Intro Essential Mat. I Fundamental Mat. I Epipolar Geom. I Corresp. I Reconst For stereo vision with horizontal camera motion on Matching in 2D space becomes 10. a B Cin 3D space A B C A B C Left image scan line Right image scan line ForA,fr. B ac and see which is the biggest and determine the correspondence E.g. If TAB, is the biggest, A corresponds to B, etc Stereo vO. a
Intro. | Essential Mat. | Fundamental Mat. | Epipolar Geom. | Corresp. | Reconst. For stereo vision with horizontal camera motion only. • Matching in 2D space becomes 1D • For A, find rA,A’ rA,B’ rA,C’ and see which is the biggest and determine the correspondence • E.g. IfrA,B’ is the biggest, A corresponds to B’, etc Stereo v0.a 11 A B C Left image scan line A’ B’ C’ Right image scan line A B C in 3D space
Intro. Essential Mat. Fundamental Mat. Epipolar Geom. Corresp. Reconst Part 2: Epipolar geometry approach of 3-D reconstruction from stereo images Assumption: the cameras can be placed at any viewing angles as long both share enough common views Epipolar approaches Essential matrix( known intrinsic parameters Fundamental unknown intrinsic parameters Stereo vO. a
Intro. | Essential Mat. | Fundamental Mat. | Epipolar Geom. | Corresp. | Reconst. Part 2: Epipolar geometry approach of 3-D reconstruction from stereo images Assumption : the cameras can be placed at any viewing angles as long both share enough common views. Epipolar Approaches: • Essential matrix (known intrinsic parameters) • Fundamental matrix (unknown intrinsic parameters) Stereo v0.a 12
Intro Essential Mat. I Fundamental Mat. I Epipolar Geom. I Corresp. I Reconst Stereo reconstruction by epipolar geometry Stereo camera pair calibration: Find fundamental matrix (F) Find 2-d to 2-d feature correspondences by cross-correlation From correspondences find f 3D model reconstruction from stereo cameras after stereo calibratⅰon Step 0: Rectification of stereo images optional it makes feature correspondences more efficient but not entirely necessary) Step 1: Find 2-D feature correspondences(by f, and rectification Step 2: 3-D Model reconstruction by triangulation Stereo vO. a
Intro. | Essential Mat. | Fundamental Mat. | Epipolar Geom. | Corresp. | Reconst. Stereo reconstruction by epipolar geometry • Stereo camera pair calibration : Find Fundamental matrix (F) – Find 2-D to 2-D feature correspondences by Cross-correlation – From correspondences find F • 3D model reconstruction from stereo cameras after stereo calibration – Step 0: Rectification of stereo images (optional: it makes feature correspondences more efficient, but not entirely necessary) – Step 1: Find 2-D feature correspondences (by F, and rectification) – Step 2: 3-D Model reconstruction by triangulation Stereo v0.a 13
Intro. KEssential Mat)l Fundamental Mat. I Epipolar Geom. I Corresp. I Reconst Essential matrix(E) for describing the geometry of a stereo camera pair Find essential matrix e Stereo vO. a
Intro. | Essential Mat. | Fundamental Mat. | Epipolar Geom. | Corresp. | Reconst. Essential matrix (E) for describing the geometry of a stereo camera pair Find Essential matrix E Stereo v0.a 14
Intro. KEssential Mat)l Fundamental Mat. I Epipolar Geom. I Corresp. I Reconst Notations: Tc of chapter 2 X and t in this chapter In chapter 2 R(P Ⅹ2 R-R*7。 Cam1 center C (reference/world) Cam2 center 0 (camera) R, T are used in this chapter This chapter, caml is at ref/world.x2=R*X,+T, R=Ro coord. cam2 is at camera coord where a vector X, in world coordinates is the Define r=r= r same vector X, in camera coordinates. X, does not change, it is only the coordinate system that X changes YRT‖Y Tc= a vector from cam 1 center to cam2 center in cam1 coordinates So-Tc is a vector from cam2 center to cam1 center in cam1 coordinates T=-R*T, or T=-R*T T=-Rc*TC=-RT. a vector from cam2 center to cam1 center in cam2 coordinates
Intro. | Essential Mat. | Fundamental Mat. | Epipolar Geom. | Corresp. | Reconst. Notations: Tc of chapter 2 and T in this chapter Stereo v0.a 15 • R,T are used in this chapter • X2=R*X1+T, R=Rc • where a vector X1 in world coordinates is the same vector X2 in camera coordinates . X1 does not change, it is only the coordinate system that changes • Tc= a vector from cam1 center to cam2 center in cam1 coordinates • So –Tc is a vector from cam2 center to cam1 center in cam1 coordinates. • T=-Rc*Tc=-R*Tc = a vector from cam2 center to cam1 center in cam2 coordinates. c c c c cam w w w c c c c c c c c w c T R T T R T Z YX R T Z YX R R R Z YX R R T Z YX P R P T * ,or * 1 0 1 1 Define coord., cam2 is at camera coord. This chapter, cam1is at ref/world , 1 0 1 * 1 ( ) In chapter 2 1 1 1 2 2 2 1 = − = − = = = − = = − − Y2 X2 Z2 Z1 X1 Y1 X Cam2 center (camera) Cam1 center TC (reference/world)