Introduction Epipolar Geometry Calibration Methods Further Readings. Stereo Camera Calibration

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1 Stereo Camera Calibration Stereo Camera Calibration Stereo Camera Calibration Stereo Camera Calibration

2 Overview Introduction Summary /

3 Motivation Depth Perception Ambiguity of Correspondence The Epipolar Constraint Motivation 3D-Vision Problems Correspondence problem Reconstruction problem

4 Motivation Depth Perception Ambiguity of Correspondence The Epipolar Constraint Motivation 3D-Vision Problems Correspondence problem Reconstruction problem

5 Motivation Depth Perception Ambiguity of Correspondence The Epipolar Constraint How does depth perception work? High level process? Objects are matched... Low level process! Proof: Random Dot Images Problem: Matching the Dots

6 Motivation Depth Perception Ambiguity of Correspondence The Epipolar Constraint The Ambiguity of Correspondence Real Correspondence Just looking similar

7 Motivation Depth Perception Ambiguity of Correspondence The Epipolar Constraint The Epipolar Constraint World Point M corresponds to point m in Image Plane I

8 Motivation Depth Perception Ambiguity of Correspondence The Epipolar Constraint The Epipolar Constraint Ray CM has Image in I'

9 Motivation Depth Perception Ambiguity of Correspondence The Epipolar Constraint The Epipolar Constraint Corresponding Point in I' can be found...

10 Motivation Depth Perception Ambiguity of Correspondence The Epipolar Constraint The Epipolar Constraint The Points C, C' and M build a Plane. e is Image of C' and v.v.

11 Motivation Depth Perception Ambiguity of Correspondence The Epipolar Constraint The Epipolar Constraint m' has epipolar line too

12 Motivation Depth Perception Ambiguity of Correspondence The Epipolar Constraint Pencils of Epipolar Lines Different World Points can have different epipolar lines.

13 Basics The Essential Matrix The Fundamental Matrix Summary Perspective Projection Matrix There are normalized and unnormalized cameras Decomposition possible:

14 Basics The Essential Matrix The Fundamental Matrix Summary Camera Intrinsic Matrix Result of decomposition

15 Basics The Essential Matrix The Fundamental Matrix Summary Skew Symmetric Matrix Mapping from 3D-Vector to 3x3 antisymmetric Matrix

16 Basics The Essential Matrix The Fundamental Matrix Summary The Essential Matrix Mapping from Point to Line

17 Basics The Essential Matrix The Fundamental Matrix Summary Defining EG with Normalized Images Relation between camera coordinate systems Normalized Coordinates

18 Basics The Essential Matrix The Fundamental Matrix Summary Defining EG with Normalized Images

19 Basics The Essential Matrix The Fundamental Matrix Summary Defining EG with Normalized Images Coplanar

20 Basics The Essential Matrix The Fundamental Matrix Summary Defining EG with Normalized Images

21 Basics The Essential Matrix The Fundamental Matrix Summary Properties of E Determined completely by the transform between the cameras det(e)=0

22 Basics The Essential Matrix The Fundamental Matrix Summary The Fundamental Matrix Fundamental Matrix F must satisfy where A and A' are intrinsic Matrices of the two Cameras

23 Basics The Essential Matrix The Fundamental Matrix Summary Properties of F det(f)=0 because det(e)=0 F is of rank 2 F has 9 Parameters but 7DOF

24 Basics The Essential Matrix The Fundamental Matrix Summary Summary Stereo Vision is about point matching E and F map points in one image to lines in the other. F satisfies 7 DOF and has between two arbitrary images can be estimated by 7 point correspondences

25 Calibration Basics 7-Point-Algorithm >8-Point-Algorithm(s) Experiments/Tricks Calibration Basics Assumptions The full perspective projection model is used. Intrinsic and Extrinsic parameters of the Cameras are unknown. Rigid transformation between the two camera systems

26 Calibration Basics 7-Point-Algorithm >8-Point-Algorithm(s) Experiments/Tricks Equivalent Problems 2 Images by a moving camera at two different moments (static). 2 Images by a fixed camera at two different moments (single object,dynamic) 2 Images by 2 Cameras at the same time 2 Images by 2 Cameras (static)

27 Calibration Basics 7-Point-Algorithm >8-Point-Algorithm(s) Experiments/Tricks Definitions The Epipolar Equation can be written as linear homogeneous equation

28 Calibration Basics 7-Point-Algorithm >8-Point-Algorithm(s) Experiments/Tricks Definitions For n point matches

29 Calibration Basics 7-Point-Algorithm >8-Point-Algorithm(s) Experiments/Tricks Not that easy Noise Least-square methods are very sensitive to false matches Enforce rank 2 constraint or epipolar lines will "smear out" find closest rank 2 approximation of F

