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Projective GeometryProjective Geometry

Projective GeometryEuclidean versus Projective Geometry nEuclidean geometry describes shapes "as they are" -Properties of objects that are unchanged by rigid motions

»Lengths

»Angles

»Parallelism

nProjective 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 GeometryOverview

nTools of algebraic geometry nInformal description of projective geometry in a plane nDescriptions of lines and points nPoints at infinity and line at infinity nProjective transformations, projectivity matrix nExample of application nSpecial projectivities: affine transforms, similarities,

Euclidean transforms

nCross-ratio invariance for points, lines, planes Projective GeometrynPlane passing through originand perpendicular to vector is locus of points such that nPlane through origin is completely defined by Tools of Algebraic Geometry 1

O),,(321xxx=x),,(cba=n

0

321=++xcxbxa

),,(cba),,(321xxx=x0=·xn ),,(cba=nx 1x 2x 3 Projective GeometrynA vector parallel to intersection of 2 planes and is obtained by cross-product Tools of Algebraic Geometry 2 O ),,(cba),,(cba )'','',''(cba)',','(cba )',','(),,()'','',''(cbacbacba´= )',','(cba

Projective GeometrynPlane passing through two points xand x' is defined byTools of Algebraic Geometry 3

O ),,(cbax'x´=),,(cba),,(321xxx=x )',','(321xxx=x'

Projective GeometryProjective Geometry in 2D

nWe are in a plane P and want to describe lines and points in P nWe consider a third dimension to make things easier when dealingwith infinity -Origin Oout of the plane, at a distance equal to 1 from plane nTo 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),,(321xxx=x Pl m),,(cba),,(321xxx=x ),,(cba=L x3x 1x 2

Projective GeometryProjective Geometry in 2D

nThe rays and are the same and are mapped to the same point mof the plane P -X is the coordinate vector of m, are its homogeneous coordinates nThe planes and are the same and are mapped to the same line l of the plane P -Lis the coordinate vector of l, are its homogeneous coordinatesO),,(321xxx=x Pl m ),,(321xxx ),,(cballl ),,(cba ),,(cba=L

Projective GeometryProperties

nPoint X belongs to line Lif L. X= 0 nEquation of line Lin projective geometry is nWe obtain homogeneous equations0321=++xcxbxaO),,(321xxx=x Pl m ),,(cba),,(cba=L Projective GeometryFrom Projective Plane to Euclidean Plane nHow 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:O),,(321xxx=x Pl m ),,(cba),,(321xxxlll=x ),,(cba=L 1

3=x,/13x=l)/,/(m3231xxxx=

0

321=++xcxbxa13=x021=++cxbxa

Projective GeometryLines and Points

nTwo lines L = (a, b, c)and L' = (a',b',c')intersect in the point nThe line through 2 points xand x'is nDuality 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'LLx´='xxL´= O ),,(cba)',','(cbaP),,(321xxx=xL L'x' Projective GeometryIdeal Points and Line at Infinity nThe points x= (x1, x2, 0) do not correspond to finite points in the plane. They are points at infinity, also called ideal points nThe line L= (0,0,1) passes through all points at infinity, since L . x= 0 nTwo parallel lines L= (a, b, c) and L'= (a, b, c') intersect at the point =(c'-c)(b, -a,0), i.e. (b, -a,0) nAny line (a, b, c) intersects the line at infinity at (b, -a,0). So the line at infinity is the set of all points at infinityO)0,,(21xx=x1l m )1,0,0(¥¥P x

3'LLx´=

Projective GeometryIdeal Points and Line at Infinity nWith 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 Plane nProjectivity -Mapping from points in plane to points in plane -3 aligned points are mapped to 3 aligned points nAlso called -Collineation -Homography

Projective GeometryProjectivityTheorem

nA mapping is a projectivityif and only if the mapping consists of a linear transformation of homogeneous coordinates with Hnon singular nProof: -If x1, x2, and x3are 3 points that lie on a line L, and x'

1= Hx1, etc, then x'1, x'2, and x'3lie on a line L'

-LTxi= 0, LTH -1H xi= 0, so points H xilie on line

H -T L

nConverse 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 coordinatesxxH'=

Projective GeometryProjectivityMatrix

nThe matrix Hcan be multiplied by an arbitrary non-zero number without altering the projective transformation nMatrix His called a "homogeneous matrix" (only ratios of terms are important) nThere are 8 independent ratios. It follows that projectivity has 8 degrees of freedom nA projectivityis simply a linear transformation of the rays÷

