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Projective 3D geometry class 4ContentMultiple View Geometry course schedule (subject to change)Last week …Last week …Fixed points and linesSingular Value DecompositionSingular Value DecompositionProjective 3D Geometry3D pointsPlanesPlanes from pointsPoints from planesLinesPoints, lines and planesPlücker matricesPlücker matricesPlücker line coordinatesPlücker line coordinatesQuadrics and dual quadricsQuadric classificationQuadric classificationTwisted cubicHierarchy of transformationsScrew decomposition2D Euclidean Motion and the screw decompositionThe plane at infinityThe absolute conicThe absolute conicThe absolute dual quadricNext classes: Parameter estimationProjective 3D geometry class 4Multiple View GeometryComp 290-089Marc PollefeysContent• Background: Projective geometry (2D, 3D), Parameter estimation, Algorithm evaluation.• Single View: Camera model, Calibration, Single View Geometry.• Two Views: Epipolar Geometry, 3D reconstruction, Computing F, Computing structure, Plane and homographies.• Three Views: Trifocal Tensor, Computing T.• More Views: N-Linearities, Multiple view reconstruction, Bundle adjustment, auto- calibration, Dynamic SfM, Cheirality, DualityMultiple View Geometry course schedule (subject to change)Jan. 7, 9 Intro & motivation Projective 2D GeometryJan. 14, 16 (no course) Projective 2D GeometryJan. 21, 23 Projective 3D GeometryParameter EstimationJan. 28, 30 Parameter Estimation Algorithm EvaluationFeb. 4, 6 Camera Models Camera CalibrationFeb. 11, 13 Single View Geometry Epipolar GeometryFeb. 18, 20 3D reconstruction Fund. Matrix Comp.Feb. 25, 27 Structure Comp. Planes & HomographiesMar. 4, 6 Trifocal Tensor Three View ReconstructionMar. 18, 20 Multiple View Geometry MultipleView ReconstructionMar. 25, 27 Bundle adjustment PapersApr. 1, 3 Auto-Calibration PapersApr. 8, 10 Dynamic SfM PapersApr. 15, 17 Cheirality PapersApr. 22, 24 Duality Project DemosLast week …()T1,0,0l =∞TTJIIJ*+=∞C⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=∞000010001*C0ml*=∞CT(orthogonality)circular points (similarities)line at infinity (affinities)Last week …pole-polar relation0xy =CT0222=±± wyxεδconjugate points & linesprojective conic classificationaffine conic classificationxl C=lx*C=0lm*=CTABCDXChasles’ theoremcross-ratioFixed points and linesλee=H(eigenvectors H =fixed points)lλl=−TH(eigenvectors H-T =fixed lines)(λ1 =λ2 ⇒ pointwise fixed line)Singular Value DecompositionTnnnmmmnm ××××= VΣUAIUU =T021≥≥≥≥nσσσLIVV =Tnm ≥XXVTXVΣTXVUΣTTTTnnnVUVUVUA222111σσσ+++= L⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡=000000000Σ21LMMMLMOMMLLnσσσSingular Value Decomposition• Homogeneous least-squares• Span and null-space• Closest rank r approximation• Pseudo inverse1X AXmin subject to =nVX solution=()nrrdiagσσσσσ,,,,,,121LL+=Σ() 0 ,, 0 ,,,,~21LLrdiagσσσ=ΣTVΣ~UA~=TVUΣA =⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡=00000000000000Σ21σσ[][]4321UU;UU==LLNS[][]4321VV;VV==RRNSTUVΣA++=() 0 ,, 0 ,,,,11211LL−−−+=ΣrdiagσσσTVUΣA =Projective 3D Geometry• Points, lines, planes and quadrics• Transformations• П∞ , ω∞ and Ω ∞3D points()TT1 ,,,1,,,X434241ZYXXXXXXX=⎟⎟⎠⎞⎜⎜⎝⎛=in R3()04≠X()TZYX ,,in