Method and system for determining projections in non-central catadioptric optical systems

• US 8,134,786 B2
• Filed: 06/30/2010
• Issued: 03/13/2012
• Est. Priority Date: 06/30/2010
• Status: Expired due to Fees

• Current Assignee: Mitsubishi Electric Research Laboratories, Inc. (Mitsubishi Electric Corporation)
• Original Assignee: Mitsubishi Electric Research Laboratories, Inc. (Mitsubishi Electric Corporation)
• Inventors: Taguchi, Yuichi, Agrawal, Amit K., Ramalingam, Srikumar
• Primary Examiner(s): Spector, David N
• Application Number:
  US12/826,823
• Publication Number:
  US 20120002304A1
• Time in Patent Office: 622 Days
• Field of Search: 359/364, 359/726, 359/727, 359/728, 359/729, 359/730, 359/731, 359/732, 359/733, 359/734, 359/735, 359/736, 382/154, 703/2
• US Class Current: 359/726
• CPC Class Codes: G02B 17/08 Catadioptric systems used in non-imaging applications G02B19/00
• CPC Class Codes: G02B 27/0012 Optical design, e.g. procedures, algorithms, optimisation routines

Abstract

Embodiment of invention discloses a system and a method for determining a three-dimensional (3D) location of a folding point of a ray between a point in a scene (PS) and a center of projection (COP) of a camera of a catadioptric system. One embodiment maps the catadioptric system, including 3D locations of the PS and the COP on a two-dimensional (2D) plane defined by an axis of symmetry of a folding optical element and the PS to produce a conic and 2D locations of the PS and COP on the 2D plane, and determines a 2D location of the folding point on the 2D plane based on the conic, the 2D locations of the PS and the COP. Next, the embodiment determines the 3D location of the folding point from the 2D location of the folding point on the 2D plane.

Claims

1. A method for determining a three-dimensional (3D) location of at least one folding point of a ray between a point in a scene (PS) and a center of projection (COP) of a camera of a catadioptric system, wherein the catadioptric system includes the camera arranged at a distance from a surface of a folding optical element, wherein the folding optical element is selected from a reflector or a refractor, wherein the folding point is on the surface and is caused by a reflection or a refraction of the ray on the surface such that the folding point partitions the ray into an incoming ray and an outgoing ray, and wherein the PS and the COP are identified by 3D locations, comprising steps of:

acquiring a configuration of the catadioptric system, wherein the catadioptric system is non central, the surface is a quadric surface rotationally symmetric around an axis of symmetry of the surface, and wherein the COP is arranged on the axis of symmetry;

mapping the surface, and the 3D locations of the PS and the COP on a two-dimensional (2D) plane defined by the axis of symmetry and the PS to produce a conic and 2D locations of the PS and COP on the 2D plane, wherein the conic is a parameterization of the surface on the 2D plane;

determining a 2D location of the folding point on the 2D plane based on the conic, the 2D locations of the PS and the COP, and at least one constraint on the incoming ray and the outgoing ray; and

determining the 3D location of the folding point from the 2D location of the folding point on the 2D plane, wherein the steps of the method are performed by a processor.

2. The method of claim 1, wherein the folding optical element is the refractor, the folding point is a point of refraction of the ray from the surface, and the constraint is a law of refraction.

3. The method of claim 2, wherein the refractor is a refractive sphere, and the axis of symmetry passes through a center of the refractive sphere and the COP.

4. The method of claim 3, wherein at least one folding point includes a first refraction point n₁=[x,y]^T and a second refraction point n₂=[x₂,y₂]^T, the constraints are

5. The method of claim 1, further comprising:

determining a 2D image projection of the PS from 3D location of the folding point according to

6. The method of claim 5, further comprising:

determining concurrently 3D points in the scene and parameters of the catadioptric system using bundle adjustment.

7. The method of claim 5, further comprising:

determining a Jacobian matrix of a reprojection error.

8. The method of claim 7, further comprising:

determining concurrently 3D points in the scene and parameters of the catadioptric system using the Jacobian matrix and bundle adjustment.

9. The method of claim 1, wherein the folding optical element is the reflector, the folding point is a point of reflection of the ray from the surface, and the constraint is a law of reflection.