èae

èae

3 21

333231232221131211

3' 2' 1 x xx hhhhhhhhh x xxxxH'= Projective GeometryExamples of Projective Transformations nCentral projection maps planar scenepoints to image plane by a projectivity -True because all points on a scene line are mapped to points on its image line nThe 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'i= H' x i, x"i= H" x i so x"i= H" H'-1x' iP' O' M y'x' yxPM'M" O" P" Projective GeometryComputing Projective Transformation nSince 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

image -Each point provides 2 independent equations -Equations are linear in the 8 unknowns h'ij=hij/ h331'''''

3231131211

333231131211

31
++++==yhxhhyhxh hyhxhhyhxh xxx1'''''

3231232221

333231232221

32
++++==yhxhhyhxh hyhxhhyhxh xxy

Projective GeometryExample of Application

nRobot going down the road nLarge squares painted on the road to make it easier nFind road shape without perspective distortion from image -Use corners of squares: coordinates of 4 points allow us to compute matrix H -Then use matrix Hto compute 3D road shape

Projective GeometrySpecial Projectivitiesú

333231232221131211

hhhhhhhhh

10022211211

xx taataa

10022211211

yx trsrstrsrs

10022211211

yx trrtrrProjectivity 8 dof

Affine transform

6 dof

Similarity

4 dof

Euclidean transform

3 dofInvariants

Collinearity,

Cross-ratios

Parallelism,

Ratios of areas,

Length ratios

Angles,

Lengths,

AreasAngles,

Length ratios

Projective GeometryProjective Space Pn

nA point in a projective space Pnis represented by a vector of n+1 coordinates nAt least one coordinate is non zero. nCoordinates are called homogeneous or projective coordinates nVector xis called a coordinate vector nTwo 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.),,,(121+=nxxxLx ),,,(121+=nxxxLx),,,(121+=nyyyLy iiyxl=

Projective GeometryProjective Geometry in 1D

nPoints malong a line nAdd up one dimension, consider origin at distance 1 from line nRepresent m as a ray from the origin (0, 0): nX = (1,0) is point at infinity nPoints can be written X = (a, 1), where a is abscissa along the line1

O),(21xx=x

m),(21xx=x)0,1(=xax 2 x 1

Projective GeometryProjectivity in 1D

nA projective transformation of a line is represented by a 2x2 matrix nTransformation has 3 degrees of freedom corresponding to the 4 elements of the matrix, minus one for overall scaling nProjectivity matrix can be determined from 3 corresponding points1

O),(21xx=x

m)0,1(=xa÷

èaeúûùêëé

èae

21

22211211

21
xx hhhh xxxxH'=

Projective GeometryCross-Ratio Invariance in 1D

nCross-ratio of 4 points A, B, C, D on a line is defined as nCross-ratio 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 get nCross-ratio is invariant under any projectivity O

1),(21xx=x

A)0,1(=xadcbcdabaA,B,C,D--¸

--= )Cross(ú =¸=2211detwith )Cross(BABA xxxxABCDCB

ADABA,B,C,D

B

Projective GeometryCross-Ratio Invariance in 1D

nFor the 4 sets of collinear points in the figure, the cross-ratio for corresponding points has the same value Projective GeometryCross-Ratio Invariance between Lines nThe cross-ratio between 4 lines forming a pencilis invariant when the point of intersection C is moved nIt is equal to the cross-ratio of the 4 pointsC C

Projective GeometryProjective Geometry in 3D

nSpace P3is called projective space nA point in 3D space is defined by 4 numbers (x1, x2 , x3 , x4 ) nA plane is also defined by 4 numbers (u1, u2 , u3 , u4 ) nEquation of plane is nThe plane at infinity is the plane (0,0,0,1). Its equation is x4=0 nThe points (x1, x2 , x3 , 0) belong to that plane in the direction (x1, x2 , x3) of Euclidean space nA line is defined as the set of points that are a linear combination of two points P1andP2 nThe cross-ratio of 4 planes is equal to the cross-ratio of the lines of intersection with a fifth plane0 4 1 =å=i iixu

Projective GeometryCentral Projection

s s is s i z yfyzxfx ==Scene point (xs, ys, zs)Image point (xi, yi, f)x z Cfy center of projectionImage plane 1 000

1000000

s ss z yx ff w vu wvywuxii/,/==If world and image points are represented by homogeneous vectors, central projection is a linear mapping between P3and P2:

Projective GeometryReferences

nMultiple View Geometry in Computer Vision, R. Hartley and A. Zisserman, Cambridge University Press, 2000 nThree-Dimensional Computer Vision: A Geometric

Approach, O. Faugeras, MIT Press, 1996

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