P3XX' H=(4x4-1=15 dof)projective transformation3D point()T4321,,,X XXXX=Planes0ππππ4321=+++ZYX0ππππ44332211=+++XXXX0Xπ =TDual: points ↔ planes, lines ↔ lines 3D plane0X~.n =+ d()T321π,π,πn =()TZYX ,,X~=14=Xd=4πEuclidean representationn/dXX' H=ππ'-TH=TransformationPlanes from points0πXXX321=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡TTT2341DX()T123124134234,,,π DDDD −−=()()()() () ()() () ()() () ()0det4342414333231323222121312111=⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡XXXXXXXXXXXXXXXX0πX 0πX 0,πX π 321===TTTandfromSolve(solve as right nullspace of )π⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡TTT321XXX[]0XXX Xdet321=Or implicitly from coplanarity condition124313422341DXDXDX+−01234124313422341=−+−DXDXDXDX13422341DXDX−Points from planes0Xπππ321=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡TTTxX M=[]321XXX=M0π =MT⎥⎦⎤⎢⎣⎡=IpM()Tdcba ,,,π =T⎟⎠⎞⎜⎝⎛=−−−adacabp ,,0Xπ 0Xπ 0,Xπ X 321===TTTandfromSolve(solve as right nullspace of )X⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡TTT321πππRepresenting a plane by its spanLines⎥⎦⎤⎢⎣⎡=TTBAWμBλA+⎥⎦⎤⎢⎣⎡=TTQPW*μQλP+22**0WWWW×==TT⎥⎦⎤⎢⎣⎡=00011000W⎥⎦⎤⎢⎣⎡=00100100W*Example: X-axis(4dof)Points, lines and planes⎥⎦⎤⎢⎣⎡=TXWM0π=M⎥⎦⎤⎢⎣⎡=TπW*M0X=MWX*WπPlücker matricesjijiijABBAl−=TTBAABL−=Plücker matrix (4x4 skew-symmetric homogeneous matrix)1. L has rank 22. 4dof3. generalization of4. L independent of choice A and B5. Transformation24*0LW×=Tyxl×=THLHL'=[][]⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡−=⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡−⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡=00010000000010001000000100011000LTExample: x-axisPlücker matricesTTQPPQL*−=Dual Plücker matrix L*-1TLHHL-'*=*12*13*14*23*42*34344223141312:::::::::: llllllllllll =XLπ*=LπX=Correspondence Join and incidence0XL*=(plane through point and line)(point on line)(intersection point of plane and line)(line in plane)0Lπ=[]0π,L,L21=K(coplanar lines)Plücker line coordinates[]T344223141312,,,,, llllll=/5P∈[]0000001000010000100001000010000100000344223141312344223141312=⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡llllllllllll()()Bˆ,Aˆ,BA,ˆ, ↔//[]()////ˆ|KˆˆˆˆˆˆˆBˆ,AˆB,A,det123413421423231442133412==+++++=Tllllllllllll0231442133412=++llllllon Klein quadric0K =//TPlücker line coordinates()[]0Bˆ,AˆB,A,detˆ| ==//()[]0Qˆ,PˆQ,P,detˆ| ==//()()()()()0BPAQBQAPˆ| =−=TTTT//()0|=//(Plücker internal constraint)(two lines intersect)(two lines intersect)(two lines intersect)Quadrics and dual quadrics(Q : 4x4 symmetric matrix)0QXX =T1. 9 d.o.f.2. in general 9 points define quadric 3. det Q=0 ↔ degenerate quadric4. pole – polar 5. (plane ∩ quadric)=conic6. transformation ⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡••••••••••=ooooooQQXπ=QMMCT=MxX:π=-1-TQHHQ'=0πQπ*=T-1*QQ =1. relation to quadric (non-degenerate)2. transformation THHQQ'**=Quadric classificationRank Sign. Diagonal Equation Realization4 4 (1,1,1,1) X2+ Y2+ Z2+1=0 No real points2 (1,1,1,-1) X2+ Y2+ Z2=1 Sphere0 (1,1,-1,-1) X2+ Y2= Z2+1 Hyperboloid (1S)3 3 (1,1,1,0) X2+


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Berkeley ELENG 290T - Projective 3D geometry

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