Projective geometry- 2D Acknowledgements Marc Pollefeys: for allowing the use of his excellent slides on this topic http://www.cs.unc.edu/~marc/mvg/ Richard Hartley and Andrew Zisserman, " Multiple View Geometry in Computer Vision " Homogeneous coordinates Homogeneous representation of lines ( ) + + = T ax by c 0 a,b,c ( ) ( ) T T + + = � � ( ka ) x ( kb ) y kc 0 , k 0 a,b,c ~ k a,b,c equivalence class of vectors, any vector is representative Set of all equivalence classes in R 3 � (0,0,0) T forms P 2 Homogeneous representation of points ( ) ( ) T T x = on = + + = x , y l a,b,c ax by c 0 if and only if ( )( ) ( ) ( ) ( ) T = x,y, = T T � k � x,y, 1 a,b,c 1 l 0 x , y , 1 ~ k x , y , 1 , 0 The point x lies on the line l if and only if x T l = l T x = 0 ( ) T x , x , x Homogeneous coordinates but only 2DOF 1 2 3 Inhomogeneous coordinates ( ) T x , y Spring 2006 Projective Geometry 2D 2 1
Points from lines and vice-versa Intersections of lines = � The intersection of two lines and is l l' x l l' Line joining two points = � x x' l x x' The line through two points and is Example = y 1 = x 1 Spring 2006 Projective Geometry 2D 3 Ideal points and the line at infinity Intersections of parallel lines ( ) ( ) ( ) T and T = = T � = b � l a , b , c l' a , b , c ' l l' , a , 0 Example = = x 1 x 2 ( ) T Ideal points x 1 x , , 0 Note that this set lies on a single line, 2 ( ) l = T Line at infinity 0 , 0 , 1 � Note that in P 2 there is no distinction = � P 2 R 2 l � between ideal points and others Spring 2006 Projective Geometry 2D 4 2
Summary The set of ideal points lies on the line at infinity , intersects the line at infinity in the ideal point A line parallel to l also intersects in the same ideal point, irrespective of the value of c’. In inhomogeneous notation, is a vector tangent to the line. It is orthogonal to (a, b) -- the line normal. Thus it represents the line direction. As the line’s direction varies, the ideal point varies over . --> line at infinity can be thought of as the set of directions of lines in the plane. Spring 2006 Projective Geometry 2D 5 A model for the projective plane Points represented by rays through origin Lines represented by planes through origin x1x2 plane represents line at infinity exactly one line through two points exaclty one point at intersection of two lines Spring 2006 Projective Geometry 2D 6 3
Duality x l = = T T x l 0 l x 0 = � = � x l l' l x x' Duality principle: To any theorem of 2-dimensional projective geometry there corresponds a dual theorem, which may be derived by interchanging the role of points and lines in the original theorem Spring 2006 Projective Geometry 2D 7 Conics Curve described by 2 nd -degree equation in the plane 2 + + 2 + + + = ax bxy cy dx ey f 0 x x or homogenized x � , y � 1 2 x x 3 3 2 2 + + + + + 2 = ax bx x cx dx x ex x fx 0 1 1 2 2 1 3 2 3 3 or in matrix form � � a b / 2 d / 2 � � T = = x C x 0 with C b / 2 c e / 2 � � � � d / 2 e / 2 f � � { } a : b : c : d : e : f 5DOF: Spring 2006 Projective Geometry 2D 8 4
Conics … http://ccins.camosun.bc.ca/~jbritton/jbconics.htm Spring 2006 Projective Geometry 2D 9 Five points define a conic For each point the conic passes through + + + + + = ax 2 bx y cy 2 dx ey f 0 i i i i i i or ( ) ( ) 2 2 = = T x , x y , y , x , y , f c 0 c a , b , c , d , e , f i i i i i i stacking constraints yields � � 2 2 x x y y x y 1 1 1 1 1 1 1 � � 2 2 x x y y x y 1 � � 2 2 2 2 2 2 � � = x 2 x y y 2 x y 1 c 0 � 3 3 3 3 3 3 � 2 2 x x y y x y 1 � � 4 4 4 4 4 4 � � 2 2 x x y y x y 1 � � 5 5 5 5 5 5 Spring 2006 Projective Geometry 2D 10 5
Tangent lines to conics The line l tangent to C at point x on C is given by l= C x l x C Spring 2006 Projective Geometry 2D 11 Dual conics * = T A line tangent to the conic C satisfies l C l 0 = C � In general ( C full rank): * 1 C C* : Adjoint matrix of C. Dual conics = line conics = conic envelopes Spring 2006 Projective Geometry 2D 12 6
