Projective 2D Geometry

Transcription

1 Projective D Geometry Multi View Geometry (Spring '08) Projective D Geometry Prof. Kyoung Mu Lee SoEECS, Seoul National University Homogeneous representation of lines and points Projective D Geometry Line equation: a + by + c 0 ka + kby + kc 0 a a ka (, y,) b 0 ( k, ky, k) b 0, or (, y,) kb 0 c c kc Homogeneous representation of lines l k( a, b, c), k 0 he set of all equivalence classes in R 3 (0,0,0) forms P Homogeneous representation of point (,y) on l ( a, b, c) ( k, ky, k) k(, y,), k 0 homogeneous point inhomogeneous point (,, 3) (, ) equivalence class of vectors homogeneous vectors 3 3

2 Homogeneous representation of lines and points Projective D Geometry 3 he point lies on the line l iff l0 he intersection of two lines l and l is the point (Note l ( l l ) l ( l l ) 0 ) E) intersection of line (-+0) and y (-y+0) l l the null space of L l 0, L l L hus, l l (,0,), (0,,) y Inhomogeneous point (,) he line passing two points and is l (Note ( ) ( ) 0 ) Ideal points and the line at infinity Intersection of parallel lines Projective D Geometry 4 Eample l ( a, b, c) l ( a, b, c ) : a + by + c 0 : a + by + c 0 i j k b l l a b c ( c c) a a b c l 0 Inhomogeneous representation b 0 a 0 Ideal points (points at infinity): (,,0) Line at infinity: l (0,0,) which satisfy (,,0) l 0 he parallel lines l and l intersect l in the ideal point (b,-a,0), where (b,-a) is the line s direction hus, the line at infinity is the set of directions of lines in the plane

3 A model for the projective plane Projective D Geometry 5 eactly one line through two points eactly one point at intersection of two lines Points and lines of P can be represented by the intersections of rays and planes through the origin by the plane 3. Duality Projective D Geometry 6 l 0 l l 0 l l' l ' Duality Principle: o any theorem of -D projective geometry there corresponds a dual theorem, which may be derived by interchanging the role of points and lines in the original theorem

4 Conics Projective D Geometry 7 Conics: Curve described by nd -degree equation in the plane hyperbola, ellipse, and parabola Conic equation (inhomogeneous coordinates): In homogeneous representation: 3, y 3 a C 0 with C b/ d / Conic coeff. matri Symmetric, 6 elements but 5 DOF (less one for scale) { a : b : c : d : e : f } b/ c e/ d / e/ f Point conic Determination of a Conic Projective D Geometry 8 How to determine the conic coeff. matric C? Using five points on the conic: c ( a, b, c, d, e, f ) By stacking these 5 constraints, we have C is the null vector of 56 matri.

5 angent Line to Conics Projective D Geometry 9 he line l tangent to C at a point on C is given by l C. l C Dual Conics Projective D Geometry 0 Conics that defines an equation on lines: C * A line l tangent to the conic C satisfies l C l 0, where C * is the adjoint matri of C. In general, for a non-singular symmetric matri, C C. Dual conics line conics conic envelopes C 0 point conic l C l 0 line conic

6 Degenerate Conics Projective D Geometry A conic is degenerate if matri C is not of full rank e.g. two lines (rank ) m l C lm + ml e.g. repeated line (rank ) C ll Degenerate line conics: points (rank ), double point (rank) * * Note that for degenerate conics C ( ) C Projective ransformations Projective D Geometry Def) A projectivity is an invertible mapping h from P to itself such that three points,, 3 lie on the same line iff h( ), h( ) and h( 3 ) do. (line to line mapping) hm) A mapping h: P P is a projectivity iff there eists a non-singular 33 matri H such that h() H. Pf) Let,, 3 lies on a line l, then l i 0, i,,3. hen, for a non-singular H 33, all points i H i lie on the same line l H - l such that ' l ( H ' i l) H l H i H l 0 i i

7 Projective ransformations Projective D Geometry 3 Def) A planar projective transformation is a linear transformation on homogeneous 3-vectors represented by a non-singular 33 matri: H Homogeneous matri H: 8 DOF (less one for scale) Projectivity collineation projective transfromation homography Mapping between Planes Projective D Geometry 4 central projection may be epressed by H (application of theorem) similarity affine projective Distortions by central projection

8 Eamples of Projective ransformations Projective D Geometry 5 Removing the projective distortions Projective D Geometry 6 Select four points in a plane with known coordinates Inhomogeneous correspondence (,y) (,y ) (linear in h ij ) the eight elements are determined by four point correspondences. ( constraints/point, 8DOF 4 points needed)

9 ransform of lines and conics Projective D Geometry 7 Point transform: ransformation of lines: Note: l l ( H ) ( l H H l H l ( l l H ) l 0 ) ransformation of conics: C H CH ' ( C HC H ) Note: Hierarchy of transformations - Isometries Projective D Geometry 8 Class I: Isometries (isosame, metricmeasure) ε : orientation-preserving ε-: reverse orientation R : rotation matri (orthogonal) t : translation -vector Planar Euclidean transform rigid body motion 3 DOF ( rotation, translation) point correspondences Invariants: length, angle and area R R I

