1	Global Alignment and Structure from Motion Computer Vision CSE576, Spring 2005 Richard Szeliski
2	Today’s lecture Rotational alignment (“3D stitching”) [Project 3] pairwise alignment (Procrustes) global alignment (linearized least squares) Calibration camera matrix (Direct Linear Transform) non-linear least squares separating intrinsics and extrinsics focal length and optic center
3	Today’s lecture Structure from Motion triangulation and pose two-frame methods factorization bundle adjustment robust statistics
4	Global rotational alignment Fully Automated Panoramic Stitching [Project 3]
5	AutoStitch [Brown & Lowe’03] Stitch panoramic image from an arbitrary collection of photographs (known focal length) Extract and (pairwise) match features Estimate pairwise rotations using RANSAC Add to stitch and re-run global alignment Warp images to sphere and blend
6	3D Rotation Model Projection equations Project from image to 3D ray (x₀,y₀,z₀) = (u₀-u_c,v₀-v_c,f) Rotate the ray by camera motion (x₁,y₁,z₁) = R₀₁ (x₀,y₀,z₀) Project back into new (source) image (u₁,v₁) = (fx₁/z₁+u_c,fy₁/z₁+v_c)
7	Pairwise alignment Absolute orientation [Arun et al., PAMI 1987] [Horn et al., JOSA A 1988], Procrustes Algorithm [Golub & VanLoan] Given two sets of matching points, compute R p_i’ = R p_iwith 3D rays p_i = N(x_i,y_i,z_i) = N(u_i-u_c,v_i-v_c,f) A = Σ_ip_i p_i’^T = Σ_ip_i p_i^T R^T = U S V^T= (U S U^T)R^T V^T= U^TR^T R = VU^T
8	Pairwise alignment RANSAC loop: Select two feature pairs (at random) p_i = N(u_i-u_c,v_i-v_c,f ), p_i’ = N(u_i’-u_c,v_i’-v_c,f ), i=0,1 Compute outer product matrix A = Σ_ip_i p_i’^T Compute R using SVD, A = U S V^T_,R = VU^T Compute inliers where f \|p_i’ - R p_i\| < ε Keep largest set of inliers Re-compute least-squares SVD estimate on all of the inliers, i=0..n
9	Automatic stitching Match all pairs and keep the good ones (# inliers > threshold) Sort pairs by strength (# inliers) Add in next strongest match (and other relevant matches) to current stitch Perform global alignment
10	Incremental selection & addition [3] [4] (3,4) (4,3) [2] (2,4) (4,2) (2,3) (3,2) [1] (1,2) (2,1) (1,4) (4,1) (1,3) (3,1) [5] (5,3) (3,5) (4,5) (5,4)
11	Global alignment Task: Compute globally consistent set of rotations {R_i} such that R_jp_ij ≈ R_kp_ik or min \|R_jp_ij - R_kp_ik\|² Initialize “first” frame R_i = I Multiply “next” frame by pairwise rotation R_ij Globally update all of the current {R_i} Q: How to parameterize and update the {R_i} ?
12	Parameterizing rotations How do we parameterize R and ΔR? Euler angles: bad idea quaternions: 4-vectors on unit sphere use incremental rotation R(I + DR) update with Rodriguez formula
13	Global alignment Least-squares solution of min \|R_jp_ij - R_kp_ik\|²orR_jp_ij - R_kp_ik = 0 Use the linearized update (I+[ω_j]_´)R_jp_ij - (I+[ω_k]_´) R_kp_ik = 0 or [q_ij]_´ω_j- [q_ik]_´ω_k = q_ij-q_ik, q_ij= R_jp_ij Estimate least square solution over {ω_i} Iterate a few times (updating the {R_i})
14	Iterative focal length adjustment (Optional) [Szeliski & Shum’97; MSR-TR-03] Simplest approach: arg min_f f \|R_jp_ij - R_kp_ik\|² More complex approach: full bundle adjustment (op. cit. & later in talk)
15	Camera Calibration
16	Camera calibration Determine camera parameters from known 3D points or calibration object(s) internal or intrinsic parameters such as focal length, optical center, aspect ratio: what kind of camera? external or extrinsic (pose) parameters: where is the camera? How can we do this?
