This preview shows page 1-2-3-19-20-38-39-40 out of 40 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 40 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 40 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 40 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 40 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 40 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 40 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 40 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 40 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 40 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

Camera Models class 8 Multiple View Geometry Comp 290 089 Marc Pollefeys Multiple View Geometry course schedule subject to change Jan 7 9 Intro motivation Projective 2D Geometry Jan 14 16 no class Projective 2D Geometry Jan 21 23 Projective 3D Geometry no class Jan 28 30 Parameter Estimation Parameter Estimation Feb 4 6 Algorithm Evaluation Camera Models Feb 11 13 Camera Calibration Single View Geometry Feb 18 20 Epipolar Geometry 3D reconstruction Feb 25 27 Fund Matrix Comp Structure Comp Planes Homographies Trifocal Tensor Three View Reconstruction Multiple View Geometry Mar 4 6 Mar 18 20 Mar 25 27 MultipleView Reconstruction Bundle adjustment Apr 1 3 Auto Calibration Papers Apr 8 10 Dynamic SfM Papers Apr 15 17 Cheirality Papers Apr 22 24 Duality Project Demos N measurements independent Gaussian noise 2 model with d essential parameters use s d and s N d i RMS residual error for ML estimator eres 2 E X X N 1 2 1 d N 1 2 ii RMS estimation error for ML estimator eest E X X N 2 n X X X 1 2 d N 1 2 Error in two images 1 2 SM n 4 eest 2n 1 2 n 4 eres 2n d 8 2n and N 4n Forward propagation of covariance f X J PJ T J v Backward propagation of covariance 1 x P J J T 1 P Over parameterization 1 x P J J T A Monte Carlo estimation of covariance f X 1 X Example 1 pixel 0 5cm Crimisi 97 Single view geometry Camera model Camera calibration Single view geom Pinhole camera model X Y Z a fX Z fY Z T X fX f Y Z a fY Z 1 f T X 0 Y 0 Z 1 0 1 Pinhole camera model fX f fY Z f x PX X 0 Y 0 Z 1 0 1 fX f fY Z f X 0 1 Y 1 0 Z 1 1 0 1 P diag f f 1 I 0 Principal point offset X Y Z a fX Z p x fY Z p y T px p y T principal point X fX Zp x f Y Z a fY Zp y Z 1 px f py 1 X 0 Y 0 Z 0 1 T Principal point offset X 0 Y 0 Z 0 1 px fX Zp x f f py fY Zp x Z 1 px f K f p y calibration matrix x K I 0 X cam 1 Camera rotation and translation X cam R X C X R RC Y R RC X cam X 1 Z 0 1 0 1 x K I 0 X cam x KR I C X x PX P K R t t RC CCD camera mx K my f 1 f px p y 1 x K y px p y 1 Finite projective camera x s K y P KR I C px p y 1 11 dof 5 3 3 non singular decompose P in K R C P M p 4 K R RQ M C M 1p 4 finite cameras P3x4 det M 0 If rank P 3 but rank M 3 then cam at infinity Camera anatomy Camera center Column points Principal plane Axis plane Principal point Principal ray Camera center null space camera projection matrix PC 0 C is camera center Image of camera center is 0 0 0 T i e undefined M 1p 4 Finite cameras C 1 d Infinite cameras C Md 0 0 Column vectors 0 1 p 2 p1p 2 p3p 4 0 0 Image points corresponding to X Y Z directions and origin Row vectors X x p y p 2 T Y T Z 0 p 3 1 1T 1T X 0 p y p 2 T Y T Z w p 3 1 note p1 p2 dependent on image reparametrization The principal point p 3 p31 p32 p33 0 principal point x 0 Pp 3 Mm3 The principal axis vector vector defining front side of camera m3 x Pcam X cam K I 0 X cam Pcam a kPcam P kKR I C M p 4 Pcam a kPcam v det M m 3 0 0 1 T v a k 4v direction unaffected v a det kM km 3 k 4 v Action of projective camera on point Forward projection x PX x PD M p 4 D Md Back projection PC 0 X P x X P x C P P PP T T 1 pseudo inverse PP I d M 1x M 1x M 1p 4 M 1 x p 4 X 1 0 1 C D Depth of points 3T w P X P 3T X C m PC 0 3T X C dot product 3 det M 0 m 1 If then m3 unit vector in positive direction X X Y Z T T sign detM w depth X P T m3 Camera matrix decomposition Finding the camera center PC 0 use SVD to find null space X det p 2 p 3 p 4 Y det p1 p 3 p 4 Z det p1 p 2 p 4 T det p1 p 2 p 3 Finding the camera orientation and internal parameters M KR use RQ ecomposition When is skew non zero x s K x px p y 1 arctan 1 s 1 for CCD CMOS always s 0 Image from image s 0 possible non coinciding principal axis resulting camera HP Euclidean vs projective general projective interpretation 1 0 0 0 P 3 3 homography 0 1 0 0 4 4 homography 0 0 1 0 Meaningfull decomposition in K R t requires Euclidean image and space Camera center is still valid in projective space Principal plane requires affine image and space Principal ray requires affine image and Euclidean space Cameras at infinity Camera center at infinity det M 0 Affine and non affine cameras Definition affine camera has P3T 0 0 0 1 Affine cameras Affine cameras r 2T P0 KR I C K r 3T 3T r d 0 r C 1T r 2T Pt K r r 3T 1T r r 2T r 3T 1T 3 C tr 3 C tr 3 C tr r C 2T r C 3T r C 1T r K r r modifying p34 corresponds to moving along principal ray 1T 2T 3T r C 2T r C d t 1T Affine cameras now adjust zoom to compensate 1T 1T r C d t d 0 r 2T 2T r C dt d0 Pt K r d t 1 r 3T 1T r1T r C dt 2T 2T K r r C d 0 3T d d d r 0 t 0 r1T r1T C 2T 2T P lim Pt K r r C t 0 d 0 Error in employing affine cameras r1 r 2 point on plane parallel with X principal plane and through origin 1 then P0 X Pt X P X r1 r 2 r 3 general points X 1 x x proj P0 X K y d 0 x proj …


View Full Document

Berkeley ELENG 290T - Camera Models

Download Camera Models
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Camera Models and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Camera Models 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?