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

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.

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.