30 Calibration Basics 7-Point-Algorithm >8-Point-Algorithm(s) Experiments/Tricks Rank 2 Constraint

31 Calibration Basics 7-Point-Algorithm >8-Point-Algorithm(s) Experiments/Tricks Rank 2 Constraint Rank 3 Rank 2

32 Calibration Basics 7-Point-Algorithm >8-Point-Algorithm(s) Experiments/Tricks 7 Point Algorithm In this case n=7 and rank(u 7 )=7 SVD 2 last columns of Y span the right null-space of U => F1 and F2

33 Calibration Basics 7-Point-Algorithm >8-Point-Algorithm(s) Experiments/Tricks 7 Point Algorithm 1 Parameter family of matrices because F is homogenous Can have 1 or 3 Solutions!

34 Calibration Basics 7-Point-Algorithm >8-Point-Algorithm(s) Experiments/Tricks 8 Point Algorithm(s) Usually more than 7 Matches Many Methods (see [1]) Linear Methods Nonlinear Methods Robust Methods

35 Calibration Basics 7-Point-Algorithm >8-Point-Algorithm(s) Experiments/Tricks Linear Methods Minimize epipolar Equation or better There is a trivial solution:

36 Calibration Basics 7-Point-Algorithm >8-Point-Algorithm(s) Experiments/Tricks Eigen Analysis Impose a constraint on norm of Get unconstrained problem through Lagrange multipliers

37 Calibration Basics 7-Point-Algorithm >8-Point-Algorithm(s) Experiments/Tricks Eigen Analysis - Minimizing First derivative = 0 deriving...

38 Calibration Basics 7-Point-Algorithm >8-Point-Algorithm(s) Experiments/Tricks Eigen Analysis - Back substitution There are 9 eigenvalues Back substituting the solution Solution is the Eigenvector of associated to the smallest Eigenvalue (smallest Singular Vector of )

39 Calibration Basics 7-Point-Algorithm >8-Point-Algorithm(s) Experiments/Tricks Problems with most linear methods The rank 2 constraint is not satisfied Minimized Quantity has no physical meaning High instability (in the numerical sense) => normalize Input Data!

40 Calibration Basics 7-Point-Algorithm >8-Point-Algorithm(s) Experiments/Tricks Iterative Linear Method Minimize the distances between points and corresponding epipolar lines Symmetrical Distance

41 Calibration Basics 7-Point-Algorithm >8-Point-Algorithm(s) Experiments/Tricks Iterative Linear Method - Weights Represent distances as weights Problem: Weights depend on F Solution: Set all weights to 1 Compute F Calculate new weights from F Repeat 'till satisfaction

42 Calibration Basics 7-Point-Algorithm >8-Point-Algorithm(s) Experiments/Tricks Iterative Linear Method - Can't get no... Pros: Easy to implement Minimizes Physical quantity Cons: Rank-2 constraint isn't satisfied No significant improvement[1] Better: Nonlinear Minimization in Parameter Space [1]

43 Calibration Basics 7-Point-Algorithm >8-Point-Algorithm(s) Experiments/Tricks Robust Methods - RANSAC Exploit heuristics Try to remove "outliers" RANdom SAmple Consensus

44 Calibration Basics 7-Point-Algorithm >8-Point-Algorithm(s) Experiments/Tricks Robust Methods - RANSAC Step 1. Extract features Step 2. Compute a set of potential matches Step 3. do Step 3.1 select minimal sample Step 3.2 compute solution(s) for F (verify hypothesis) Step 3.3 determine inliers until Γ(#inliers,#samples) 95% Step 4. Compute F based on all inliers Step 5. Look for additional matches Step 6. Refine F based on all correct matches (generate hypothesis)

45 Calibration Basics 7-Point-Algorithm >8-Point-Algorithm(s) Experiments/Tricks Robust Methods - RANSAC Propability that at least one random sample is free of outliers

46 Calibration Basics 7-Point-Algorithm >8-Point-Algorithm(s) Experiments/Tricks Experiments / Tricks Image Rectification [2]

47 Calibration Basics 7-Point-Algorithm >8-Point-Algorithm(s) Experiments/Tricks Experiments / Tricks Image Rectification [2] Demos online

48 " in Stereo Motion and Object Recognition" Many (if not all) algorithms Gang Xu and Zhengyou Zhang You don't want to buy it... Get it from the library... [1]

49 "Three-Dimensional Computer Vision - A Geometric Viewpoint" Olivier Faugeras Good Basics, Interesting Chapter about Edge-Detection. Many Images. [2]

50 "Multiple View Geometry in Computer Vision" Richard Hartley and Andrew Zisserman [3]

51 Web (matlab functions) (demo) (matlab toolbox) [3]

52 That's it Thanks

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