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UCSB ECE 181B - Projective geometry- 2D

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Projective geometry- 2DAcknowledgementsMarc Pollefeys: for allowing the use of his excellent slides on this topichttp://www.cs.unc.edu/~marc/mvg/Richard Hartley and Andrew Zisserman, "Multiple View Geometry in Computer Vision"04/01/2004 Projective Geometry 2D 2Homogeneous coordinates0=++ cbyax( )Ta,b,c0,0)()( ≠∀=++ kkcykbxka( ) ( )TTa,b,cka,b,c ~Homogeneous representation of linesequivalence class of vectors, any vector is representativeSet of all equivalence classes in R3−(0,0,0)T forms P2Homogeneous representation of points0=++ cbyax( )Ta,b,c=l( )Tyx,x =onif and only if( )( ) ( )0l 11 == x,y,a,b,cx,y,T( ) ( )0,1,,~1,, ≠∀kyxkyxTTThe point x lies on the line l if and only if xTl=lTx=0Homogeneous coordinatesInhomogeneous coordinates( )Tyx,( )T321,, xxxbut only 2DOF04/01/2004 Projective Geometry 2D 3Points from lines and vice-versal'lx ×=Intersections of lines The intersection of two lines and is ll'Line joining two pointsThe line through two points and is x'xl ×=xx'Example1=x1=y04/01/2004 Projective Geometry 2D 4Ideal points and the line at infinity( )T0,,l'l ab −=×Intersections of parallel lines ( ) ( )TTand ',,l' ,,l cbacba ==Example1=x2=xIdeal points( )T0,,21xxLine at infinity( )T1,0,0l =∞∞∪= l22RPNote that in P2 there is no distinction between ideal points and othersNote that this set lies on a single line,04/01/2004 Projective Geometry 2D 5SummaryThe set of ideal points lies on the line at infinity, intersects the line at infinity in the ideal pointA line parallel to l also intersects in the same idealpoint, 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 theplane.04/01/2004 Projective Geometry 2D 6A model for the projective planeexactly one line through two pointsexaclty one point at intersection of two linesPoints represented by rays through originLines represented by planes through originx1x2 plane represents line at infinity04/01/2004 Projective Geometry 2D 7Dualityxl0xl =T0lx =Tl'lx ×=x'xl ×=Duality principle:To any theorem of 2-dimensional projective geometrythere corresponds a dual theorem, which may bederived by interchanging the role of points and lines inthe original theorem04/01/2004 Projective Geometry 2D 8ConicsCurve described by 2nd-degree equation in the plane022=+++++ feydxcybxyax0233231222121=+++++ fxxexxdxcxxbxax3231,xxyxxx aaor homogenized0xx =CTor in matrix form=fedecbdba2/2/2/2/2/2/Cwith{ }fedcba :::::5DOF:04/01/2004 Projective Geometry 2D 9Five points define a conicFor each point the conic passes through022=+++++ feydxcyybxaxiiiiiior( )0,,,,,22=cfyxyyxxiiiiii( )Tfedcba ,,,,,=c0111115525552544244424332333232222222211211121=cyxyyxxyxyyxxyxyyxxyxyyxxyxyyxxstacking constraints yields04/01/2004 Projective Geometry 2D 10Tangent lines to conicsThe line l tangent to C at point x on C is given by l=CxlxC04/01/2004 Projective Geometry 2D 11Dual conics0ll*=CTA line tangent to the conic C satisfiesDual conics = line conics = conic envelopes1* −= CCIn general (C full rank):04/01/2004 Projective Geometry 2D 12Degenerate conicsA conic is degenerate if matrix C is not of full rankTTmllm +=Ce.g. two lines (rank 2)e.g. repeated line (rank 1)Tll=CllmDegenerate line conics: 2 points (rank 2), double point (rank1)( )CC ≠**Note that for degenerate conics04/01/2004 Projective Geometry 2D 13Projective transformationsA projectivity is an invertible mapping h from P2 to itselfsuch that three points x1,x2,x3 lie on the same line if andonly if h(x1),h(x2),h(x3) do.Definition:A mapping h:P2→P2 is a projectivity if and only if thereexist a non-singular 3x3 matrix H such that for any pointin P2 reprented by a vector x it is true that h(x)=HxTheorem:Definition: Projective transformation=321333231232221131211321'''xxxhhhhhhhhhxxxxx' H=or8DOFprojectivity=collineation=projective transformation=homography04/01/2004 Projective Geometry 2D 14Mapping between planescentral projection may be expressed by x’=Hx(application of theorem)04/01/2004 Projective Geometry 2D 15Removing projective distortion33323113121131'''hyhxhhyhxhxxx++++==33323123222132'''hyhxhhyhxhxxy++++==( )131211333231' hyhxhhyhxhx ++=++( )232221333231' hyhxhhyhxhy ++=++select four points in a plane with know coordinates(linear in hij)(2 constraints/point, 8DOF ⇒ 4 points needed)Remark: no calibration at all necessary,better ways to compute (see later)04/01/2004 Projective Geometry 2D 16Transformation of lines and conicsTransformation for linesll'-TH=Transformation for conics-1-TCHHC ='Transformation for dual conicsTHHCC**' =xx' H=For a point transformation04/01/2004 Projective Geometry 2D 17Distortions under center projectionSimilarity: squares imaged as squares.Affine: parallel lines remain parallel; circles become ellipses.Projective: Parallel lines converge.04/01/2004 Projective Geometry 2D 18Class I: Isometries(iso=same, metric=measure)−=1100cossinsincos1''yxttyxyxθθεθθε1±=ε1=ε1−=εorientation preserving:orientation reversing:x0xx'==1tTRHEIRR =Tspecial cases: pure rotation, pure translation3DOF (1 rotation, 2 translation)Invariants: length, angle, area04/01/2004 Projective Geometry 2D 19Class II: Similarities(isometry + scale)−=1100cossinsincos1''yxtsstssyxyxθθθθx0xx'==1tTRHsSIRR =Talso know as equi-form (shape preserving)metric structure = structure up to similarity (in literature)4DOF (1 scale, 1 rotation, 2 translation)Invariants: ratios of length, angle, ratios of areas, parallel lines04/01/2004 Projective Geometry 2D 20Class III: Affine transformations=11001''22211211yxtaataayxyxx0xx'==1tTAHAnon-isotropic scaling! (2DOF: scale ratio and orientation)6DOF (2 scale, 2 rotation, 2 translation)Invariants: parallel lines, ratios of parallel lengths, ratios of areas( ) (


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