Projective Geometry and Camera Models

1. Projective Geometry and Camera Models Lu Zhang Imaging, Robotics, & Intelligent Systems Laboratory The University of Tennessee March 6, 2005

2. Outline • Projective Geometry - Basic principles in Projective Geometry - The projective line - The projective plane - The projective space • Camera Models - Modeling cameras - Changing coordinate systems - Calibrating cameras

3. Projective Geometry • Why projective geometry 1. Euclidean geometry is a special case of projective geometry. 2. Any system of lenses can be approximated by a system that realize a perspective projection of the world onto a plane 3. Projective spaces are extremely useful to understand the structure of the set of 3-D rotations 4. It’s used as the basis of many techniques in robotics and computer graphics

4. Projective Geometry • Basic principles in projective spaces: • Points: A point of a dimensional projective space is represented by an n+1 vector of coordinates , are called the homogeneous or projective coordinates of the points, x is called a a coordinate vector. If for the two n+1 vectors represent the same point.

5. Projective Geometry • Collineations: A (n+1)*(n+1) matrix A defines a linear transformation or collineation from into itself The matrix associate with a given collineation is defined up to a non zero scale factor • Projective basis: It is a set of n+2 points of , no n+1 of them are linearly dependent. • Change of projective basis: If there are two sets of n+2 points, there exists a unique collineation that maps the first set of points onto the second.

6. Projective Geometry • The Projective Line The space is known as projective line. The standard projective basis of projective line is and . A point on the line is , and are not both=0. • The point at infinity: If let , when=0, -> infinity, we call this point ‘point at infinity’

7. Projective Geometry • Cross ratio: {a,b;c,d}= The significant of the cross-ratio is that it is invariant under collineations of , and it is also independent of the choice of coordinates in

8. Projective Geometry • Projective plane: • Points • Lines • Principle of duality: Any theorem or statement that is true for the projective plane can be reworded by substituting points for lines and lines for Points.

9. Projective Geometry • Useful equations: 1. the point x lies on the line l if and only if ; 2. given two lines l and l’, the intersection of two lines is the point ; 3. the line through two points x and x’ is ; 4. three points p1, p2, and p3 lie on the same line if ; 5. three lines u1, u2, and u3 intersect at the same point if .

10. Projective Geometry • Line at infinity: The points with =0 are said to be at infinity or ideal points. The set of all ideal points may be written (X; Y;0). The set of all ideal points lies on a single line, the line at infinity. • Conics: A conic is a curve defined by the locus of points of the projective plane that satisfy the equation

11. Projective Geometry for all i,j and form a 3*3 symmetric matrix A, therefore A depend on 5 independent parameters. • Intersection of a conic with a line: Q,R are two points of the plane. A variant point on the line<Q,R> is if the point on this conic, it can be written as then we can get in which

12. Projective Geometry When the line <Q,R> is tangent to s. • Affine transform of the plane: There is a one-to-one correspondence between the usual affine plane and the projective plane minus the line at infinity. x’=Bx+b, B is a 2*2 matrix of rank 2, b is a 2*1 vector.

13. Projective Geometry If let A be the matrix of a collineation that leaves line at infinity invariant Then Further specialize the set of affine transformation let also preserve two special points-absolute points (1,+/-i,0)

14. Projective Geometry We have the equation Therefore This is called similarity transformation.

15. Projective Geometry • Projective space Points: Planes: • The principle of duality: • The principle of duality exists in between points and planes. A point x can be represented by u, Lines: A line is defined as the set of points that are linearly dependent on two points.

16. Projective Geometry Points on projective space are represented as The line between two points is Because ,only six of these numbers are independent,

17. Projective Geometry • The plane at infinity Among all possible planes, the one whose points are is called the plane at infinity.

18. Camera models • A simple camera model C- is the optical center; R- indicate the retinal plane in which the image is formed through an operation called a perspective projection; M-is a real world point f- is the focal length of the optical system; C- is the projective of optical center on retinal plane; Cc- the optical axis;

19. Camera models Such a system can accurately model the geometry and optics of most of the modern cameras, for example CCD, CID and Vidicon. • For a world point (X,Y,Z) and the corresponding image point (x,y). Using the,a linear projection equation is

20. Camera models • The perspective projection matrix If consider the effect of focal length f, the relationship between image coordinate and 3-D space coordinate can be written as therefore u=U/S v=V/S if S≠0, m=PM

21. Camera models This projective transformation can preserve the property of the points at infinity plane. • Changing coordinate systems -Changing coordinates in the retinal plane first consider the effect on the perspective projection matrix P of changing the origin of the image and the units on the u,v axes. This units are determined by the property of camera itself.

22. Camera models • Intrinsic calibration Let’s consider a simple example. The graph shows the transformation from retinal plane to itself. The 3*3 matrix H is given by

23. Camera models Cause We have and thus

24. Camera models If denote the coordinate of t by u,v ,the most general matrix P is Given a matrix P, if change the retinal coordinate system so that matrix P can be written

25. Camera models Let and The corresponding matrix H is There are 4 intrinsic parameters for each camera.

26. Camera models • Camera Motion (Extrinsic parameters) If we go from the old coordinate system centered at C to the new coordinate system centered at O by a rotation R followed by a translation T, in projective coordinates The 4*4 matrix K is The similar to get intrinsic parameter, K represent a collineation that preserve the plane at infinity.

27. Camera models We have Therefore The combination of intrinsic parameters and extrinsic parameters is

28. Camera models • The general form of Matrix P if write the matrix P as where Qs are 4*1 vectors, then we can rewrite P as

29. Camera models • Conference 1. 'Three Dimensional Computer Vision' by Olivier Faugeras 2.‘Self-Calibration and Metric 3D Reconstruction from Uncalibrated image sequences’ by Dr. Marc Pollefeys 3. 'An Introduction to Project Theory (for computer vision)' by Stan Birchfield

30. Thanks for your attention Any questions?