Projective Geometry. Projective Geometry

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2 Euclidean versus Euclidean geometry describes sapes as tey are Properties of objects tat are uncanged by rigid motions» Lengts» Angles» Parallelism Projective geometry describes objects as tey appear Lengts, angles, parallelism become distorted wen we look at objects Matematical model for ow images of te 3D world are formed.

3 Overview Tools of algebraic geometry Informal description of projective geometry in a plane Descriptions of lines and points Points at infinity and line at infinity Projective transformations, projectivity matri Eample of application Special projectivities: affine transforms, similarities, Euclidean transforms Cross-ratio invariance for points, lines, planes

4 Tools of Algebraic Geometry 1 n,, n Plane passing troug origin and perpendicular to vector is locus of points suc tat > a ( b 2 + c 3 Plane troug origin is completely defined by 3 ( 1, 2, 3 2 O 1 n

5 Tools of Algebraic Geometry 2 A vector parallel to intersection of 2 planes is obtained by cross-product ( a'', b'', c'' and ( a', b', c' ( a '', b' ', c'' ( a', b', c' O ( a', b', c'

6 Tools of Algebraic Geometry 3 Plane passing troug two points and is defined by ' ( 1, 2, 3 ' ( 1 ', 2', 3 ' O

7 in 2D We are in a plane P and want to describe lines and points in P We consider a tird dimension to make tings easier wen dealing wit infinity Origin O out of te plane, at a distance equal to 1 from plane To eac point m of te plane P we can associate a single ray To eac line l of te plane P we can associate a single plane ( 1, 2, 3 L 3 ( 1, 2, 3 l m P 2 O 1

8 in 2D,, λ, λ, λ ( 1 3 ( 1 3 Te rays 2 and 2 are te same and are mapped to te same point m of te plane P X is te coordinate vector of m, ( 1, 2, 3 are its omogeneous coordinates Te planes and ( λ a, λ λ are te same and are mapped to te same line l of te plane P L is te coordinate vector of l, are its omogeneous coordinates L ( 1, 2, 3 l m P O

9 Properties Point X belongs to line L if L. X Equation of line L in projective geometry is We obtain omogeneous equations a 1 + b 2 + c 3 L ( 1, 2, 3 l m P O

10 From Projective Plane to Euclidean Plane How do we land back from te projective world to te 2D world of te plane? ( λ 1, λ 2, λ 3 1 1/, 3 λ 3 m ( 1 / 3, 2 / 3 1 a 1 + b 2 + c 3 3 a + b + c For point, consider intersection of ray wit plane > For line, intersection of plane wit plane is line l: 1 2 L ( 1, 2, 3 l m P O

11 Lines and Points Two lines L (a, and L (a,b,c intersect in te point L L' Te line troug 2 points and is L Duality principle: To any teorem of 2D projective geometry, tere corresponds a dual teorem, wic may be derived by intercanging te roles of points and lines in te original teorem ' ( 1, 2, 3 L L P O ( a', b', c'

12 Ideal Points and Line at Infinity Te points ( 1, 2, do not correspond to finite points in te plane. Tey are points at infinity, also called ideal points Te line L (,,1 passes troug all points at infinity, since L. Two parallel lines L (a, and L (a, c intersect at te point (c -( -a,, i.e. ( -a, L L' Any line (a, intersects te line at infinity at ( -a,. So te line at infinity is te set of all points at infinity 3 P l 1 m ( 1, 2, (,,1 O

13 Ideal Points and Line at Infinity Wit projective geometry, two lines always meet in a single point, and two points always lie on a single line. Tis is not true of Euclidean geometry, were parallel lines form a special case.

14 Projective Transformations in a Plane Projectivity Mapping from points in plane to points in plane 3 aligned points are mapped to 3 aligned points Also called Collineation Homograpy

15 Projectivity Teorem A mapping is a projectivity if and only if te mapping consists of a linear transformation of omogeneous coordinates Proof: ' H wit H non singular If 1, 2, and 3 are 3 points tat lie on a line L, and 1 H 1, etc, ten 1, 2, and 3 lie on a line L L T i, L T H -1 H i, so points H i lie on line H -T L Converse is ard to prove, namely if all collinear sets of points are mapped to collinear sets of points, ten tere is a single linear mapping between corresponding points in omogeneous coordinates

