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

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

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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)

slide-1
SLIDE 1

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Fast and Global 3D Registration

  • f Points, Lines and Planes

Adapted from "Convex Global 3D Registration with Lagrangian Duality" (CVPR17) Jesus Briales

MAPIR Group University of Malaga

LPM Workshop Sep 28, 2017

slide-2
SLIDE 2

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Generalized Absolute Pose problem

3D-3D registration: Find optimal pose T ⋆ aligning measured points to model

slide-3
SLIDE 3

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Generalized Absolute Pose problem

3D-3D registration: Find optimal pose T ⋆ aligning measured points to model 2D-3D registration

slide-4
SLIDE 4

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Generalized Absolute Pose problem

3D-3D registration: Find optimal pose T ⋆ aligning measured points to model 2D-3D registration Generic registration

slide-5
SLIDE 5

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Problem formulation

Optimization problem

T ⋆ = arg min

T∈SE(3) m

  • i=1

dPi(T ⊕ xi)2 where

  • xi: Measured points
  • T ⊕ xi: Transformed point
  • Pi: Primitive in the model
  • d(·, ·): Euclidean distance
slide-6
SLIDE 6

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Problem formulation

Optimization problem

T ⋆ = arg min

T∈SE(3) m

  • i=1

dPi(T ⊕ xi)2

Assumptions:

  • Known correspondences

{xi ↔ Pi}m

i=1

slide-7
SLIDE 7

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Problem formulation

Optimization problem

T ⋆ = arg min

T∈SE(3) m

  • i=1

dPi(T ⊕ xi)2

Assumptions:

  • Known correspondences

{xi ↔ Pi}m

i=1

  • No outliers
slide-8
SLIDE 8

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Problem formulation

Optimization problem

T ⋆ = arg min

T∈SE(3) m

  • i=1

dPi(T ⊕ xi)2

Assumptions:

  • Known correspondences

{xi ↔ Pi}m

i=1

  • No outliers

Challenge:

  • Non-convexity of R ∈ SO(3)
slide-9
SLIDE 9

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Unified formulation

Optimization objective

f(T) =

m

  • i=1

dPi(T ⊕ xi)2

slide-10
SLIDE 10

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Unified formulation

Optimization objective

f(T) =

m

  • i=1

dPi(T ⊕ xi)2

Mahalanobis distance:

Pi ≡ {yi, Ci} dPi(x)2 = x − yi2

Ci

= (x − yi)⊤Ci(x − yi), x − y2

I3

x − y2

(I−vv⊤)

x − y2

nn⊤

slide-11
SLIDE 11

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Unified formulation

Optimization objective

f(T) =

m

  • i=1

(T ⊕ xi − yi)⊤Ci(T ⊕ xi − yi)

Mahalanobis distance:

Pi ≡ {yi, Ci} dPi(x)2 = x − yi2

Ci

= (x − yi)⊤Ci(x − yi), x − y2

I3

x − y2

(I−vv⊤)

x − y2

nn⊤

slide-12
SLIDE 12

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Unified formulation

Optimization objective:

f(T) =

m

  • i=1

(T ⊕ xi − yi)⊤Ci(T ⊕ xi − yi)

Vectorization:

T ⊕ xi − yi =

  • ˜

x⊤

i ⊗ I3| − yi

  • ˜

τ ˜ xi = xi 1

  • ˜

τ = vec(T) 1

  • vec(T) =

vec(R) t

slide-13
SLIDE 13

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Unified formulation

Optimization objective:

f(T) =

m

  • i=1

˜ τ ⊤ ˜ Mi ˜ τ = ˜ τ ⊤ m

  • i=1

˜ Mi

  • ˜

M

˜ τ. ˜ Mi =

  • ˜

x⊤

i ⊗ I3| − yi

⊤ Ci

  • ˜

x⊤

i ⊗ I3| − yi

  • Vectorization:

T ⊕ xi − yi =

  • ˜

x⊤

i ⊗ I3| − yi

  • ˜

τ ˜ xi = xi 1

  • ˜

τ = vec(T) 1

  • vec(T) =

vec(R) t

slide-14
SLIDE 14

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Unified formulation

Optimization objective:

f(T) =

m

  • i=1

˜ τ ⊤ ˜ Mi ˜ τ = ˜ τ ⊤ m

  • i=1

˜ Mi

  • ˜

M

˜ τ. ˜ Mi =

  • ˜

x⊤

i ⊗ I3| − yi

⊤ Ci

  • ˜

x⊤

i ⊗ I3| − yi

  • Vectorization:

