Fast and Global 3D Registration of Points, Lines and of Points, - PowerPoint PPT Presentation

Fast and Global 3D Registration Fast and Global 3D Registration of Points, Lines and of Points, Lines and Planes Planes Jesus Briales Adapted from "Convex Global 3D Registration with Lagrangian Duality" Problem (CVPR17) Formulation Unification Marginalization RCQP Jesus Briales QCQP SDP MAPIR Group University of Malaga Dual of RCQP Experiments LPM Workshop Conclusion Sep 28, 2017 Summary

Fast and Global 3D Generalized Absolute Pose problem Registration of Points, Lines and 3D-3D registration: Find optimal pose T ⋆ aligning measured points to model Planes Jesus Briales Problem Formulation Unification Marginalization RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Fast and Global 3D Generalized Absolute Pose problem Registration of Points, Lines and 3D-3D registration: Find optimal pose T ⋆ aligning measured points to model Planes Jesus Briales Problem Formulation Unification Marginalization RCQP QCQP SDP Dual of RCQP 2D-3D registration Experiments Conclusion Summary

Fast and Global 3D Generalized Absolute Pose problem Registration of Points, Lines and 3D-3D registration: Find optimal pose T ⋆ aligning measured points to model Planes Jesus Briales Problem Formulation Unification Marginalization RCQP QCQP SDP Dual of RCQP 2D-3D registration Generic registration Experiments Conclusion Summary

Fast and Global 3D Problem formulation Registration of Points, Lines and Planes Optimization problem Jesus Briales m Problem � T ⋆ = arg min d P i ( T ⊕ x i ) 2 Formulation T ∈ SE ( 3 ) Unification i = 1 Marginalization RCQP where QCQP SDP • x i : Measured points Dual of RCQP • T ⊕ x i : Transformed point Experiments • P i : Primitive in the model Conclusion Summary • d ( · , · ) : Euclidean distance

Fast and Global 3D Problem formulation Registration of Points, Lines and Planes Optimization problem Jesus Briales m Problem � T ⋆ = arg min d P i ( T ⊕ x i ) 2 Formulation Unification T ∈ SE ( 3 ) i = 1 Marginalization RCQP QCQP Assumptions: SDP Dual of RCQP • Known correspondences Experiments { x i ↔ P i } m i = 1 Conclusion Summary

Fast and Global 3D Problem formulation Registration of Points, Lines and Planes Optimization problem Jesus Briales m Problem � T ⋆ = arg min d P i ( T ⊕ x i ) 2 Formulation Unification T ∈ SE ( 3 ) i = 1 Marginalization RCQP QCQP Assumptions: SDP Dual of RCQP • Known correspondences Experiments { x i ↔ P i } m i = 1 Conclusion • No outliers Summary

Fast and Global 3D Problem formulation Registration of Points, Lines and Planes Optimization problem Jesus Briales m Problem � T ⋆ = arg min d P i ( T ⊕ x i ) 2 Formulation Unification T ∈ SE ( 3 ) i = 1 Marginalization RCQP QCQP Assumptions: SDP Dual of RCQP • Known correspondences Experiments { x i ↔ P i } m i = 1 Conclusion • No outliers Summary Challenge: • Non-convexity of R ∈ SO ( 3 )

Fast and Global 3D Unified formulation Registration of Points, Lines and Planes Optimization objective Jesus Briales Problem m � d P i ( T ⊕ x i ) 2 Formulation f ( T ) = Unification Marginalization i = 1 RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Fast and Global 3D Unified formulation Registration of Points, Lines and Planes Optimization objective Mahalanobis distance: Jesus Briales Problem m P i ≡ { y i , C i } � d P i ( T ⊕ x i ) 2 Formulation f ( T ) = d P i ( x ) 2 = � x − y i � 2 Unification C i Marginalization i = 1 RCQP = ( x − y i ) ⊤ C i ( x − y i ) , QCQP SDP Dual of RCQP Experiments Conclusion Summary � x − y � 2 � x − y � 2 � x − y � 2 ( I − vv ⊤ ) I 3 nn ⊤

Fast and Global 3D Unified formulation Registration of Points, Lines and Planes Optimization objective Mahalanobis distance: Jesus Briales Problem m P i ≡ { y i , C i } � ( T ⊕ x i − y i ) ⊤ C i ( T ⊕ x i − y i ) Formulation f ( T ) = d P i ( x ) 2 = � x − y i � 2 Unification C i Marginalization i = 1 RCQP = ( x − y i ) ⊤ C i ( x − y i ) , QCQP SDP Dual of RCQP Experiments Conclusion Summary � x − y � 2 � x − y � 2 � x − y � 2 ( I − vv ⊤ ) I 3 nn ⊤

Fast and Global 3D Unified formulation Registration of Points, Lines and Planes Optimization objective: Vectorization: Jesus Briales � � Problem m x ⊤ ˜ T ⊕ x i − y i = i ⊗ I 3 | − y i τ ˜ � ( T ⊕ x i − y i ) ⊤ C i ( T ⊕ x i − y i ) Formulation f ( T ) = Unification Marginalization i = 1 � x i � RCQP ˜ x i = 1 QCQP � vec ( T ) � SDP τ = ˜ Dual of RCQP 1 Experiments � vec ( R ) � Conclusion vec ( T ) = t Summary