10. The method of claim 9, wherein the reflector is a spherical mirror, and the axis of symmetry passes through a center of the spherical mirror and the COP.

11. The method of claim 9, wherein the constraint is

v_r^Tn=−

v_i^Tn,

and a second coordinate y of the 2D location of the folding point is determined according to a forward projection (FP) equation

u²K₁²(y)+K₂²(y)(Ay²+By−

C)=0, and

a first coordinate x of the 2D location of the folding point is determined based on the second coordinate y according to

x=sign(u)√

{square root over (C−

By−

Ay²)},

wherein v_i is a vector representing the incoming ray, v_r is a vector representing the outgoing ray, T is a transpose operator, n is a vector representing a normal of the surface at the folding point, u is a first coordinate of the 2D location of the PS, K₁(y) and K₂(y) are reflection polynomials in y, A, B, C are parameters for the conic.

12. The method of claim 11, further comprising:

determining the 3D location of the folding point according to

xz₁/∥

z₁∥

+yz₂/∥

z₂∥

,

wherein z₁ and z₂ are axes determining a coordinate system of the 2D plane.

13. The method of claim 11, wherein the folding optical element is a spherical mirror and the FP equation is

u²(r²(d+y)−

2dy²)²−

(r²−

y²)(r²(d+v)−

2dvy)²=0,

wherein (u, v) is the 2D location of the PS, y is a second coordinate of the 2D location of the folding point, r is a radius of the spherical mirror, d is a distance between the COP and a center of the sphere.

14. A catadioptric system configured for determining a three-dimensional (3D) location of at least one folding point of a ray between a point in a scene (PS) and a center of projection (COP) of the catadioptric system, wherein the PS and the COP are identified by 3D locations, and the catadioptric system is non central, comprising:

a folding optical element, wherein the folding point is caused by folding of the ray on a surface of the folding optical element, wherein the surface is a quadric surface rotationally symmetric around an axis of symmetry;

a camera arranged at a distance from the surface, wherein the COP of the catadioptric system is a COP of the camera and arranged on the axis of symmetry;

means for mapping the surface, and the 3D locations of the PS and the COP on a two-dimensional (2D) plane defined by the axis of symmetry and the PS to produce a conic and 2D locations of the PS and COP on the 2D plane, wherein the conic is a parameterization of the surface on the 2D plane;

a processor for determining a 2D location of the folding point on the 2D plane based on the conic, the 2D locations of the PS and the COP; and

means for determining the 3D location of the folding point from the 2D location of the folding point on the 2D plane.

15. The catadioptric system of claim 14, wherein the folding optical element is a reflector, the folding point is a point of reflection of the ray from the surface.

16. The catadioptric system of claim 14, wherein the folding optical element is a refractor, the folding point is a point of refraction of ray from the surface.

17. The catadioptric system of claim 14, further comprising:

means for determining a 2D image projection of the PS from 3D location of the folding point according to

18. The catadioptric system of claim 14, further comprising:

means for determining concurrently 3D points in the scene and parameters of the catadioptric system using bundle adjustment.

19. A method for determining a three-dimensional (3D) location of at least one folding point of a ray between a point in a scene (PS) and a center of projection (COP) of a catadioptric system, wherein the PS and the COP are identified by 3D locations, and the catadioptric system is non central, comprising steps of:

providing a folding optical element, wherein the folding point is caused by folding of the ray on a surface of the folding optical element, wherein the surface is a quadric surface rotationally symmetric around an axis of symmetry;

providing a camera arranged at a distance from the surface, wherein the COP of the catadioptric system is a COP of the camera and arranged on the axis of symmetry;

mapping the surface, and the 3D locations of the PS and the COP on a two-dimensional (2D) plane defined by the axis of symmetry and the PS to produce a conic and 2D locations of the PS and COP on the 2D plane, wherein the conic is a parameterization of the surface on the 2D plane;

determining a 2D location of the folding point on the 2D plane based on the conic, the 2D locations of the PS and the COP; and

determining the 3D location of the folding point from the 2D location of the folding point on the 2D plane, wherein the steps of the method are performed by a processor.

20. The method of claim 19, wherein the folding optical element is selected from a reflector and a refractor.