Analytical Forward Projection for Axial Non-Central Dioptric and Catadioptric Cameras
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1 Analytical Forward Projection for Axial Non-Central Dioptric and Catadioptric Cameras Amit Agrawal Yuichi Taguchi Srikumar Ramalingam Mitsubishi Electric Research Labs (MERL)
2 Perspective Cameras (Central) Perspective Camera Single Viewpoint (Central) Single Viewpoint
3 Single-Viewpoint Catadioptric Cameras Mirror + Perspective Camera Wide Field of View Perspective Camera Single Viewpoint (Central) Hyperbolic Mirror Virtual Viewpoint
4 Single-Viewpoint Catadioptric Cameras [Baker & Nayar 99] Camera Foci Hyperbola/Parabola/Ellipse Virtual Viewpoint Only a few single-viewpoint configurations Other configurations lead to non-single viewpoint Spherical mirror Camera not on foci Multiple mirrors
5 Non-Central Catadioptric Cameras Perspective Camera Camera looking into four spherical mirrors Spherical Mirror Locus of Viewpoint (Caustic) Can we analytically model the projection of 3D points to pixels?
6 Yuichi
7 Non-Central Dioptric Camera Looking through a refractive glass sphere Google Crystal Ball Photography Refractive Sphere Perspective Camera
8 Goal Exact modeling of non-central cameras Rotationally Symmetric Conic Mirrors & Refractive Sphere Axial configuration: Camera placed on the axis Avoid approximations in modeling Central Approximation General linear cameras (GLC) approximation Yu and McMillan, ECCV 2004 Fast processing Similar computational complexity as perspective camera
9 Why are non-central cameras difficult to model? Following two operations are essential for any camera Back Projection What is the 3D ray corresponding to a pixel? Generic Camera Calibration Grossberg and Nayar, ICCV 2001 Sturm & Ramalingam, ECCV 2004 Ramalingam et al. CVPR 2005 Forward Projection What is the projection of a 3D point? Inverse Ray Tracing camera Compute the Light-Path pixel p 3D point P
10 Forward Projection: Easy for Perspective Camera P = [X,Y,Z] x,y x = f*x/z y = f*y/z COP Imaging Plane
11 Easy for Central Catadioptric Camera Pinhole Pixel p P = [X,Y,Z] Mirror Intersection Point Effective COP
12 But, difficult for Non-Central Camera No effective COP Pinhole Pixel p P = [X,Y,Z] How to find the Mirror Intersection Point?
13 Forward Projection for Non-Central Camera Given a 3D point P What is the corresponding image pixel p? Difficult, no closed form solution Can do optimization (Micusik and Pajdla, CVPR 2004) Iterative Forward Projection Slow Can we obtain analytical solution for forward projection for non-central cameras?
14 Analytical Forward Projection for Non-Central Cameras Axial Configuration Camera lies on the axis of rotationally symmetric mirror For conic mirrors Solve 6 th degree equation in one unknown Reduces to 4 th degree equation for spherical mirror Closed Form Solution For refractive sphere Solve 10 th degree equation in one unknown 100 times speed up 3D reconstruction using bundle adjustment
15 Finding the Mirror Intersection Point Mirror Equation
16 [x,y] lies on the mirror Constraints Mirror Equation
17 Constraints Planarity Incoming ray (v i ), normal (n) and reflected ray (v r ) lie on same plane Angle constraint θ 1 = θ 2 Use vector form of law of reflection θ 1 θ 2
18 Constraint The reflected ray should pass through the given point Cross(v r, p m) = 0
19 Solution Mirror Equation
20 Solution 6 th degree equation in y 6 solutions Get the correct solution by checking law of reflection Obtain x using 20 lines of Matlab Code
21 Mirror Shape Degree of Forward Projection Equation for Various Configurations
22 25 20 Normal COP P 5 Mirror Intersection Point Visualization for Spherical Mirror
23 COP 15 P Normal 10 5 Mirror Axis 0 Mirror Intersection Point Visualization for Hyperbolic Mirror
24 Forward Projection for Spherical Mirror Also known as Alhazen problem or Circular Billiard Problem One of the classical problem in geometry Can be traced back to Ptolemy Optics (AD 150) Described in Book of Optics (~1000 A.D) by Alhazen References Marcus Baker, Alhazen problem, American Journal of Mathematics, Vol. 4, No. 1, 1881, pp Heinrich Dorrie, 100 Great Problems of Elementary Mathematics Four solutions Intersection of circle and hyperbola Forward projection for general mirrors Extension of Alhazen problem
25 Related Work Optimization for Forward Projection Micusik and Pajdla, CVPR 2004 Iterative Forward Projection (IFP) Local perturbation method Chen and Arvo, SIGGRAPH 2000 Approximation for fast rendering GLC Approximation Ding et al. ICCV 2009 Bertrand Vandeportaele Thesis, 2006 (in French) Analysis in 3D Higher degree (8 th degree) equation for ellipsoid
26 Results: Fast Projection of 3D Points Project randomly generated 100,000 points (spherical mirror) Matlab on standard PC 1120 seconds for IFP 13.8 seconds for AFP Speed up ~80 Similar speed up for hyperbolic, elliptical and parabolic mirrors
27 Sparse 3D Reconstruction: Simulation Single perspective camera looking at 4 spherical mirrors Perturb sphere centers and add image noise Bundle Adjustment of sphere centers and 3D points
28 Sparse 3D Reconstruction: PovRay Simulation Feature extraction using SIFT Iterate bundle adjustment and outlier removal
29 Sparse 3D Reconstruction: Real Data Stainless Steel Balls
30 Epipolar Curves (Spherical Mirror) Perspective Cameras Epipolar geometry Epipolar lines Central Catadioptric Cameras Epipolar curves (2 nd order) Svoboda & Pajdla, IJCV 2002 Non-central Catadioptric Camera (Spherical Mirror) Projection of a line == 4 th order curve (Quartic curves) Derivation similar to Strum & Barreto, ECCV 2008 Back-projection matrix using lifted image coordinates
31 Forward Projection for Refractive Sphere Refractive index μ Key idea 1: Analysis can be done in a plane
32 Key idea 2 v i θ 1 n Snells Law: μ 1 sin(θ 1 ) = μ 2 sin(θ 2 ) Medium 1 Medium 2 θ 2 v r Use vector form of the refraction equation
33 Solution
34 Solution 10 th degree equation in y
35 Visualizing Solutions Radius r=1 Refractive Index μ = 1.5 D = 5 (distance of camera from refractive sphere) 8 real solutions Removing invalid refraction points Checking Snell s Law
36 3D Reconstruction using Refractive Sphere Single camera looking through 4 refractive spheres Perturb sphere centers and add image noise Jointly optimize sphere centers and 3D points
37 Summary Analytical Forward Projection Axial Non-Central Catadioptric cameras Hyperbolic, Elliptical, Spherical and Parabolic Mirrors Refractive Sphere Poor man s fish-eye lens Avoid central and GLC approximation Can use exact non-central model 100 times speed up over iterative approach Sparse 3D reconstruction using bundle adjustment Epipolar curves Quartic curves for Spherical Mirror
38 Acknowledgments Peter Sturm, INRIA MERL Jay Thornton, Keisuke Kojima, John Barnwell Mitsubishi Electric, Japan Haruhisa Okuda, Kazuhiko Sumi
39 Analytical Forward Projection for Non-Central Cameras Axial Catadioptric Camera Solve 6 th degree Equation Refractive Sphere Solve 10 th degree Equation
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