Fast and Global 3D Unified formulation Registration of Points, Lines and Planes Optimization objective: Vectorization: Jesus Briales � m � � � Problem m x ⊤ ˜ T ⊕ x i − y i = i ⊗ I 3 | − y i τ ˜ � � τ ⊤ ˜ ˜ τ ⊤ Formulation f ( T ) = ˜ M i ˜ τ = ˜ M i τ . ˜ Unification Marginalization i = 1 i = 1 � x i � � �� � RCQP ˜ x i = ˜ M 1 QCQP � � ⊤ � � � vec ( T ) � ˜ x ⊤ x ⊤ SDP ˜ ˜ M i = i ⊗ I 3 | − y i C i i ⊗ I 3 | − y i τ = ˜ Dual of RCQP 1 Experiments � vec ( R ) � Conclusion vec ( T ) = t Summary

Fast and Global 3D Unified formulation Registration of Points, Lines and Planes Optimization objective: Vectorization: Jesus Briales � m � � � Problem m x ⊤ ˜ T ⊕ x i − y i = i ⊗ I 3 | − y i τ ˜ � � τ ⊤ ˜ ˜ τ ⊤ Formulation f ( T ) = ˜ M i ˜ τ = ˜ M i τ . ˜ Unification Marginalization i = 1 i = 1 � x i � � �� � RCQP ˜ x i = ˜ M 1 QCQP � � ⊤ � � � vec ( T ) � ˜ x ⊤ x ⊤ SDP ˜ ˜ M i = i ⊗ I 3 | − y i C i i ⊗ I 3 | − y i τ = ˜ Dual of RCQP 1 Experiments � vec ( R ) � Conclusion vec ( T ) = t Summary

Fast and Global 3D Unified formulation Registration of Points, Lines and Planes Optimization objective: Vectorization: Jesus Briales � m � � � Problem m x ⊤ ˜ T ⊕ x i − y i = i ⊗ I 3 | − y i τ ˜ � � τ ⊤ ˜ ˜ τ ⊤ Formulation f ( T ) = ˜ M i ˜ τ = ˜ M i τ . ˜ Unification Marginalization i = 1 i = 1 � x i � � �� � RCQP ˜ x i = ˜ M 1 QCQP � � ⊤ � � � vec ( T ) � ˜ x ⊤ x ⊤ SDP ˜ ˜ M i = i ⊗ I 3 | − y i C i i ⊗ I 3 | − y i τ = ˜ Dual of RCQP 1 Experiments � vec ( R ) � Conclusion vec ( T ) = t Summary

Fast and Global 3D Translation marginalization Registration of Points, Lines and Planes Complete problem Jesus Briales Problem   vec ( R ) Formulation f ⋆ = τ ⊤ ˜ T ∈ SO ( 3 ) ˜ min M ˜ τ = ˜ t Unification τ ,   Marginalization 1 RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Fast and Global 3D Translation marginalization Registration of Points, Lines and Planes Complete problem Jesus Briales Problem   vec ( R ) Formulation f ⋆ = τ ⊤ ˜ T ∈ SO ( 3 ) ˜ min M ˜ τ = ˜ t Unification τ ,   Marginalization 1 RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Fast and Global 3D Translation marginalization Registration of Points, Lines and Planes Complete problem Schur complement Jesus Briales Problem   vec ( R ) Formulation f ⋆ = τ ⊤ ˜ T ∈ SO ( 3 ) ˜ min M ˜ τ = ˜ t Unification τ ,   Marginalization 1 RCQP QCQP SDP Dual of RCQP Experiments Conclusion Q = ˜ ˜ M ! t , ! t − ˜ M ! t , t M − 1 t , t ˜ M t , ! t , Summary

Fast and Global 3D Translation marginalization Registration of Points, Lines and Planes Complete problem Schur complement Jesus Briales Problem   vec ( R ) Formulation f ⋆ = τ ⊤ ˜ T ∈ SO ( 3 ) ˜ min M ˜ τ = ˜ t Unification τ ,   Marginalization 1 RCQP QCQP SDP Marginalized problem Dual of RCQP Experiments Conclusion � vec ( R ) � Q = ˜ ˜ M ! t , ! t − ˜ M ! t , t M − 1 t , t ˜ r ⊤ ˜ M t , ! t , f ⋆ = ˜ Q ˜ ˜ min r , r = Summary 1 R ∈ SO ( 3 )

Fast and Global 3D Rotation-Constrained Quadratic Program: Registration of Points, Lines and RCQP Planes Jesus Briales � vec ( R ) � f ⋆ = min r ⊤ ˜ R ˜ Q ˜ ˜ r , r = , s.t. R ∈ SO ( 3 ) Problem 1 Formulation Unification Marginalization • Quadratic objective RCQP • Single rotation constraint QCQP SDP Dual of RCQP Experiments Conclusion Summary

Fast and Global 3D Rotation-Constrained Quadratic Program: Registration of Points, Lines and RCQP Planes Jesus Briales � vec ( R ) � f ⋆ = min r ⊤ ˜ R ˜ Q ˜ ˜ r , r = , s.t. R ∈ SO ( 3 ) Problem 1 Formulation Unification Marginalization • Quadratic objective RCQP • Single rotation constraint QCQP SDP Flexible formulation: We could also consider Dual of RCQP Experiments • Non-isotropic measurement noise Conclusion • 3D normal-to-normal correspondences Summary • 3D line-to-plane correspondences • 3D plane-to-plane correspondences