Projective Geometry for Computer Vision A Projective geometry is all geometry. Arthur Cayley (1821–1895) We are familiar with the concept and measurements of Euclidean geometry, which is a good approximation to the properties of a general physical space. However, when we con- sider the imaging process of a camera, the Euclidean geometry becomes insufficient since parallelism, lengths, and angles are no longer preserved in images. In this appendix, we will briefly survey some basic concepts and properties of projective geometry which are extensively used in computer vision. For further information, readers may refer to [1, 3, 7]. Euclidean geometry is actually a subset of the projective geometry, which is more gen- eral and least restrictive in the hierarchy of fundamental geometries. Just like Euclidean geometry, projective geometry exists in any number of dimensions, such as a line in one- dimensional projective space, denoted as P1, corresponds to 1D Euclidean space R1;the projective plane in P2 is analogous to 2D Euclidean plane; the three-dimensional projective space P3 is related to 3D Euclidean space. A.1 2D Projective Geometry A.1.1 Points and Lines In Euclidean space R2, a point can be denoted as x¯ =[x,y]T , a line passing through the point can be represented as l1x + l2y + l3 = 0 (A.1) If we multiply the same nonzero scalar w on both sides of (A.1), we have l1xw + l2yw + l3w = 0 (A.2) G. Wang, Q.M.J. Wu, Guide to Three Dimensional Structure and Motion Factorization, 183 Advances in Pattern Recognition, DOI 10.1007/978-0-85729-046-5, © Springer-Verlag London Limited 2011 184 A Projective Geometry for Computer Vision =[ ]T =[ ]T A Clearly, (A.1) and (A.2) represent the same line. Let x xw,yw,w , l l1,l2,l3 , then the line (A.2) is represented as xT l = lT x = 0 (A.3) where the line is represented by the vector l, and any point on the line is denoted by x. We call the 3-vector x the homogeneous coordinates of a point in P2, which represents the same point as inhomogeneous coordinates x¯ =[xw/w,yw/w]T =[x,y]T . Similarly, we call l the homogeneous representation of the line, since for any nonzero scalar k, l and kl represent the same line. From (A.3), we find that there is actually no difference between the representation of a line and the representation of a point. This is known as the duality principal. Given =[ ]T =[ ]T two lines l l1,l2,l3 and l l1,l2,l3 , their intersection defines a point that can be computed from x = l × l =[l]×l (A.4) where ‘×’ denotes the cross product of two vectors, ⎡ ⎤ 0 −t3 t2 ⎣ ⎦ [l]× = t3 0 −t1 −t2 t1 0 denotes the antisymmetric matrix of vector l. Similarly, a line passing through two points x and x can be computed from l = x × x =[x]×x =[x ]×x (A.5) Any point with homogeneous coordinates x =[x,y,0]T corresponds to a point at infin- ity, or ideal point. Whereas its corresponding inhomogeneous point x¯ =[x/0,y/0]T makes no sense. In space plane, all ideal points can be written as [x,y,0]T . The set of these points lies on a single line l∞, which is called the line at infinity. From (A.3), it is easy to obtain the coordinates of the line at infinity l∞ =[0, 0, 1]T . A.1.2 Conics and Duel Conics In Euclidean plane, the equation of a conic in inhomogeneous coordinates is written as ax2 + bxy + cy2 + dx + ey + f = 0 (A.6) If we adopt homogeneous coordinates and denote any point on the conic by x = T [x1,x2,x3] , then the conic (A.6) can be written as the following quadratic homogeneous expression. 2 + + 2 + + + = ax1 bx1x2 cx2 dx1x3 ex2x3 fx3 0 (A.7) A.1 2D Projective Geometry 185 Fig. A.1 A point conic (a) and its dual line conic (b). x is a point on the conic xT Cx = 0, l is a line tangent to C at point x which satisfies lT C∗l = 0 or in a matrix form as ⎡ ⎤ ⎡ ⎤ ab/2 d/2 x1 T ⎣ ⎦ ⎣ ⎦ x Cx=[x1,x2,x3] b/2 ce/2 x2 = 0 (A.8) d/2 e/2 f x3 where C is the conic coefficient matrix which is symmetric. A conic has five degrees of freedom since multiplying C by any nonzero scalar does not affect the above equation. Therefore, five points in P2 at a general position (no three points are collinear) can uniquely determine a conic. Generally, a conic matrix C is of full rank. In degenerate cases, it may degenerate to two