16 Projectivity Matri ' 1 ' 2 ' ' H Te matri H can be multiplied by an arbitrary non-zero number witout altering te projective transformation Matri H is called a omogeneous matri (only ratios of terms are important Tere are 8 independent ratios. It follows tat projectivity as 8 degrees of freedom A projectivity is simply a linear transformation of te rays

17 Eamples of Projective Transformations Central projection maps planar scene points to image plane by a projectivity P O True because all points on a scene line are mapped to points on its image line Te image of te same planar scene from a second camera can be obtained from te image from te first camera by a projectivity True because i H i, i H i O M y P P M M y so i H H -1 i

18 Computing Projective Transformation Since matri of projectivity as 8 degrees of freedom, te mapping between 2 images can be computed if we ave te coordinates of 4 points on one image, and know were tey are mapped in te oter image Eac point provides 2 independent equations ' ' ' y + y ' 11 ' + ' 12 + ' 31 y + ' y y' ' ' y y ' 21 + ' 22 ' + ' y + ' y Equations are linear in te 8 unknowns ij ij / 33

19 Eample of Application Robot going down te road Large squares painted on te road to make it easier Find road sape witout perspective distortion from image Use corners of squares: coordinates of 4 points allow us to compute matri H Ten use matri H to compute 3D road sape

20 Special Projectivities Projectivity 8 dof Invariants Collinearity, Cross-ratios Affine transform 6 dof a a a a t t 1 Parallelism, Ratios of areas, Lengt ratios Similarity 4 dof s r s r s r s r t t y 1 Angles, Lengt ratios Euclidean transform 3 dof r r r r t t y 1 Angles, Lengts, Areas

21 Projective Space P n A point in a projective space P n is represented by a vector of n+1 coordinates ( 1, 2, L, n+ 1 At least one coordinate is non zero. Coordinates are called omogeneous or projective coordinates Vector is called a coordinate vector Two vectors ( 1, 2, L, n+ 1 and y ( y1, y2, L, yn+ 1 represent te same point if and only if tere eists a scalar l suc tat λ i y i Te correspondence between points and coordinate vectors is not one to one.

22 in 1D Points m along a line Add up one dimension, consider origin at distance 1 from line Represent m as a ray from te origin (, : X (1, is point at infinity ( 1, 2 Points can be written X (a, 1, were a is abscissa along te line 2 1 a m (, 1 2 O (1, 1

23 Projectivity in 1D A projective transformation of a line is represented by a 22 matri ' ' Transformation as 3 degrees of freedom corresponding to te 4 elements of te matri, minus one for overall scaling Projectivity matri can be determined from 3 corresponding points ' H 1 O a m (, 1 2 (1,

24 Cross-Ratio Invariance in 1D Cross-ratio of 4 points A, B, C, D on a line is defined as AB CB A1 B 1 Cross( A,B,C,D wit AB det AD CD A2 B2 Cross-ratio is not dependent on wic particular omogeneous representation of te points is selected: scales cancel between numerator and denominator. For A (a, 1, B ( 1, etc, we get a b c b Cross( A,B,C,D a d c d Cross-ratio is invariant under any projectivity 1 O a A (, 1 2 B (1,

25 Cross-Ratio Invariance in 1D For te 4 sets of collinear points in te figure, te cross-ratio for corresponding points as te same value

26 Cross-Ratio Invariance between Lines Te cross-ratio between 4 lines forming a pencil is invariant wen te point of intersection C is moved It is equal to te cross-ratio of te 4 points C C

27 in 3D Space P 3 is called projective space A point in 3D space is defined by 4 numbers ( 1, 2, 3, 4 A plane is also defined by 4 numbers (u 1, u 2, u 3, u 4 Equation of plane is 4 u i Te plane at infinity is ite 1 plane (,,,1. Its equation is 4 Te points ( 1, 2, 3, belong to tat plane in te direction ( 1, 2, 3 of Euclidean space A line is defined as te set of points tat are a linear combination of two points P 1 and P 2 Te cross-ratio of 4 planes is equal to te cross-ratio of te lines of intersection wit a fift plane i

28 Central Projection s s i s s i z y f y z f Scene point ( s, y s, z s Image point ( i, y i, f z C f y center of projection Image plane 1 1 s s s z y f f w v u w v y w u i i /, / If world and image points are represented by omogeneous vectors, central projection is a linear mapping between P 3 and P 2 :

29 References Multiple View Geometry in Computer Vision, R. Hartley and A. Zisserman, Cambridge University Press, 2 Tree-Dimensional Computer Vision: A Geometric Approac, O. Faugeras, MIT Press, 1996