17	Camera calibration – approaches Possible approaches: linear regression (least squares) non-linear optimization vanishing points multiple planar patterns panoramas (rotational motion)
18	Image formation equations
19	Calibration matrix Is this form of K good enough? non-square pixels (digital video) skew radial distortion
20	Camera matrix Fold intrinsic calibration matrix K and extrinsic pose parameters (R,t) together into a camera matrix M = K [R \| t ] (put 1 in lower r.h. corner for 11 d.o.f.)
21	Camera matrix calibration Directly estimate 11 unknowns in the M matrix using known 3D points (X_i,Y_i,Z_i) and measured feature positions (u_i,v_i)
22	Camera matrix calibration Linear regression: Bring denominator over, solve set of (over-determined) linear equations. How? Least squares (pseudo-inverse) Is this good enough?
23	Levenberg-Marquardt Iterative non-linear least squares [Press’92] Linearize measurement equations Substitute into log-likelihood equation: quadratic cost function in Dm
24	Levenberg-Marquardt Iterative non-linear least squares [Press’92] Solve for minimum Hessian: error:
25	Levenberg-Marquardt What if it doesn’t converge? Multiply diagonal by (1 + l), increase l until it does Halve the step size Dm (my favorite) Use line search Other ideas? Uncertainty analysis: covariance S = A^-1 Is maximum likelihood the best idea? How to start in vicinity of global minimum?
26	Camera matrix calibration Advantages: very simple to formulate and solve can recover K [R \| t] from M using QR decomposition [Golub & VanLoan 96] Disadvantages: doesn't compute internal parameters more unknowns than true degrees of freedom need a separate camera matrix for each new view
27	Separate intrinsics / extrinsics New feature measurement equations Use non-linear minimization Standard technique in photogrammetry, computer vision, computer graphics [Tsai 87] – also estimates k₁ (freeware @ CMU) http://www.cs.cmu.edu/afs/cs/project/cil/ftp/html/v-source.html [Bogart 91] – View Correlation
28	Intrinsic/extrinsic calibration Advantages: can solve for more than one camera pose at a time potentially fewer degrees of freedom Disadvantages: more complex update rules need a good initialization (recover K [R \| t] from M)
29	Vanishing Points Determine focal length f and optical center (u_c,v_c) from image of cube’s (or building’s) vanishing points [Caprile ’90][Antone & Teller ’00]
30	Vanishing point calibration Advantages: only need to see vanishing points (e.g., architecture, table, …) Disadvantages: not that accurate need rectahedral object(s) in scene
31	Multi-plane calibration Use several images of planar target held at unknown orientations [Zhang 99] Compute plane homographies Solve for K^-TK^-1 from H_k’s 1plane if only f unknown 2 planes if (f,u_c,v_c) unknown 3+ planes for full K Code available from Zhang and OpenCV
32	Rotational motion Use pure rotation (large scene) to estimate f estimate f from pairwise homographies re-estimate f from 360º “gap” optimize over all {K,R_j} parameters [Stein 95; Hartley ’97; Shum & Szeliski ’00; Kang & Weiss ’99] Most accurate way to get f, short of surveying distant points
33	Pose estimation and triangulation
34	Pose estimation Once the internal camera parameters are known, can compute camera pose [Tsai87] [Bogart91] Application: superimpose 3D graphics onto video How do we initialize (R,t)?