10 Similarity transformations Projective D Geometry 9 Class II: Similarity transformations (isometry + scale): s : isotropic scaling R R I Equi-form transformation, preserves shape. 4 DOF ( scale, rotation, translation) point correspondences Invariants: angles, ratio of lengths and areas, parallel lines Metric structure structure defined up to a similarity Affine transformations Projective D Geometry 0 Calss III: Affine transformations: A : non-singular matri 6 DOF ( scale, rotation, translation) hree point correspondences Invariants: parallel lines, ratio of lengths of parallel line segments, ratio of areas non-isotropic scaling (DOF: scale ratio and orientation)

11 Projective transformations Projective D Geometry Case IV: projective transformations: v ( v v ) 8 DOF ( scale, rotation, translation, line at infinity) Action non-homogeneous over the plane 4 point correspondences Invariants: cross ratio (ratio of ratios) of four collinear points, Summary Projective D Geometry

12 Projective D Geometry 3 Action of affinities and projectivities on line at infinity Line at infinity (ideal point) stays at infinity for affine transform, but points move along line Parallel line are still parallel Line at infinity (ideal point) becomes finite for perspective transform, allows to observe vanishing points, horizon v A A t + 0 v v v v A A t Projective D Geometry 4 Decomposition of a projective transformation H can be decomposed as E A : non-singular matri, A srk + tv K : upper-triangular matri with det K v 0, s is positive

13 Number of invariants Projective D Geometry 5 he number of functional invariants is equal to, or greater than, the number of degrees of freedom of the configuration less the number of degrees of freedom of the transformation e.g. configuration of 4 points in general position has 8 dof (/pt) and so 4 similarity, affinity and zero projective invariants Recovery of affine and metric properties from images Under a projective transform H, since ideal points are mapped to finite points, the line at infinity l is mapped to a finite line. However, l is a fied line iff H is an affinity. 0 0 A 0 l H A l 0 0 l t A Note however that since Projective D Geometry 6 a point on l is not mapped to the same point on l A, ) k(, ) ( unless

14 Recovery of affine properties from images projection rectification Projective D Geometry 7 If l ( l is the imaged line at infinity with l 3 0,, l, l3) following H p maps l back to l (0,0,) ' H p H A 0 l l 3 l H ' p (, l, l3 l ) (0,0, ) l Determining imaged line at infinity vanishing line Projective D Geometry 8 v v l l l 3 l l 4 v l3 l 4 v l l l v v

15 Distance ratios Projective D Geometry 9 ( a, b ): d( b,c ) a : b d ( 0,),( a,),( a + b, ) H a, b,c v' H(,0 ) he circular points Projective D Geometry 30 Circular points:

16 he circular points Projective D Geometry 3 Any circle intersects l in the circular points circular points l + + d3 + e3 + f3 I 3 0 (,0,0 ) + i( 0,, 0) 0 + I J Algebraically, encodes orthogonal directions 0 (, i,0) (, i,0) Conic dual to the circular points Projective D Geometry 3 he conic dual to the circular points:

17 Angles Projective D Geometry 33 For lines l ( l, l, l ) and ( m, m, m ) the angle between 3 m 3 them is (b,-a) Euclidean: (a,b) his can be rewritten by l(a,b,c) Projective: his is invariant to projective transform since for H and l H l ( l l H ) * l C m 0 orthogonal Recovery of metric properties from images Projective D Geometry 34 We can find a projective transform that maps the imaged circular points to their canonical positions (,±i,0), then rectify the image using it. OR, metric rectification using : For point transform C ' * H C * ( H HPH HAH HS ) C ( H HPH HAH HS ) * ( H HPH HA ) H HSC H HS ( H HPH HA ) * ( H H ) C ( H H ) HP HA KK v KK HP HA KK v v KK v

18 Recovery of metric properties from images Projective D Geometry 35 Rectifying homography using SVD: aking SVD of ' C 0 0 H 0 0 H H U hen the rectifying projectivity is similarity, since H C r H ( H U U C ' H r U U 0 0 U U 0 0 r ) up to a Metric from affine Projective D Geometry 36 Affine to metric rectification using orthogonality constraints m KK 0 s s ( l l l3 ) m KK 0 s s 0 0 ' m 3 l m, l m + l m, l m s, s, s s ( s, s s ) is the null-vector of 3 matri, ( )( ) 0 C s KK K ' C SVD UC * U H U

19 Metric from projective Projective D Geometry 37 General metric rectification using 5 orthogonality constraints m KK K v ( l l l 3 ) m 0 v K v v ' m C 3 ( l m 0.5( l m + l m ), l m, 0.5( l m + l m ), 0. ( l m + l m ), l m ) c 0, c is the null-vector of 56 matri ' c C SVD UC * U H U Pole-polar relationship Projective D Geometry 38 he polar line lc of the point with respect to the conic C intersects the conic in two points. he two lines tangent to C at these points intersect at y C 0 Conjugate Pole of l Polar of