Parameter estimation. Christiano Gava Gabriele Bleser

Transcription

1 Parameter estimation Christiano Gava Gabriele Bleser

2 Introduction Previous lectures: P-matrix 2D projective transformations Estimation (direct linear transform) Today Estimation techniques Linear Nonlinear Robust Rotation matrix representation 11/22/2013 Lecture 3D Computer Vision 2

3 Motivation Projection Matrix Homography Estimation techniques Triangulation Fundamental Matrix X x x / F C C / 11/22/2013 Lecture 3D Computer Vision 3

4 Context: general computer vision pipeline Model-based tracking Simultaneous localisation and mapping, structure from motion (next lectures) Hybrid tracking (e.g. visual-inertial) sensor fusion Additional measurements Camera image Image processing 2D/3D point correspondences Pose estimation Camera pose Scene model Structure estimation 11/22/2013 Lecture 3D Computer Vision 4

5 Linear estimation Today Estimation techniques Linear Nonlinear Robust Rotation matrix representation 11/22/2013 Lecture 3D Computer Vision 5

6 Reminder: direct linear transform Many computer vision problems can be solved with DLT: examples? Projection matrix estimation from 2D/3D point correspondences Homography estimation from 2D/2D point correspondences 11/22/2013 Lecture 3D Computer Vision 6

7 Direct linear transform There is more to come (next lectures): Fundamental matrix estimation from 2D/2D point correspondences Triangulation from camera views Camera view= camerapose+ 2D point location 11/22/2013 Lecture 3D Computer Vision 7

8 Reminder: general algorithm Objective Given a sufficientnumber, n, of measurements determine X such that Algorithm (i) Set up linear equation system (m is number of parameters): (ii) If b is zero solve with SVD. (iii) Else solve with pseudo-inverse. (iv) Determine X from x. Today: Howdoesthisworkin detail? What comes after DLT? 11/22/2013 Lecture 3D Computer Vision 8

9 Solution for b equals zero Over-determined system(more measurements than needed) 0 1 Additional constraint to avoid trivial solution Exact solution (null space of A) usually does not exist: Why? Due to measurement noise How does this show? A has fullrank Howdo wesolvethis? Find approximatesolution: How do degenerate measurement configurations show? A has rank < m Constrained optimization problem, can be solved with SVD 11/22/2013 Lecture 3D Computer Vision 9

10 Homogeneous least squares How is this minimized? Set first derivative to zero x is an eigenvector of A T A x iseigenvectorof A T A correspondingto smallest eigen value 11/22/2013 Lecture 3D Computer Vision 10

11 Reminder SVD Howdo wegettheeigenvectorof A T A correspondingto the smallest eigen value? Column of V corresponding to the smallest singular value of A 11/22/2013 Lecture 3D Computer Vision 11

12 Solution forb notzero For example: inhomogeneous solution to homography estimation (previous lecture) Least squares solution: Set derivativeto zero: Also called pseudo-inverse Matlab operator: x=a\b 11/22/2013 Lecture 3D Computer Vision 12

13 Summary: direct linear estimation Advantages: No initialization required 1 iteration Disadvantages: Sensitive to imprecise measurements and outliers Usually no minimal parametrization Constraints enforced afterwards(e.g. rotation matrix orthogonalization) 11/22/2013 Lecture 3D Computer Vision 13

14 Iterative nonlinear estimation Today Estimation techniques Linear Nonlinear Robust Rotation matrix representation 11/22/2013 Lecture 3D Computer Vision 14

15 Iterative minimization Properties: Often slower than DLT Requires initialization typically use linear solution or sample parameter space No guaranteed convergence, local minima Stopping criterion required Many optimization algorithms exist: Newton s method Levenberg-Marquardt Simplex method In this lecture treated as black box 11/22/2013 Lecture 3D Computer Vision 15

16 Iterative minimization General nonlinear problem formulation for n measurements: In this lecture: Estimate Parameter vector Which cost function for computing the residuals? How to represent the parameter vector? Residual/reprojection error Now back to the concrete problem of camera pose estimation from 2D/3D correspondences Parameters to estimate: camera pose(rotation and translation) 11/22/2013 Lecture 3D Computer Vision 16

17 Collinearity constraints Basis for cost functions for pose and structure estimation (s represents the extrinsic camera parameters with 6 DOF) Image collinearity constraint (ICC) Homogeneous collinearity constraint (HCC) 11/22/2013 Lecture 3D Computer Vision 17

18 Collinearity constraints Homogeneous: Error in homogeneous space Dependson depthof 3D point Distant 3D points have more influence on estimation Image: Error in image space (normalized image/pixel error) Independent of 3D point depth Division through z avoided better suited for linearization Well suited for nonlinear estimation Intuitive error measure for inlier/outlier decisions 11/22/2013 Lecture 3D Computer Vision 18

