Projective GeometryProjective Geometry - PDF document

Projective GeometryProjective Geometry Projective GeometryEuclidean versus Projective GeometrynEuclidean geometry describes shapes “as they are”–Properties of objects that are unchanged by rigid motionsProjective geometry describes objects “as they appear”–Lengths, angles, parallelism become “distorted” when we look at objects–Mathematical model for how images of the 3D world are formed. Projective GeometryOverviewTools of algebraic geometrynInformal description of projective geometry in a planenDescriptions of lines and pointsnPoints at infinity and line at infinitynProjective transformations, projectivity matrixExample of applicationnSpecial projectivities: affine transforms, similarities, Euclidean transformsnratio invariance for points, lines, planes Projective GeometrynPlane passing through originand perpendicular to vectoris locus of points such thatnPlane through origin is completely defined by Tools of Algebraic Geometry 1 O ,21xx,a 21++cx,a,21xx x,a 1x2x3 Projective GeometrynA vector parallel to intersection of 2 planes and is obtained by cross-product Tools of Algebraic Geometry 2 O,a,a''' a),)'''aaa Projective GeometrynPlane passing through two points xand x’ is defined by Tools of Algebraic Geometry 3 O,a ,a ,21xx21xx Projective GeometryProjective Geometry in 2DnWe are in a plane P and want to describe lines and points in PWe consider a third dimension to make things easier when dealingwith infinity–Origin Oout of the plane, at a distance equal to 1 from planenTo each point mof the plane P we can associate a single ray nTo each line l of the plane Pwe can associate a single plane O ,21xx , ,21xx,a 3 x1 x2 Projective GeometryProjective Geometry in 2DnThe rays and are the same and are mapped to the same point mof the plane PX is the coordinate vector of m, are its homogeneous coordinatesnThe planes and are the same and are mapped to the same line l of the plane Pis the coordinate vector of lare its homogeneous coordinates O ,21xx ,a, ,21xx,21xxll=,21xx, ,a,a Projective GeometryPropertiesPoint X belongs to line Lif L= 0nEquation of line Lin projective geometry is nWe obtain homogeneous equations021++cx ,21xx , ,a Projective GeometryFrom Projective Plane to Euclidean PlanenHow do we “land” back from the projective world to the 2D world of the plane?–For point, consider intersection of ray �with plane = nFor line, intersection of plane with plane is line l: ,21xx ,a,21xxll=,a 231xxx 21++cx11++x Projective GeometryLines and PointsnTwo lines L = (a, b, c)and L’ = (a’,b’,c’intersect in the point nThe line through 2 points xand x’Duality principle: To any theorem of 2D projective geometry, there corresponds a dual theorem, which may be derived by interchanging the roles ofpoints and lines in the original theorem ' L L x ´ = ' x x L ´ = O,a a ,21xx Projective GeometryIdeal Points and Line at InfinitynThe points x= (x2, 0) do not correspond to finite points in the plane. They are points at infinity, also called ideal pointsnThe line L= (0,0,1) passes through all points at infinity, since L . x= 0nTwo parallel lines L= (a, b, c) and L’= (a, b, c’) intersect at the point =(b, -0), i.e. (b, -Any line (a, b, c) intersects the line at infinity at (b, -0). So the line at infinity is the set of all points at infinity O ,1x 3 Projective GeometryIdeal Points and Line at InfinitynWith projective geometry, two lines always meet in a single point, and two points always lie on a single line. nThis is not true of Euclidean geometry, where parallel lines form a special case. Projective GeometryProjective Transformations in a PlanenMapping from points in plane to points in plane–3 aligned points are mapped to 3 aligned pointsnAlso called– Projective GeometryProjectivityA mapping is a projectivityif and only if the mapping consists of a linear transformation of homogeneous coordinateswith Hnon singularn: –If xx, and xare 3 points that lie on a line L, and x’= H, etc, then x’, and x’lie on a line L’= 0, LH -H x= 0, so points H xlie on line H -T LConverse is hard to prove, namely if all collinear sets of points are mapped to collinear sets of points, then there is a single linear mapping between corresponding points in homogeneous coordinates x x H Projective GeometryProjectivityThe matrix Hcan be multiplied by an arbitrary non-zero number without altering the projective transformationnMatrix His called a “homogeneous matrix” (only