35	Pose estimation Previous initialization techniques: vanishing points [Caprile 90] planar pattern [Zhang 99] Other possibilities Through-the-Lens Camera Control [Gleicher92]: differential update 3+ point “linear methods”: [DeMenthon 95][Quan 99][Ameller 00]
36	Triangulation Problem: Given some points in correspondence across two or more images (taken from calibrated cameras), {(u_j,v_j)}, compute the 3D location X
37	Triangulation Method I: intersect viewing rays in 3D, minimize: X is the unknown 3D point C_j is the optical center of camera j V_j is the viewing ray for pixel (u_j,v_j) s_j is unknown distance along V_j Advantage: geometrically intuitive
38	Triangulation Method II: solve linear equations in X advantage: very simple Method III: non-linear minimization advantage: most accurate (image plane error)
39	Structure from Motion
40	Structure from motion Given many points in correspondence across several images, {(u_ij,v_ij)}, simultaneously compute the 3D location x_i and camera (or motion) parameters (K, R_j, t_j) Two main variants: calibrated, and uncalibrated (sometimes associated with Euclidean and projective reconstructions)
41	Structure from motion How many points do we need to match? 2 frames: (R,t): 5 dof + 3n point locations £ 4n point measurements Þ n ³ 5 k frames: 6(k–1)-1 + 3n £ 2kn always want to use many more
42	Two-frame methods Two main variants: Calibrated: “Essential matrix” E use ray directions (x_i, x_i’ ) Uncalibrated: “Fundamental matrix” F [Hartley & Zisserman 2000]
43	Essential matrix Co-planarity constraint: x’ ≈ R x + t [t]_´ x’ ≈ [t]_´ R x x’^T [t]_´ x’ ≈ x’ ^T [t]_´ R x x’ ^T E x = 0 with E =[t]_´ R Solve for E using least squares (SVD) t is the least singular vector of E R obtained from the other two s.v.s
44	Fundamental matrix Camera calibrations are unknown x’ F x = 0 with F = [e]_´ H = K’[t]_´ R K^-1 Solve for F using least squares (SVD) re-scale (x_i, x_i’ ) so that \|x_i\|≈1/2 [Hartley] e (epipole) is still the least singular vector of F H obtained from the other two s.v.s “plane + parallax” (projective) reconstruction use self-calibration to determine K [Pollefeys]
45	Multi-frame Structure from Motion
46	Factorization [Tomasi & Kanade, IJCV 92]
47	Structure [from] Motion Given a set of feature tracks, estimate the 3D structure and 3D (camera) motion. Assumption: orthographic projection Tracks: (u_fp,v_fp), f: frame, p: point Subtract out mean 2D position… u_fp = i_f^Ts_p i_f: rotation, s_p: position v_fp = j_f^Ts_p
48	Measurement equations Measurement equations u_fp = i_f^Ts_p i_f: rotation, s_p: position v_fp = j_f^Ts_p Stack them up… W = R S R = (i₁,…,i_F, j₁,…,j_F)^T S = (s₁,…,s_P)
49	Factorization W = R₂_F_´₃ S₃_´_P SVD W = U Λ V Λ must be rank 3 W’ = (U Λ ^1/2)(Λ^1/2 V) = U’ V’ Make R orthogonal R = QU’ , S = Q^-1V’ i_f^TQ^TQi_f = 1 …
50	Results Look at paper figures…
51	Extensions Paraperspective [Poelman & Kanade, PAMI 97] Sequential Factorization [Morita & Kanade, PAMI 97] Factorization under perspective [Christy & Horaud, PAMI 96] [Sturm & Triggs, ECCV 96] Factorization with Uncertainty [Anandan & Irani, IJCV 2002]
52	Bundle Adjustment What makes this non-linear minimization hard? many more parameters: potentially slow poorer conditioning (high correlation) potentially lots of outliers gauge (coordinate) freedom
53	Lots of parameters: sparsity Only a few entries in Jacobian are non-zero
54	Sparse Cholesky (skyline) First used in finite element analysis Applied to SfM by [Szeliski & Kang 1994] structure \| motion fill-in
55	Conditioning and gauge freedom Poor conditioning: use 2^nd order method use Cholesky decomposition Gauge freedom fix certain parameters (orientation) or zero out last few rows in Cholesky decomposition
56	Robust error models Outlier rejection use robust penalty applied to each set of joint measurements for extremely bad data, use random sampling [RANSAC, Fischler & Bolles, CACM’81]
57	Structure from motion: limitations Very difficult to reliably estimate metric structure and motion unless: large (x or y) rotation or large field of view and depth variation Camera calibration important for Euclidean reconstructions Need good feature tracker
58	Bibliography M.-A. Ameller, B. Triggs, and L. Quan. Camera pose revisited -- new linear algorithms. http://www.inrialpes.fr/movi/people/Triggs/home.html, 2000. M. Antone and S. Teller. Recovering relative camera rotations in urban scenes. In IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'2000), volume 2, pages 282--289, Hilton Head Island, June 2000. S. Becker and V. M. Bove. Semiautomatic {3-D model extraction from uncalibrated 2-d camera views. In SPIE Vol. 2410, Visual Data Exploration and Analysis {II, pages 447--461, San Jose, CA, February 1995. Society of Photo-Optical Instrumentation Engineers. R. G. Bogart. View correlation. In J. Arvo, editor, Graphics Gems II, pages 181--190. Academic Press, Boston, 1991.