lines when rank(C) = 2, or one repeated line when rank(C) = 1. The conic defined in (A.8) is defined by points in P2, which is usually termed as point conic. According to duality principal, we can obtain the dual line conic as ∗ lT C l = 0 (A.9) where the notation C∗ stands for the adjoint matrix of C. The dual conic is also called conic envelope, as shown in Fig. A.1, which is formulated by lines tangent to C. For conics (A.8) and (A.9), we have the following Results. Result A.1 The line l tangent to the non-degenerate conic C at point x is given by l = Cx. In duality, the tangent point x to the non-degenerate line conic C∗ at line l is given by x = C∗l. Result A.2 For non-degenerate conic C and its duality C∗, we have C∗ = C−1, and (C∗)∗ = C. The line conics may degenerate to two points when rank(C∗) = 2, or one repeated point when rank(C∗) = 1, and (C∗)∗ = C in degenerate cases. Any point x and a conic C define a line l = Cx, as shown in Fig. A.2. Then x and l forms a pole-polar relationship. The point x is called the pole of line l with respect to conic C, and the line l is called the polar of point x with respect to conic C. It is easy to verify the following Result. 186 A Projective Geometry for Computer Vision Fig. A.2 The pole-polar A relationship. The line l = Cx is the polar of point x with respect to conic C,andthe point x = C−1l is the pole of l with respect to conic C Result A.3 The polar line l = Cx intersects the conic C at two points x1 and x2, then the two lines l1 = x × x1 and l2 = x × x2 are tangent to the conic C. If the point x is on the conic, then the polar is the tangent line to C at x. Result A.4 Any two points x and y satisfying xT Cy = 0 are called conjugate points with respect to C. The set of all conjugate points of x forms the polar line l. If x is on the polar of x, then x is also on the polar of x since xT Cx = xT Cx = 0. In duality, two lines l and l are conjugate with respect to C if lT Cl = 0. Result A.5 There are a pair of conjugate ideal points ⎡ ⎤ ⎡ ⎤ 1 1 i = ⎣i⎦ , j = ⎣−i⎦ 0 0 on the line at infinity l∞. We call i and j the canonical forms of circular points. Essentially, the circular points are the intersection of any circle with the line at infinity. Thus three additional points can uniquely determine a circle, which is equivalent to the fact that five general points can uniquely determine a general conic. The dual of the circular point forms ∗ T T a degenerated line conic given by C∞ = ij + ji . A.1.3 2D Projective Transformation Two dimensional projective transformation is an invertible linear mapping H : P2 → P2 which is a 3 × 3 matrix. The transformation is also known as projectivity, or homography. T The mapping of a point x =[x1,x2,x3] can be written as ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ x1 h11 h12 h13 x1 ⎣ ⎦ = ⎣ ⎦ ⎣ ⎦ x2 h21 h22 h23 x2 (A.10) x3 h31 h32 h33 x3 A.2 3D Projective Geometry 187 or more briefly as x = Hx. This is a homogeneous transformation which is defined up to scale. Thus there are only 8 degrees of freedom on H. Four pairs of corresponding points can uniquely determine the transformation if no three points are collinear. The transforma- tion (A.10) is defined by points. Result A.6 Under a point transformation x = Hx, a line l is transformed to l via − l = H 1l (A.11) A conic C is transformed to C via − − C = H T CH 1 (A.12) and a dual conic C∗ is transformed to C∗ via ∗ ∗ C = HC HT (A.13) All projective transformations form a group which is called projective linear group. There are some specializations or subgroups of the transformation, such as affine group, Euclidean group, and oriented Euclidean group. Different transformations have different geometric invariance and properties. For example, length and area are invariant under Euclidean transformation; parallelism and line at infinity are invariant under affine trans- formation; general projective transformation preserves concurrency, collinearity, and cross ratio. A.2 3D Projective Geometry A.2.1 Points, Lines, and Planes In 3D space P3, the homogeneous coordinates of a point is represented by a 4-vector T as X =[x1,x2,x3,x4] , which is defined up to a scale since X and sX (s = 0) repre- sent the same point.