# Projective 3D geometry class 4 Multiple View Geometry Slides modified from Marc Pollefeys Comp 290-089

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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)

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 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

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