n 1 n 1 n 1 i 2 i i XB Y b m 1 x 1 x y y b m 1 x y E Normal equation n i i i b m x y E 1 2 ) ( Y X XB X T T Least squares methods -fitting a line - Y X X X B T T 1 (XB ) (XB ) Y 2(XB ) Y Y XB ) (Y XB ) (Y T T T T Silvio Savarese

7 Least squares methods -fitting a line - Ax b More equations than unknowns Look for solution which minimizes Ax-b = (Ax-b) T (Ax-b) T Solve ( Ax b) ( Ax b) 0 x i LS solution x ( A T A ) 1 A T b Silvio Savarese

8 Least squares methods What if (A t A) not invertible? -fitting a line - Solving x ( A t A ) 1 A t b A t 1 ( A A ) A t = pseudo-inverse of A A U t V = SVD decomposition of A A V 1 1 U A V U 1 with equal to for all nonzero singular values and zero otherwise Silvio Savarese

9 Singular Value Decomposition A = U D V T m x n m x n n x n n x n Column orthogonal matrix U T U =I m m V T V =I n n spans column space of A spans row space of A

10 Singular Value Decomposition A = U D V T m x n m x n n x n n x n Column orthogonal matrix U T U =I m m V T V =I n n Singular value matrix D diag{,,, } where = 1 2 n 1 2 n 0

11 Example I A = [u,d,v] = svd(a) u = d = v =

12 Example I A = u*d*v' ans = Reconstruction

13 SVD of this?

14 [u,d,v] = svd(i); semilogy(diag(d(1:20,1:20)),'x-') Im2 = u(:,1:20)*d(1:20,1:20)*v(:,1:20)';

15 U(:,1:4) D(1:4,1:4) V(:,1:4)

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18 Images as Vectors n = m n*m

19 Vector Mean n = = m I1 + I2 = mean image n*m n*m

20 Average face

21 Eigenfaces Eigenfaces look somewhat like generic faces.

22 Eigen-images of Berlin

23 Eigen-images Average of 16 induviduals trasformed via biometrical data of different ethnics

24 Average of 16 induviduals trasformed via biometrical data of different ages

25 Example II (Rotation) Z 1 e= i j k R= X 3 >> [u,d,v] = svd(r) u = v = d = Note that singular values are always one for a rotation matrix

26 Rank A = U D V T m x n m x n n x n n x n rank(a) = r min(m,n) D r Rank is the same as the number of nonzero singular values.

27 Nullspace A = U D V T m x n m x n n x n n x n null(a) = V r:n A V r:n =0

28 Example III (Fundamental Matrix) F = 1.0e+003 * [u,d,v] = svd(f) u = d = 1.0e+004 * v = Rank(F) = 2 d(1,1) ans = e+004 d(2,2) ans = d(3,3) ans = e-016

29 Matrix Inversion with SVD A = U D V T m x n m x n n x n n x n A = V D 1 U T AA=I 1 D = 1, 2,, n diag{1/ 1/ 1/ } if 0, otherwise zero. i

30 Linear Inhomogeneous Equations 1) rank(a) = r < n : infinite number of solutions = A x b 1 T x=vd Ub+ r+1 V r nvn Particular solution Homogeneous solution A=UDV T V= V1 V n where and. m x n n x 1 m x 1

31 Linear Inhomogeneous Equations = A x b m x n n x 1 m x 1 1) 2) rank(a) = r < n : infinite number of solutions 1 T x=vd Ub+ r+1 V r nvn Particular solution Homogeneous solution A=UDV T where and. rank(a) = n : exact solution x=a 1 b V= V1 V n

32 Linear Inhomogeneous Equations = A x b m x n n x 1 m x 1 1) 2) 3) rank(a) = r < n : infinite number of solutions 1 T x=vd Ub+ r+1 V r nvn Particular solution Homogeneous solution A=UDV T where and. rank(a) = n : exact solution x=a 1 b V= V1 V n n m : no exact solution in general (needs least squares) min Ax - b 2 x or x=a\b -1 T T x= A A A b in MATLAB.

