"Multi-View Geometry (Cont.)"
Multi-View Geometry (Cont.) Stereo Constraints (Review) p p’ ? Given p in left image, where can the corresponding point p’ in right image be? Stereo Constraints (Review) M Image plane Epipolar Line Y1 p p’ Y2 X2 O1 Z1 X1 O2 Z2 Focal plane Epipole Epipolar Constraint (Review) From Geometry to Algebra (Review) P p p’ O O’ From Geometry to Algebra (Review) P p p’ O O’ Linear Constraint: Should be able to express as matrix multiplication. The Essential Matrix (Review) The Essential Matrix • Based on the Relative Geometry of the Cameras • Assumes Cameras are calibrated (i.e., intrinsic parameters are known) • Relates image of point in one camera to a second camera (points in camera coordinate system). • Is defined up to scale • 5 independent parameters The Essential Matrix T Similarly p is the epipolar line corresponding to p in the right camera The Essential Matrix e t Re 0 Similarly, e R t e R t e 0 T T T T Essential Matrix is singular with rank 2 e’ Small Motions and Epipolar Constraint Motion Models (Review) X 1 X TX Y 1 Y TY 3D Rigid Motion Z 1 Z TZ X 0 1 0 0 X TX Y 0 1 0 Y T 0 Y VX Z 0 0 0 1 Z TZ V Velocity Vector Y VZ X X 0 X TX Y Y VTX 0 Y TY Translational Z Z 0 Z TZ VTY Component of Velocity VT Z VX 0 Z Y X VT X V X Y VT x 0 Angular Velocity Y Z Y Y VZ Y X 0 Z VTZ Z Small Motions t tv R I t p p tp VX p T p 0 p VY Velocity Vector VZ p v I t p tp 0 T VTX v VTY Translational VT Component of Velocity Z p T v p p p .v 0 X Y Angular Velocity Z Translating Camera pT v p p p .v 0 0 p p .v 0 p, p, and v are coplanar Focus of expansion (FOE): Under pure translation, the motion field at every point in the image points toward the focus of expansion FOE for Translating Camera FOE from Basic Equations of Motion VTZ x VTX f X xy Y x 2 px Y f Z y Z f f VTZ y VTY f Y xy X y 2 py X f Z x Z f f 0 q VTZ x VTX f q p px Z p VTZ y VTY f py Z O v VTX e x0 , y0 x0 f VTZ VTZ p x x x0 Z VTY VTZ y0 f p y y y0 VTZ Z What if Camera Calibration is not known Review: Intrinsic Camera Parameters Y M X C , Y C , Z C Image plane i ku I j Z C v X j kv J J i I u m u , v , f C C Focal plane u , v I I X C U new f u 0 u0 0 C new Y f u fku V 0 fv v0 0 C Z S 0 0 f v fkv 0 1 1 P Fundamental Matrix p T p 0 p and p are in cameracoordinatesystem If u and u’ are corresponding image coordinates then we have uPp 1 p P 1u 1 u P2 p p P21u P u T P T 1 2 u 0 1 uT Fu 0 F P T P21 1 Fundamental Matrix u Fu 0 T F P T P21 1 Fundamental Matrix is singular with rank 2 In principal F has 7 parameters up to scale and can be estimated from 7 point correspondences Direct Simpler Method requires 8 correspondences Estimating Fundamental Matrix The 8-point algorithm u Fu 0 T Each point correspondence can be expressed as a linear equation F11 F12 F13 u u v 1 F21 F22 F23 v 0 F31 F32 F33 1 F11 F 12 F13 F21 uu uv u uv vv v u v 1 F22 0 F23 F 31 F32 F 33 The 8-point Algorithm Shape from Stereo Pinhole Camera Model X xf Z Basic Stereo Derivations Derive expression for Z as a function of x1, x2, f and B Basic Stereo Derivations X X B B x1 f x2 f x1 f Z Z Z fB Z x1 x2 Basic Stereo Derivations Define the disparity: d x1 x2 fB Z d