19 Least squares(ls) estimation Typical problem formulation(error in normalized image space): All measurements are incorporated with equal weight Ifinformationaboutthequalityof measurementsisgiven, how to incorporate this? Point projected with current pose estimate d n m n m n 19 m w 11/22/2013 Lecture 3D Computer Vision 19

20 Weighted least squares(wls) estimation Incorporation of simple stochasticmodel: parametersand measurements modelled as Gaussian random variables Mahalanobis distance Joint covarianceof 2D/3D correspondences (first order error propagation) 11/22/2013 Lecture 3D Computer Vision 20

21 Weighted least squares(wls) estimation Allows for: Incorporating measurement uncertainties Recursive estimation(filtering) Representation/estimation of errors But Still notrobust in caseof wrong correspondences(outliers) 11/22/2013 Lecture 3D Computer Vision 21

22 Robust estimation Today Estimation techniques Linear Nonlinear Robust Rotation matrix representation 11/22/2013 Lecture 3D Computer Vision 22

23 M-estimators Minimize the sum of a function of residuals: Function appearsas weight Assumptions: Only few outliers Initialization close to solution Example: F-Matrix estimation 11/22/2013 Lecture 3D Computer Vision 23

24 Robust estimation M-estimators usually are not able to cope with this More rigorous method required: RANdom Sampling Consensus (RANSAC) 11/22/2013 Lecture 3D Computer Vision 24

25 Robust estimation Using RANSAC... 11/22/2013 Lecture 3D Computer Vision 25

26 RANSAC line fitting video How can we use RANSAC in computer vision? 11/22/2013 Lecture 3D Computer Vision 26

27 RANSAC pose estimation Algorithm: Choose a minimum random subset of correspondences and estimate the pose(how many?) Check theconsistencyof all correspondenceswiththisposeand find the inliers(how?) Iterate until enough inliers are found Problems: Computational intensive False negatives 11/22/2013 Lecture 3D Computer Vision 27

28 RANSAC (general formulation) Objective Robust fit of model to data set S, which contains outliers Algorithm (i) (ii) (iii) Randomly select a sample of s data points from S and instantiate the model from this subset. Determine the set of data points S i which are within a distance threshold t of the model. The set S i is the consensus set of samples and defines the inliers of S. If the subset of S i is greater than some threshold T, re-estimate the model using all the points in S i and terminate. (iv) If the size of S i is less than T, select a new subset and repeat the above. (v) After N trials the largest consensus set S i is selected, and the model is re-estimated using all the points in the subset S i 11/22/2013 Lecture 3D Computer Vision 28

29 Example: homography estimation 11/22/2013 Lecture 3D Computer Vision 29

30 Rotation representation Today Estimation techniques Linear Nonlinear Robust Rotation matrix representation 11/22/2013 Lecture 3D Computer Vision 30

31 Parametrization Example: minimization over camera pose How is the camera pose modelled? For DLT: rotation matrix R, translation vector t Problem: R has 9 parameters, but only 3 DOF orthogonalization required Reminder: orthogonalizationwithsvd: R =UWV T R=UV T Better solution: re-parametrize R Euler angles(3 parameters, second lecture) Axis angle (3 parameters) Quaternions(4 parameters) 11/22/2013 Lecture 3D Computer Vision 31

32 Euler angle representation Sequence not standardized! + Minimal parametrization, nice geometric interpretation - Periodicity, Gimbal lock 11/22/2013 Lecture 3D Computer Vision 32

33 Axis angle representation A rotation canberepresentedas a 3D vector direction rotation axis length rotation angle Conversion to rotation matrix by Rodrigues formula: r + Minimal parametrization - Singularity at θ=0, conversion required to rotate vectors 11/22/2013 Lecture 3D Computer Vision 33

34 Quaternion representation Compromise between rotation matrix and minimal representation A quaternionisa 4D vector: Conversion from axis angle to rotation matrix: + Easier conversion to rotation matrix + Singularity-free, can be interpolated - Not minimal, 4 parameters, unit magnitude constraint 11/22/2013 Lecture 3D Computer Vision 34

35 Rotation parametrization All parametrizations have their advantages and disadvantages Choose the parametrization according to the task: Initialization with DLT rotation matrix(linear) Nonlinearoptimization Euler angles, axisangle (minimal parametrization) Recursive estimation, interpolation quaternions Nicepaperon thistopic: GrassiaS. F.: Practical parameterization of rotations using the exponential map, Journal of Graphics Tools, A. K. Peters, Ltd., 1998, 3,

36 References General textbook: Hartley R. and ZissermanA.: Multiple View Geometry in Computer Vision. Robust homography/pose estimation Brown and Lowe: Recognizing Panoramas, ICCV, Bleseret al: Real-time Vision-based Tracking and Reconstruction, Journal of Real-Time Image Processing 2 (2007), 2-3, pp , Berlin, Heidelberg, New York : Springer Verlag. Rotation representation Shuster M. D.: A Survey of Attitude Representations, The Journal of the Astronautical Sciences, 1993, 41, Images used in the F-Matrix estimation example: Original RANSAC video: 36

37 Thank you!