59	Bibliography D. C. Brown. Close-range camera calibration. Photogrammetric Engineering, 37(8):855--866, 1971. B. Caprile and V. Torre. Using vanishing points for camera calibration. International Journal of Computer Vision, 4(2):127--139, March 1990. R. T. Collins and R. S. Weiss. Vanishing point calculation as a statistical inference on the unit sphere. In Third International Conference on Computer Vision (ICCV'90), pages 400--403, Osaka, Japan, December 1990. IEEE Computer Society Press. A. Criminisi, I. Reid, and A. Zisserman. Single view metrology. In Seventh International Conference on Computer Vision (ICCV'99), pages 434--441, Kerkyra, Greece, September 1999.
60	Bibliography L. {de Agapito, R. I. Hartley, and E. Hayman. Linear calibration of a rotating and zooming camera. In IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'99), volume 1, pages 15--21, Fort Collins, June 1999. D. I. DeMenthon and L. S. Davis. Model-based object pose in 25 lines of code. International Journal of Computer Vision, 15:123--141, June 1995. M. Gleicher and A. Witkin. Through-the-lens camera control. Computer Graphics (SIGGRAPH'92), 26(2):331--340, July 1992. R. I. Hartley. An algorithm for self calibration from several views. In IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'94), pages 908--912, Seattle, Washington, June 1994. IEEE Computer Society.
61	Bibliography R. I. Hartley. Self-calibration of stationary cameras. International Journal of Computer Vision, 22(1):5--23, 1997. R. I. Hartley, E. Hayman, L. {de Agapito, and I. Reid. Camera calibration and the search for infinity. In IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'2000), volume 1, pages 510--517, Hilton Head Island, June 2000. R. I. Hartley. and A. Zisserman. Multiple View Geometry. Cambridge University Press, 2000. B. K. P. Horn. Closed-form solution of absolute orientation using unit quaternions. Journal of the Optical Society of America A, 4(4):629--642, 1987.
62	Bibliography S. B. Kang and R. Weiss. Characterization of errors in compositing panoramic images. Computer Vision and Image Understanding, 73(2):269--280, February 1999. M. Pollefeys, R. Koch and L. Van Gool. Self-Calibration and Metric Reconstruction in spite of Varying and Unknown Internal Camera Parameters. International Journal of Computer Vision, 32(1), 7-25, 1999. [pdf] L. Quan and Z. Lan. Linear N-point camera pose determination. IEEE Transactions on Pattern Analysis and Machine Intelligence, 21(8):774--780, August 1999. G. Stein. Accurate internal camera calibration using rotation, with analysis of sources of error. In Fifth International Conference on Computer Vision (ICCV'95), pages 230--236, Cambridge, Massachusetts, June 1995.
63	Bibliography Stewart, C. V. (1999). Robust parameter estimation in computer vision. SIAM Reviews, 41(3), 513–537. R. Szeliski and S. B. Kang. Recovering 3D Shape and Motion from Image Streams using Nonlinear Least Squares Journal of Visual Communication and Image Representation, 5(1):10-28, March 1994. R. Y. Tsai. A versatile camera calibration technique for high-accuracy {3D machine vision metrology using off-the-shelf {TV cameras and lenses. IEEE Journal of Robotics and Automation, RA-3(4):323--344, August 1987. Z. Zhang. Flexible camera calibration by viewing a plane from unknown orientations. In Seventh International Conference on Computer Vision (ICCV'99), pages 666--687, Kerkyra, Greece, September 1999.