ratios of terms are important)There are 8 independent ratios. It follows that projectivity has 8 degrees of freedomnA projectivityis simply a linear transformation of the rays÷÷÷øöçççèæúúûùêêëé÷÷øöçççèæ21333231232221131211xxhhhhhhhhxx Projective GeometryExamples of Projective TransformationsnCentral projection maps planar scenepoints to image plane by aprojectivityTrue because all points on a scene line are mapped to points on its image linenThe image of the same planar scene from a second camera can be obtained from the image from the first camera by a projectivity –True because x’= H’ x i= H” x iso x”= H” H’-x’ i Projective GeometryComputing Projective TransformationnSince matrix of projectivity has 8 degrees of freedom, the mapping between 2 images can be computed if we have the coordinates of 4 points on one image, and know where they are mapped in the other imageEach point provides 2 independent equations–Equations are linear in the 8 unknowns h’/ h '''''''3113121133323113121131+ + + =++ + + ==hhhxx '''''''3123222133323123222132+ + + =++ + + ==hhhxy Projective GeometryExample of ApplicationnRobot going down the roadnLarge squares painted on the road to make it easiernFind road shape without perspective distortion from image–Use corners of squares: coordinates of 4 points allow us to compute matrix HThen use matrix Hto compute 3D road shape Projective GeometrySpecial Projectivities úúûùêêêëé3231232221131211hhhhhhhhúúûùêêêëé00211211xaaúûùêêëé00211211xsrssrsúúûùêêêëé00211211xrr8 dofAffine transform6 dof4 dofEuclidean transform3 dofRatios of areas,Length ratiosAngles,Length ratios Projective GeometryProjective Space PA point in a projective space Pis represented by a vector of n+1 coordinates nAt least one coordinate is non zero.nCoordinates are called homogeneous or projective coordinatesnVector xis called a coordinate vectornTwo vectors and represent the same point if and only if there exists a scalar lsuch that The correspondence between points and coordinate vectors is not one to one.),,21xxx,,21xxx,,21yyyix= Projective GeometryProjective Geometry in 1DnPoints malong a linenAdd up one dimension, consider origin at distance 1 from linenRepresent m as a ray from the origin (0, 0):nX = (1,0) is point at infinitynPoints can be written X = (a, 1), where a is abscissa along the line 1 1x1x x2x1 Projective GeometryProjectivity in 1DnA projective transformation of a line is represented by a 2x2 matrixTransformation has 3 degrees of freedom corresponding to the 4 elements of the matrix, minus one for overall scalingnProjectivity matrix can be determined from 3 corresponding points 1 1x ÷øöççèæúûùêëé÷øöççèæ12221121121'xhhhxx Projective GeometryCrossRatio Invariance in 1Dnratio of 4 points A, B, C, D on a line is defined as nratio is not dependent on which particular homogeneous representation of the points is selected: scales cancel between numerator and denominator. For A = (a, 1), B = (b, 1), etc, we getratio is invariant under any projectivity 1x a - ¸- - = ) ûùêëé¸=211with )ABAxxx Projective GeometryCrossRatio Invariance in 1DnFor the 4 sets of collinear points in the figure, the cross-ratio for corresponding points has the same value Projective GeometryCrossRatio Invariance between LinesnThe cross-ratio between 4 lines forming a pencilis invariant when the point of intersection C is movednIt is equal to the cross-ratio of the 4 points CC Projective GeometryProjective Geometry in 3DnSpace Pis called projective spacenA point in 3D space is defined by 4 numbers (1, x2 , x3 , x4 )A plane is also defined by 4 numbers (, u2 , u3 , u4 )Equation of plane is nThe plane at infinity is the plane (0,0,0,1). Its equation is x4The points (1, x2 , x3 , ) belong to that plane in the direction (1, x2 , x3) of Euclidean spacenA line is defined as the set of points that are a linear combination of two points PThe cross-ratio of 4 planes is equal to the cross-ratio of the lines of intersection with a fifth plane01ii Projective GeometryCentral Projection sss= Scene point(Image point(, f)x center of projection Image planeúúûùêêëéúúúûùêêêëéúúûùêêêëé0001000000ssxfyi=If world and image points are represented by homogeneous vectors, central projection is a linear mapping between Pand P Projective GeometryReferencesMultiple View Geometry in Computer Vision, R. Hartley and A. Zisserman, Cambridge University Press, 2000nDimensional Computer Vision: A Geometric Approach, O. Faugeras, MIT Press, 1996