T ⊕ xi − yi =

  • ˜

x⊤

i ⊗ I3| − yi

  • ˜

τ ˜ xi = xi 1

  • ˜

τ = vec(T) 1

  • vec(T) =

vec(R) t

slide-15
SLIDE 15

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Unified formulation

Optimization objective:

f(T) =

m

  • i=1

˜ τ ⊤ ˜ Mi ˜ τ = ˜ τ ⊤ m

  • i=1

˜ Mi

  • ˜

M

˜ τ. ˜ Mi =

  • ˜

x⊤

i ⊗ I3| − yi

⊤ Ci

  • ˜

x⊤

i ⊗ I3| − yi

  • Vectorization:

T ⊕ xi − yi =

  • ˜

x⊤

i ⊗ I3| − yi

  • ˜

τ ˜ xi = xi 1

  • ˜

τ = vec(T) 1

  • vec(T) =

vec(R) t

slide-16
SLIDE 16

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Translation marginalization

Complete problem

f ⋆ = min

T∈SO(3) ˜

τ ⊤ ˜ M ˜ τ, ˜ τ =   vec(R) t 1  

slide-17
SLIDE 17

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Translation marginalization

Complete problem

f ⋆ = min

T∈SO(3) ˜

τ ⊤ ˜ M ˜ τ, ˜ τ =   vec(R) t 1  

slide-18
SLIDE 18

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Translation marginalization

Complete problem

f ⋆ = min

T∈SO(3) ˜

τ ⊤ ˜ M ˜ τ, ˜ τ =   vec(R) t 1  

Schur complement

˜ Q = ˜ M!t,!t − ˜ M!t,tM−1

t,t ˜

Mt,!t,

slide-19
SLIDE 19

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Translation marginalization

Complete problem

f ⋆ = min

T∈SO(3) ˜

τ ⊤ ˜ M ˜ τ, ˜ τ =   vec(R) t 1  

Marginalized problem

f ⋆ = min

R∈SO(3)

˜ r⊤ ˜ Q˜ r, ˜ r = vec(R) 1

  • Schur complement

˜ Q = ˜ M!t,!t − ˜ M!t,tM−1

t,t ˜

Mt,!t,

slide-20
SLIDE 20

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Rotation-Constrained Quadratic Program: RCQP

f ⋆ = min

R ˜

r⊤ ˜ Q˜ r, ˜ r = vec(R) 1

  • ,

s.t. R ∈ SO(3)

  • Quadratic objective
  • Single rotation constraint
slide-21
SLIDE 21

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Rotation-Constrained Quadratic Program: RCQP

f ⋆ = min

R ˜

r⊤ ˜ Q˜ r, ˜ r = vec(R) 1

  • ,

s.t. R ∈ SO(3)

  • Quadratic objective
  • Single rotation constraint

Flexible formulation: We could also consider

  • Non-isotropic measurement noise
  • 3D normal-to-normal correspondences
  • 3D line-to-plane correspondences
  • 3D plane-to-plane correspondences
slide-22
SLIDE 22

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Rotation-Constrained Quadratic Program: RCQP

f ⋆ = min

R ˜

r⊤ ˜ Q˜ r, ˜ r = vec(R) 1

  • ,

s.t. R ∈ SO(3)

  • Quadratic objective
  • Single

rotation constraint → non-convex → how to solve globally?

Flexible formulation: We could also consider

  • Non-isotropic measurement noise
  • 3D normal-to-normal correspondences
  • 3D line-to-plane correspondences
  • 3D plane-to-plane correspondences
slide-23
SLIDE 23

