1 / 41

One view geometry---camera modeling

X’. P3. u. X. u’. u. v. O. P2. One view geometry---camera modeling. Modeling. ‘abstract’ camera: projection from P3 to P2. Math: central proj. Physics: pin-hole. As lines are preserved so that it is a linear transformation and can be represented by a 3*4 matrix.

wendyn
Download Presentation

One view geometry---camera modeling

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. X’ P3 u X u’ u v O P2 One view geometry---camera modeling

  2. Modeling ‘abstract’ camera: projection from P3 to P2 Math: central proj. Physics: pin-hole As lines are preserved so that it is a linear transformation and can be represented by a 3*4 matrix This is the most general camera model without considering optical distortion

  3. X’ P3 u X u’ u v O P2

  4. Properties of the 3*4 matrix P • d.o.f.? • Rank(P) = ? • ker(P)? • row vectors, planes • column vectors, directions • principal plane: w=0 • calibration, how many pts? • decomposition by QR, • K intrinsic (5). R, t, extrinsic (6) • geometric interpretation of K, R, t (backward from u/x=v/y=f/z to P) • internal parameters and absolute conic

  5. Geometric description From concrete phy. paramters to algebraic param. Central projection in Cartesian coordinates • Camera coordinate frame • image coordinate frame • world coordinate frame

  6. Z u Y y X x X f x v O Camera coordinate frame

  7. In more familiar matrix form:

  8. Z u u y Y y X x o x X f x v v O Image coordinate frame

  9. Image coordiante frame: intrinsic parameters Camera calibration matrix • Focal length in horizontal/vertical pixels (2) (aspect ratio) • the principal point (2) • the skew (1) ? • one rough example: 135 film

  10. Z Zw u Y Xw Yw y X x X f x v O World (object) coordinate frame Xw

  11. World coordinate frame: extrinsic parameters Finally, we should count properly ... Relation between the abstract algebraic and geometric models is in the intrinsic/extrinsic parameters!

  12. Summary of camera modelling • 3 coordinate frame • projection matrix • decomposition • intrinsic/extrinsic param

  13. What is the calibration matrix K? It is the image of the absolute conic, prove it first! Point conic: The dual conic:

  14. It turns the camera into an spherical one, or angular/direction sensor! Direction vector: Angle between two rays ...

  15. Don’t forget: when the world is planar … A general plane homography!

  16. Camera calibration Given from image processing or by hand  • Estimate C • decompose C into intrinsic/extrinsic

  17. Calibration set-up: 3D calibration object

  18. The remaining pb is how to solve this ‘trivial’ system of equations!

  19. Review of some basic numerical algorithms • linear algebra • non-linear optimisation • statistics Go to see the slides ‘calibration.ppt’ for math review

  20. Linear algebra in 5 mins • Gaussian elimination • LU decomposition • Choleski (sym positive) LL^T • orthogonal decomposition • QR (Gram-Schmidt) • SVD (the high(est)light of linear algebra!) • row space: first Vs • null space: last Vs • col space: first Us • null space of the trans : last Us

  21. Linear methods of computing P • p34=1 • ||p||=1 • ||p3||=1 Geometric interpretation of these constraints

  22. Non-linear … • ||p3||=1 and • no skew (p1 X p3) . (p2 X p3) = 0 It’s constrained optimizaion, but non linear …

  23. Decomposition • analytical by equating K(R,t)=P • QR (more exactly it is RQ)

  24. Linear, but non-optimal,but we want optima, but non-linear, methods of computing P Each observation has a (0,sigma) independent Gaussian distribution: The probability of obtaining all measurements x given that the camera matrix is P: MLE (maximum likelihood estimate) is:

  25. Even linear models, but end up with non-linear optimization …Non-linear models for optimization, always end-up iterative linear computing!Yesterday, closed-form, algebraic methods (gradients), for small scale, Today, everything numerical in big scale.

  26. How to solve this non-linear system of equations?

  27. Non-linear iterative optimisation • J d = r from vector F(x+d)=F(x)+J d • minimize the square of y-F(x+d)=y-F(x)-J d = r – J d • normal equation is J^T J d = J^T r (Gauss-Newton) • (H+lambda I) d = J^T r (LM) Note: F is a vector of functions, i.e. min f=(y-F)^T(y-F)

  28. General non-linear optimisation • 1-order , d gradient descent d= g and H =I • 2-order, • Newton step: H d = -g • Gauss-Newton for LS: f=(y-F)^T(y-F), H=J^TJ, g=-J^T r • ‘restricted step’, trust-region, LM: (H+lambda W) d = -g R. Fletcher: practical method of optimisation f(x+d) = f(x)+g^T d + ½ d^T H d Note: f is a scalar valued function here.

  29. statistics • ‘small errors’ --- classical estimation theory • analytical based on first order appoxi. • Monte Carlo • ‘big errors’ --- robust stat. • RANSAC • LMS • M-estimators Talk about it later

  30. (Non-linear) Optical distorsion Radial distorsion could be modeled by • alpha_i and eventually (xc,yc) are distorsion parameters • you may have any degree for r, but probably the correction is not sufficient for first degree • x is normalised coordinates, x’ is actual distorted measures (people are confusing x and x’  • normalised as calibrated for ‘photogrammetrists’ (more difficult to write down the cost function), while normalised just to [-1,1] to compute r (easy to integrate into the cost function, still pixels)

  31. Two approaches: • add these terms into the non-linear projection equations using a calibration object • estimate them independently using the ‘colinearity’ constraint (see Devernay, photogrammetry) to fit lines

  32. Solution uniqueness? Rank of the matrix A?

  33. Using a planar pattern Why? it is more convenient to have a planar calibration pattern than a 3D calibration object, so it’s very popular now for amateurs. Cf. the paper by Zhenyou Zhang (ICCV99), Sturm and Maybank (CVPR99) Homework: read these papers.

  34. first estimate the plane homogrphies Hi from u and x, 1. How to estimate H? 2. Why one may not be sufficient? • extract parameters from the plane homographies

  35. How to extract intrinsic parameters? The absolute conic in image The (transformed) absolute conic in the plane: The circular points of the Euclidean plane (i,1,0) and (-i,1,0) go thru this conic: two equations on K!

  36. Summary of calibration • Get image-space points • Solve the linear system • Optimal sol. by non-linear method • Decomposition by RQ

  37. Where is the camera? Given rom image processing or by hand  • Camera pose (given K) • where is the camera?

  38. Camera pose: 3-point algorithm • fundamental Euclidean constraint • 3 point algorithm • quarternion for rotation?

  39. Where is the camera? A projective setting • on a plane, 1D camera, • Circle by 3 pts and a constant angle • Chasles conics • 2D camera • calibration singularities

More Related