Projective Geometry Shao-Yi Chien Department of Electrical - PowerPoint PPT Presentation

Projective Geometry 簡韶逸 Shao-Yi Chien Department of Electrical Engineering National Taiwan University Fall 2019 1

Outline • Projective 2D geometry • Projective 3D geometry [Slides credit: Marc Pollefeys] 2

Projective 2D Geometry • Points, lines & conics • Transformations & invariants • 1D projective geometry and the cross-ratio 3

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  Homogeneous representation of points     x      T T on x , y l a,b,c 0 if and only if ax by c           x,y,   k  T T T 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 The point x= (𝑦 1 , 𝑦 2 , 𝑦 3 )T represent the point (𝑦 1 /𝑦 3 , 𝑦 2 /𝑦 3 )T in ℝ 2 4

Points and Lines  Intersections of lines   The intersection of two lines and is x l l' l l'  Line joining two points   The line through two points and is x l x x' x' Example  y 1  x 1 5

Ideal Points and the Line at Infinity  Intersections of parallel lines       T and   b    T T l l' , a , 0 l a , b , c l' a , b , c ' Example   b  tangent vector (line’s direction) , a   a , b normal direction   x 1 x 2   T Ideal points x 1 x , , 0 2    T Line at infinity l 0 , 0 , 1  Note that in P 2 there is no distinction   2 2 P R l  between ideal points and others 6

A Model for the Projective Plane x 3 =1 exactly one line through two points exaclty one point at intersection of two lines 7

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 8

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 with x C x 0 C b / 2 c e / 2       d / 2 e / 2 f symmetric   5DOF: a : b : c : d : e : f 9

Five Points Define a Conic For each point the conic passes through       2 2 ax bx y cy dx ey f 0 i i i i i i or       T 2 2 , , , , , c 0 x x y y x y f 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    2 2 c 0 x x y y x y 1   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 10

Tangent Lines to Conics The line l tangent to C at point x on C is given by l= C x l x C 11

Dual Conics *  T A line tangent to the conic C satisfies l C l 0   C * 1 In general ( C full rank): C Dual conics = line conics = conic envelopes 12

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 reprented 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 14

Mapping between Planes central projection may be expressed by x’=Hx (application of theorem) 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     1 11 12 13 2 21 22 23 x ' y '     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 16

More Examples 17

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 18

A Hierarchy of Transformations  Projective linear group  Affine group (last row (0,0,1))  Euclidean group (upper left 2x2 orthogonal)  Oriented Euclidean group (upper left 2x2 det 1) Alternative, characterize transformation in terms of elements or quantities that are preserved or invariant e.g. Euclidean transformations leave distances unchanged Affine Projective Similarity 19

Class I: Isometries           x ' cos sin t x     x              y ' sin cos t y 1   y             1 0 0 1 1   (Euclidean transform) orientation preserving: 1    1 orientation reversing:   t R    x' H E x   x T R R I T   1 0 3DOF (1 rotation, 2 translation), can be computed from 2 point correspondences special cases: pure rotation, pure translation Invariants: length, angle, area 20

Class II: Similarities          x ' s cos s sin t x     x          y ' s sin s cos t y   y             1 0 0 1 1   t s R   x' H x   x  T R R I S T   1 0 4DOF (1 scale, 1 rotation, 2 translation), can be computed from 2 point correspondences also know as equi-form (shape preserving) Invariants: ratios of length, angle, ratios of areas, parallel lines 21

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   t A     x' H A x x T  1  0    0             1 A R R DR D      0 2 6DOF (2 scale, 2 rotation, 2 translation), can be computed from 3 point correspondences non-isotropic scaling! (2DOF: scale ratio and orientation) Invariants: parallel lines, ratios of parallel lengths, ratios of areas 22

Class VI: Projective Transformations   t A      T x' H x   x v v 1 , v P T 2   v v 8DOF (2 scale, 2 rotation, 2 translation, 2 line at infinity) can be computed from 4 point correspondences Action non-homogeneous over the plane Invariants: cross-ratio of four points on a line, (ratio of ratio) 23

Action of Affinities and Projectivities on Line at Infinity       x x       1   t 1 A A          x   x 2 T    2  0 v       0 0 Line at infinity stays at infinity, but points move along line       x x       1   t 1 A A          x   x 2 T    2  v v      v x v x   0 1 1 2 2 Line at infinity becomes finite, allows to observe vanishing points, horizon 24