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 3DVision Problems Correspondence problem Reconstruction problem
4 Motivation Depth Perception Ambiguity of Correspondence The Epipolar Constraint Motivation 3DVision 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 3DVector 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 7PointAlgorithm >8PointAlgorithm(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 7PointAlgorithm >8PointAlgorithm(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 7PointAlgorithm >8PointAlgorithm(s) Experiments/Tricks Definitions The Epipolar Equation can be written as linear homogeneous equation
28 Calibration Basics 7PointAlgorithm >8PointAlgorithm(s) Experiments/Tricks Definitions For n point matches
29 Calibration Basics 7PointAlgorithm >8PointAlgorithm(s) Experiments/Tricks Not that easy Noise Leastsquare 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 7PointAlgorithm >8PointAlgorithm(s) Experiments/Tricks Rank 2 Constraint
31 Calibration Basics 7PointAlgorithm >8PointAlgorithm(s) Experiments/Tricks Rank 2 Constraint Rank 3 Rank 2
32 Calibration Basics 7PointAlgorithm >8PointAlgorithm(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 nullspace of U => F1 and F2
33 Calibration Basics 7PointAlgorithm >8PointAlgorithm(s) Experiments/Tricks 7 Point Algorithm 1 Parameter family of matrices because F is homogenous Can have 1 or 3 Solutions!
34 Calibration Basics 7PointAlgorithm >8PointAlgorithm(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 7PointAlgorithm >8PointAlgorithm(s) Experiments/Tricks Linear Methods Minimize epipolar Equation or better There is a trivial solution:
36 Calibration Basics 7PointAlgorithm >8PointAlgorithm(s) Experiments/Tricks Eigen Analysis Impose a constraint on norm of Get unconstrained problem through Lagrange multipliers
37 Calibration Basics 7PointAlgorithm >8PointAlgorithm(s) Experiments/Tricks Eigen Analysis  Minimizing First derivative = 0 deriving...
38 Calibration Basics 7PointAlgorithm >8PointAlgorithm(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 7PointAlgorithm >8PointAlgorithm(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 7PointAlgorithm >8PointAlgorithm(s) Experiments/Tricks Iterative Linear Method Minimize the distances between points and corresponding epipolar lines Symmetrical Distance
41 Calibration Basics 7PointAlgorithm >8PointAlgorithm(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 7PointAlgorithm >8PointAlgorithm(s) Experiments/Tricks Iterative Linear Method  Can't get no... Pros: Easy to implement Minimizes Physical quantity Cons: Rank2 constraint isn't satisfied No significant improvement[1] Better: Nonlinear Minimization in Parameter Space [1]
43 Calibration Basics 7PointAlgorithm >8PointAlgorithm(s) Experiments/Tricks Robust Methods  RANSAC Exploit heuristics Try to remove "outliers" RANdom SAmple Consensus
44 Calibration Basics 7PointAlgorithm >8PointAlgorithm(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 7PointAlgorithm >8PointAlgorithm(s) Experiments/Tricks Robust Methods  RANSAC Propability that at least one random sample is free of outliers
46 Calibration Basics 7PointAlgorithm >8PointAlgorithm(s) Experiments/Tricks Experiments / Tricks Image Rectification [2]
47 Calibration Basics 7PointAlgorithm >8PointAlgorithm(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 "ThreeDimensional Computer Vision  A Geometric Viewpoint" Olivier Faugeras Good Basics, Interesting Chapter about EdgeDetection. 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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