33 Digression (but will become relevant) Power method algorithm, for computing eigenvalues, eigenvectors. Bob Collins

34 Two types of Least Square Problem: (i) Unconstrained affine Linear Least Squares min kax xr bk2 with kbk 6= 0 n (ii) Norm-constrained homogeneous Linear Least Squares min xr n kaxk2 s.t. kxk =1

35 Linear Homogeneous Equations Linear least square solve produces a trivial solution: T T x= A A A b An additional constraint on x=0 x to avoid the trivial solution: x =1 A x = 0 m x n n x 1 m x 1

36 Let E(x) =kaxk 2 be the cost function Let C(x) =1 kxk 2 be the constraint The lagrangian is L(x) =E(x)+C(x), with alagrangemultiplier We must have: L x = 0 and L =0

37 We get: L x =2AT Ax 2x From which: and L =1 kxk2 = C(x) A T Ax = x and C(x) =0 is an eigenvalue of A T A and x an (normalized) eigenvector is a squared singular value of A and x asingularvector Let an SVD (Singular Value Decomposition) of matrix A be: A svd UV T We have x = v i with i {1,...,n}

38 Plug x = v i back in the cost function E(x) =kaxk 2 : E(v i )=kav i k 2 = kuv T v i k 2 Since V is an orthonormal matrix, V T v i = e i (e i is a zero vector with one at the i-th entry): E(v i )=kue i k 2 = ku i e i k 2 Let U =(u 1 u n ), we get: E(v i )=k i u i k 2 = 2 i ku i k 2 Given that U is orthonormal: E(v i )= 2 i We therefore choose for x the singular vector associated to the smallest singular value, which concludes the proof.

39 Linear Homogeneous Equations Linear least square solve produces a trivial solution: T T x= A A A b An additional constraint on x=0 x to avoid the trivial solution: x =1 A x = 0 1) rank(a) = r < n-1 : infinite number of solutions n where 2 r+1 r+1 n n i i=r+1 x= V + + V 1 m x n n x 1 m x 1

40 Linear Homogeneous Equations Linear least square solve produces a trivial solution: T T x= A A A b An additional constraint on x=0 x to avoid the trivial solution: x =1 A x = 0 1) rank(a) = r < n-1 : infinite number of solutions x= V + + V r+1 r+1 n n where n 2 i i=r+1 1 m x n n x 1 m x 1 2) rank(a) = n -1 x=v n : one exact solution

41 Linear Homogeneous Equations Linear least square solve produces a trivial solution: T T x= A A A b An additional constraint on x=0 x to avoid the trivial solution: x =1 A x = 0 1) rank(a) = r < n-1 : infinite number of solutions x= V + + V r+1 r+1 n n where n 2 i i=r+1 1 m x n n x 1 m x 1 2) 3) rank(a) = n -1 x=v n : one exact solution n m : no exact solution in general (needs least squares) min Ax 2 x subject to x =1 x=v n

42 Least squares methods -fitting a line - E n i 1 ( y i m x i b ) 2 y=mx+b B T 1 T X X X Y m B b (x i, y i ) Limitations Not rotation-invariant Fails completely for vertical lines

43 Least squares methods -fitting a line - Distance between point (x n, y n ) and line ax+by=d ax+by=d Find (a, b, d) to minimize the sum of squared perpendicular distances E n i 1 ( a x i b y i d ) 2 (x i, y i ) ( U T U ) N 0 data model parameters

44 Least squares methods -fitting a line - A h = 0 Minimize A h subject to h 1 A UDV T h last column of V

45 (i) Unconstrained affine Linear Least Squares min kax xr bk2 with kbk 6= 0 n Solution with the pseudo-inverse x = A b Closed-form expression (for well-posed problems) A = A T A 1 A T Regularized expression (for all cases) A = V U T with A = UV T an SVD contains the reciprocal singular values = 1 if >², 0otherwise (ii) Norm-constrained homogeneous Linear Least Squares min xr kaxk2 s.t. kxk =1 n x is given by the last column of V

46 Homography Linear Estimation H h h h h h h h h h x =Hx 2 1 x u2 v x u1 v 1 1 1

47 Homography Linear Estimation x =Hx 2 1 x2 Hx1 0 u2 h1 v2h3x1 h2x1 v2 h2 x1 hx 1 1 u2h3x1 0 1 h u h x v hx T T T 0 13 x1 v 2x1 h1 T T T x u 2x1h20 T T T v2x1 u2x h3 3 x 9 because x 2 rank( ) = 2 is a rank 2 matrix. x u1 v Therefore, 4 point correspondences are required to estimate a homography.

