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Projective 3D geometryclass 4Multiple View GeometrySlides modified from Marc Pollefeys Comp 290-089

Last week

(orthogonality)circular points (similarities)line at infinity (affinities)

Last week pole-polar relationconjugate points & linesprojective conic classificationaffine conic classificationChasles theorem

cross-ratio

Fixed points and lines

(eigenvectors H =fixed points)

(1=2 pointwise fixed line)

Singular Value Decomposition

Singular Value DecompositionHomogeneous least-squares

Span and null-space

Closest rank r approximation

Pseudo inverse

Projective 3D GeometryPoints, lines, planes and quadrics

Transformations

, and

3D pointsin R3 in P3 (4x4-1=15 dof)projective transformation3D point

PlanesDual: points planes, lines lines 3D plane

Planes from pointsOr implicitly from coplanarity condition

Points from planesParameterizing points on a plan by representing a plane by its span

LinesExample: X-axis(4dof)

Points, lines and planes

Plcker matricesPlcker matrix (4x4 skew-symmetric homogeneous matrix)L has rank 24dofgeneralization of L independent of choice A and BTransformationExample: x-axis

Plcker matricesDual Plcker matrix L*Correspondence Join and incidence(plane through point and line)(point on line)(intersection point of plane and line)(line in plane)(coplanar lines)

Plcker line coordinates

Plcker line coordinates(Plcker internal constraint)(two lines intersect)(two lines intersect)(two lines intersect)

Quadrics and dual quadrics(Q : 4x4 symmetric matrix)9 d.o.f.in general 9 points define quadric det Q=0 degenerate quadricpole polar (plane quadric)=conictransformation

Quadric classification

Rank Sign.DiagonalEquationRealization44(1,1,1,1)X2+ Y2+ Z2+1=0No real points2(1,1,1,-1)X2+ Y2+ Z2=1Sphere0(1,1,-1,-1)X2+ Y2= Z2+1Hyperboloid (1S)33(1,1,1,0)X2+ Y2+ Z2=0Single point1(1,1,-1,0)X2+ Y2= Z2Cone 22(1,1,0,0)X2+ Y2= 0Single line0(1,-1,0,0)X2= Y2Two planes11(1,0,0,0)X2=0Single plane

Quadric classificationProjectively equivalent to sphere:hyperboloid of two sheetsparaboloidsphereellipsoid

Twisted cubic3 intersection with plane (in general)12 dof (15 for A 3 for reparametrisation (1 23)2 constraints per point on cubic, defined by 6 pointsprojectively equivalent to (1 23)Horopter & degenerate case for reconstruction

Hierarchy of transformationsProjective15dofAffine12dofSimilarity7dofEuclidean6dofIntersection and tangencyParallellism of planes,Volume ratios, centroids,The plane at infinity The absolute conic

Volume

Screw decompositionAny particular translation and rotation is equivalent to a rotation about a screw axis and a translation along the screw axis.

The plane at infinity

The plane at infinity is a fixed plane under a projective transformation H iff H is an affinity

canical positioncontains directions two planes are parallel line of intersection in line // line (or plane) point of intersection in

The absolute conic

The absolute conic is a fixed conic under the projective transformation H iff H is a similarity

The absolute conic is a (point) conic on . In a metric frame: or conic for directions:(with no real points) is only fixed as a setCircle intersect in two pointsSpheres intersect in

The absolute conic

Euclidean:

Projective:

(orthogonality=conjugacy)

plane

normal

The absolute dual quadric

The absolute conic * is a fixed conic under the projective transformation H iff H is a similarity

8 dofplane at infinity is the nullvector of Angles:

Next classes:Parameter estimationDirect Linear TransformIterative EstimationMaximum Likelihood Est.Robust Estimation

*A,B on line AW*=0, BW*=0, 4dof, skew symmetric 6dof, scale -1, rank2 -1Prove independence of choice by filling in C=A+muB, AA-AA disappears, Prove transform based on A=HA, B=HB,

*Indicate 12-34 {1234} always present

Prove join (AB+BA)P=A (if AP=0) which is general since L independent of A and B

Example intersection X-axis with X=1: X=counterdiag(-1 0 0 1)(1 0 0 -1)=(1 0 0 1)*L consists of non-zero elements*3. and thus defined by less points4. On quadric=> tangent, outside quadric=> plane through tangent point5. Derive XQX=xMQMx=0

*Signature sigma= sum of diagonal,e.g. +1+1+1-1=2,always more + than -, so always positive

*Ruled quadric: two family of lines, called generators.Hyperboloid of 1 sheet topologically equivalent to torus!*Represents 3DOF between projective and affine*Represent 5 DOF between affine and similarity*Orthogonality is conjugacy with respect to Absolute Conic