Degenerate conics A conic is degenerate if matrix C is not of full rank m e.g. two lines (rank 2) l = lm + T T C ml e.g. repeated line (rank 1) = T C ll l Degenerate line conics: 2 points (rank 2), double point (rank1) ( ) * � * Note that for degenerate conics C C Spring 2006 Projective Geometry 2D 13 Projective transformations Definition: A projectivity is an invertible mapping h from P 2 to itself such that three points x 1 ,x 2 ,x 3 lie on the same line if and only if h (x 1 ), h (x 2 ), h (x 3 ) do. Theorem: A mapping h : P 2 � P 2 is a projectivity if and only if there exist a non-singular 3x3 matrix H such that for any point in P 2 represented by a vector x it is true that h (x)= H x Definition: Projective transformation � x ' � � h h h � � x � � � � � 1 11 12 13 1 � � = � x ' � h h h � x � = x' H x or � � 2 21 22 23 2 � � � � � � x ' h h h x � � � � � � 8DOF 3 31 32 33 3 projectivity=collineation=projective transformation=homography Spring 2006 Projective Geometry 2D 14 7
Mapping between planes central projection may be expressed by x’=Hx (application of theorem) Spring 2006 Projective Geometry 2D 15 Removing projective distortion select four points in a plane with know coordinates + + + + x ' h x h y h x ' h x h y h = = = = x ' 1 11 12 13 y ' 2 21 22 23 + + + + x ' h x h y h x ' h x h y h 3 31 32 33 3 31 32 33 ( ) + + = + + x ' h x h y h h x h y h 31 32 33 11 12 13 (linear in h ij ) ( ) + + = + + y ' h x h y h h x h y h 31 32 33 21 22 23 (2 constraints/point, 8DOF � 4 points needed) Remark: no calibration at all necessary, better ways to compute (see later) Spring 2006 Projective Geometry 2D 16 8
Transformation of lines and conics For a point transformation = x' H x Transformation for lines = -T l' H l Transformation for conics C = -T CH -1 ' H Transformation for dual conics ' = * * T C HC H Spring 2006 Projective Geometry 2D 17 Distortions under center projection Similarity: squares imaged as squares. Affine: parallel lines remain parallel; circles become ellipses. Projective: Parallel lines converge. Spring 2006 Projective Geometry 2D 18 9
Class I: Isometries ( iso =same, metric =measure) � � � � � x ' � � cos sin t � � x � � � � � x � � = � � � � = ± � y ' � sin cos t � y � 1 � � y � � � � � � 1 0 0 1 1 � � � � � � � = orientation preserving: 1 � = � orientation reversing: 1 � R t � = = x' H E x x = R T R I � � T 0 1 � � 3DOF (1 rotation, 2 translation) special cases: pure rotation, pure translation Invariants: length, angle, area Spring 2006 Projective Geometry 2D 19 Class II: Similarities � � � ( isometry + scale ) � x ' � � s cos s sin t � � x � � � x � � � � = � � � y ' � s sin s cos t � y � � � y � � � � � � 1 0 0 1 1 � � � � � � � s R t � = = x' H x x � � T = R R I S T 0 1 � � 4DOF (1 scale, 1 rotation, 2 translation) also know as equi-form (shape preserving) metric structure = structure up to similarity (in literature) Invariants: ratios of length, angle, ratios of areas, parallel lines Spring 2006 Projective Geometry 2D 20 10
Class III: Affine transformations � x ' � � a a t � � x � � � 11 12 x � � � � = � y ' � a a t � y � � � 21 22 y � � � � � � 1 0 0 1 1 � � � � � � � A t � = = x' H A x x � � T 0 1 � � � � 0 � ( ) ( ) ( ) = � � � � = 1 A R R DR D � � � 0 � � 2 6DOF (2 scale, 2 rotation, 2 translation) non-isotropic scaling! (2DOF: scale ratio and orientation) Invariants: parallel lines, ratios of parallel lengths, ratios of areas Spring 2006 Projective Geometry 2D 21 Class VI: Projective transformations � A t � ( ) = T = = v v 1 , v x' H x x � � 2 P T v v � � 8DOF (2 scale, 2 rotation, 2 translation, 2 line at infinity) Action non-homogeneous over the plane Invariants: cross-ratio of four points on a line (ratio of ratio) Spring 2006 Projective Geometry 2D 22 11
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