48 Fundamental Matrix Estimation x u2 v T xfx x u1 v 1 1 1

49 Fundamental Matrix Estimation T xfx T u2 f11 f12 f13u1 v2 f21 f22 f23 v1 0 1 f f f uuf 2 111uvf 2 112uf 2 13vuf 2 121vvf 2 122vf 2 23uf 131vf 132f330 f11 f12 9 f 13 uu 2 1 uv 2 1 u2 vu 2 1 vv 2 1 v2 u1 v1 1 f21 f22 0 f23 f 31 f32 f33 Therefore, 8 point correspondences are required to estimate a fundamental matrix.

50 Point Triangulation X ( X,Y,Z ) Given camera poses, reconstruct 3D points. x X x X =P 1 1 P x P 1 X 0 1 x rank( P) = At least 2 views are required to reconstruct a 3D point.

51 Least squares: Robustness to noise outlier! Problem: squared error heavily penalizes outliers Silvio Savarese

52 Fitting Goal: Choose a parametric model to fit a certain quantity from data Techniques: Least square methods RANSAC Hough transform Silvio Savarese

53 Basic philosophy (voting scheme) Data elements are used to vote for one (or multiple) models Robust to outliers and missing data Assumption1: Noise features will not vote consistently for any single model ( few outliers) Assumption2: there are enough features to agree on a good model ( few missing data) Silvio Savarese

54 RANSAC (RANdom SAmple Consensus) : Learning technique to estimate parameters of a model by random sampling of observed data Fischler & Bolles in 81. f : I P, such that: ( P, ) O min O f ( P, ) Model parameters T 1 T P P P Silvio Savarese

55 RANSAC Sample set = set of points in 2D Algorithm: 1. Select random sample of minimum required size to fit model 2. Compute a putative model from sample set 3. Compute the set of inliers to this model from whole data set Repeat 1-3 until model with the most inliers over all samples is found Silvio Savarese

56 RANSAC Sample set = set of points in 2D Algorithm: 1. Select random sample of minimum required size to fit model [?] =[2] 2. Compute a putative model from sample set 3. Compute the set of inliers to this model from whole data set Repeat 1-3 until model with the most inliers over all samples is found Silvio Savarese

57 RANSAC Sample set = set of points in 2D Algorithm: 1. Select random sample of minimum required size to fit model [?] =[2] 2. Compute a putative model from sample set 3. Compute the set of inliers to this model from whole data set Repeat 1-3 until model with the most inliers over all samples is found Silvio Savarese

58 RANSAC Sample set = set of points in 2D Algorithm: O = Select random sample of minimum required size to fit model [?] =[2] 2. Compute a putative model from sample set 3. Compute the set of inliers to this model from whole data set Repeat 1-3 until model with the most inliers over all samples is found Silvio Savarese

59 RANSAC (RANdom SAmple Consensus) : Fischler & Bolles in 81. Algorithm: O = 6 1. Select random sample of minimum required size to fit model [?] 2. Compute a putative model from sample set 3. Compute the set of inliers to this model from whole data set Repeat 1-3 until model with the most inliers over all samples is found Silvio Savarese

60 How many samples? Number of samples N Choose N so that, with probability p, at least one random sample is free from outliers (e.g. p=0.99) (outlier ratio: e ) Initial number of points s Typically minimum number needed to fit the model Distance threshold Choose so probability for inlier is p (e.g. 0.95) Zero-mean Gaussian noise with std. dev. : t 2 = N log 1 p/ log1 1 e s proportion of outliers e s 5% 10% 20% 25% 30% 40% 50% Source: M. Pollefeys

61 Estimating H by RANSAC Silvio Savarese H 8 DOF Need 4 correspondences Sample set = set of matches between 2 images Algorithm: 1. Select a random sample of minimum required size [?] 2. Compute a putative model from these 3. Compute the set of inliers to this model from whole sample space Repeat 1-3 until model with the most inliers over all samples is found

62 Estimating F by RANSAC Silvio Savarese Outlier matches F 7 DOF Need 7 (8) correspondences Sample set = set of matches between 2 images Algorithm: 1. Select a random sample of minimum required size [?] 2. Compute a putative model from these 3. Compute the set of inliers to this model from whole sample space Repeat 1-3 until model with the most inliers over all samples is found

63 RANSAC - conclusions Good: Simple and easily implementable Successful in different contexts Bad: Many parameters to tune Trade-off accuracy-vs-time Cannot be use if ratio inliers/outliers is too small Silvio Savarese

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