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1.Epipolar Geometry and Stereo Vision Computer Vision Jia-Bin Huang, Virginia Tech Many slides from S. Seitz and D. Hoiem

2.Last class: Image Stitching Two images with rotation/zoom but no translation f f . x x X

3.This class: Two-View Geometry Epipolar geometry Relates cameras from two positions Stereo depth estimation Recover depth from two images

4.This class: Two-View Geometry Epipolar geometry Relates cameras from two positions Stereo depth estimation Recover depth from two images

5.This class: Two-View Geometry Epipolar geometry Relates cameras from two positions Stereo depth estimation Recover depth from two images

6.Correspondence Problem We have two images taken from cameras with different intrinsic and extrinsic parameters How do we match a point in the first image to a point in the second? How can we constrain our search? x ?

7.Key idea: Epipolar constraint

8.Potential matches for x have to lie on the corresponding line l’ . Potential matches for x’ have to lie on the corresponding line l . Key idea: Epipolar constraint x x’ X x’ X x’ X

9. Epipolar Plane – plane containing baseline (1D family) Epipoles = intersections of baseline with image planes = projections of the other camera center Baseline – line connecting the two camera centers Epipolar geometry: notation X x x’

10. Epipolar Lines - intersections of epipolar plane with image planes (always come in corresponding pairs) Epipolar geometry: notation X x x’ Epipolar Plane – plane containing baseline (1D family) Epipoles = intersections of baseline with image planes = projections of the other camera center Baseline – line connecting the two camera centers

11.Example: Converging cameras

12.Example: Motion parallel to image plane

13.Example: Forward motion What would the epipolar lines look like if the camera moves directly forward?

14.e e’ Example: Forward motion Epipole has same coordinates in both images. Points move along lines radiating from e: “Focus of expansion”

15.X x x’ Epipolar constraint: Calibrated case Given the intrinsic parameters of the cameras : Convert to normalized coordinates by pre-multiplying all points with the inverse of the calibration matrix; set first camera’s coordinate system to world coordinates Homogeneous 2d point (3D ray towards X) 2D pixel coordinate (homogeneous) 3D scene point 3D scene point in 2 nd camera’s 3D coordinates

16.X x x’ Epipolar constraint: Calibrated case Given the intrinsic parameters of the cameras: Convert to normalized coordinates by pre-multiplying all points with the inverse of the calibration matrix; set first camera’s coordinate system to world coordinates Define some R and t that relate X to X’ as below for some scale factor

17.Epipolar constraint: Calibrated case x x’ X (because and are co-planar)

18.Essential Matrix (Longuet-Higgins, 1981) Essential matrix X x x’

19.X Properties of the Essential matrix E x’ is the epipolar line associated with x’ ( l = E x’ ) E T x is the epipolar line associated with x ( l’ = E T x ) E e ’ = 0 and E T e = 0 E is singular (rank two) E has five degrees of freedom (3 for R, 2 for t because it’s up to a scale) Drop ^ below to simplify notation x x’ Skew-symmetric matrix

20.Epipolar constraint: Uncalibrated case If we don’t know K and K’ , then we can write the epipolar constraint in terms of unknown normalized coordinates: X x x’

21.The Fundamental Matrix Fundamental Matrix ( Faugeras and Luong , 1992) Without knowing K and K’, we can define a similar relation using unknown normalized coordinates

22.Properties of the Fundamental matrix F x’ is the epipolar line associated with x’ ( l = F x’ ) F T x is the epipolar line associated with x ( l’ = F T x ) F e’ = 0 and F T e = 0 F is singular (rank two): det (F)=0 F has seven degrees of freedom: 9 entries but defined up to scale, det (F)=0 X x x’

23.Estimating the Fundamental Matrix 8-point algorithm Least squares solution using SVD on equations from 8 pairs of correspondences Enforce det (F)=0 constraint using SVD on F 7-point algorithm Use least squares to solve for null space (two vectors) using SVD and 7 pairs of correspondences Solve for linear combination of null space vectors that satisfies det (F)=0 Minimize reprojection error Non-linear least squares Note: estimation of F (or E) is degenerate for a planar scene.

24.8-point algorithm Solve a system of homogeneous linear equations Write down the system of equations = 0

25.8-point algorithm Solve a system of homogeneous linear equations Write down the system of equations Solve f from A f = 0 using SVD Matlab : [U, S, V] = svd (A); f = V(:, end); F = reshape(f, [3 3])’;

26.Need to enforce singularity constraint

27.8-point algorithm Solve a system of homogeneous linear equations Write down the system of equations Solve f from A f = 0 using SVD Resolve det (F) = 0 constraint using SVD Matlab : [U, S, V] = svd (A); f = V(:, end); F = reshape(f, [3 3])’; Matlab : [U, S, V] = svd (F); S(3,3) = 0; F = U*S*V’;

28.8-point algorithm Solve a system of homogeneous linear equations Write down the system of equations Solve f from A f = 0 using SVD Resolve det (F) = 0 constraint by SVD Notes: Use RANSAC to deal with outliers (sample 8 points) How to test for outliers? Solve in normalized coordinates mean=0 s tandard deviation ~= (1,1,1) just like with estimating the homography for stitching

29.Homography (No Translation) Fundamental Matrix (Translation) Correspondence Relation Normalize image coordinates RANSAC with 8 points Initial solution via SVD Enforce by SVD De-normalize: Correspondence Relation Normalize image coordinates RANSAC with 4 points Solution via SVD De-normalize: Comparison of homography estimation and the 8-point algorithm Assume we have matched points x x ’ with outliers