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

QCQP and Lagrangian relaxation

QCQP problem

f ⋆ = min

x

˜ x⊤ ˜ Q˜ x, ˜ x ≡ x 1

  • ,

s.t. ˜ x⊤ ˜ Pi ˜ x = 0, ∀i ∈ C

slide-24
SLIDE 24

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

QCQP and Lagrangian relaxation

Primal problem

f ⋆ = min

x

˜ x⊤ ˜ Q˜ x, ˜ x ≡ x 1

  • ,

s.t. ˜ x⊤ ˜ Pi ˜ x = 0, ∀i ∈ C

slide-25
SLIDE 25

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

QCQP and Lagrangian relaxation

Primal problem

f ⋆ = min

x

˜ x⊤ ˜ Q˜ x, ˜ x ≡ x y

  • ,

s.t. ˜ x⊤ ˜ Pi ˜ x = 0, ∀i ∈ C, y2 = 1

slide-26
SLIDE 26

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

QCQP and Lagrangian relaxation

Primal problem

f ⋆ = min

x

˜ x⊤ ˜ Q˜ x, ˜ x ≡ x y

  • ,

s.t. ˜ x⊤ ˜ Pi ˜ x = 0, ∀i ∈ ˜ C

slide-27
SLIDE 27

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

QCQP and Lagrangian relaxation

Primal problem

f ⋆ = min

x

˜ x⊤ ˜ Q˜ x, ˜ x ≡ x y

  • ,

s.t. ˜ x⊤ ˜ Pi ˜ x = 0, ∀i ∈ ˜ C

Lagrangian relaxation

L(˜ x, ˜ λ) = γ + ˜ x⊤( ˜ Q +

  • i∈ ˜

C

λi ˜ Pi)˜ x,

slide-28
SLIDE 28

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

QCQP and Lagrangian relaxation

Primal problem

f ⋆ = min

x

˜ x⊤ ˜ Q˜ x, ˜ x ≡ x y

  • ,

s.t. ˜ x⊤ ˜ Pi ˜ x = 0, ∀i ∈ ˜ C

Lagrangian relaxation

L(˜ x, ˜ λ) = γ + ˜ x⊤( ˜ Q +

  • i∈ ˜

C

λi ˜ Pi)˜ x, d(˜ λ) = min

˜ x

L(˜ x, ˜ λ)

slide-29
SLIDE 29

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

QCQP and Lagrangian relaxation

Primal problem

f ⋆ = min

x

˜ x⊤ ˜ Q˜ x, ˜ x ≡ x y

  • ,

s.t. ˜ x⊤ ˜ Pi ˜ x = 0, ∀i ∈ ˜ C

Lagrangian relaxation

L(˜ x, ˜ λ) = γ + ˜ x⊤( ˜ Q +

  • i∈ ˜

C

λi ˜ Pi)˜ x, d(˜ λ) = min

˜ x

L(˜ x, ˜ λ) ≤ f ⋆

slide-30
SLIDE 30

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

QCQP and Lagrangian relaxation

Primal problem

f ⋆ = min

x

˜ x⊤ ˜ Q˜ x, ˜ x ≡ x y

  • ,

s.t. ˜ x⊤ ˜ Pi ˜ x = 0, ∀i ∈ ˜ C

Lagrangian relaxation

L(˜ x, ˜ λ) = γ + ˜ x⊤( ˜ Q +

  • i∈ ˜

C

λi ˜ Pi)˜ x, d(˜ λ) = γ s.t. ˜ Q +

  • i∈ ˜

C

λi ˜ Pi 0

slide-31
SLIDE 31

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

QCQP and Lagrangian relaxation

Primal problem

f ⋆ = min

x

˜ x⊤ ˜ Q˜ x, ˜ x ≡ x y

  • ,

s.t. ˜ x⊤ ˜ Pi ˜ x = 0, ∀i ∈ ˜ C

Lagrangian relaxation

L(˜ x, ˜ λ) = γ + ˜ x⊤( ˜ Q +

  • i∈ ˜

C

λi ˜ Pi)˜ x, d(˜ λ) = γ s.t. ˜ Q +

  • i∈ ˜

C

λi ˜ Pi 0

Dual problem

d⋆ = max

˜ λ

d(˜ λ)

slide-32
SLIDE 32

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

QCQP and Lagrangian relaxation

Primal problem

f ⋆ = min

x

˜ x⊤ ˜ Q˜ x, ˜ x ≡ x y

  • ,

s.t. ˜ x⊤ ˜ Pi ˜ x = 0, ∀i ∈ ˜ C

Lagrangian relaxation

L(˜ x, ˜ λ) = γ + ˜ x⊤( ˜ Q +

  • i∈ ˜

C

λi ˜ Pi)˜ x, d(˜ λ) = γ s.t. ˜ Q +

  • i∈ ˜

C

λi ˜ Pi 0

Dual problem

d⋆ = max

˜ λ

d(˜ λ)

  • Weak duality (always): d⋆ ≤ f ⋆
slide-33
SLIDE 33

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

QCQP and Lagrangian relaxation

Primal problem

f ⋆ = min

x

˜ x⊤ ˜ Q˜ x, ˜ x ≡ x y

  • ,

s.t. ˜ x⊤ ˜ Pi ˜ x = 0, ∀i ∈ ˜ C

Lagrangian relaxation

L(˜ x, ˜ λ) = γ + ˜ x⊤( ˜ Q +

  • i∈ ˜

C

λi ˜ Pi)˜ x, d(˜ λ) = γ s.t. ˜ Q +

  • i∈ ˜

C

λi ˜ Pi 0

Dual problem

d⋆ = max

˜ λ

d(˜ λ)

  • Weak duality (always): d⋆ ≤ f ⋆
  • Strong duality (sometimes): d⋆ = f ⋆
slide-34
SLIDE 34

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

QCQP and Lagrangian relaxation

Primal problem

f ⋆ = min

x

˜ x⊤ ˜ Q˜ x, ˜ x ≡ x y

  • ,

s.t. ˜ x⊤ ˜ Pi ˜ x = 0, ∀i ∈ ˜ C

Lagrangian relaxation

L(˜ x, ˜ λ) = γ + ˜ x⊤( ˜ Q +

  • i∈ ˜

C

λi ˜ Pi)˜ x, d(˜ λ) = γ s.t. ˜ Q +

  • i∈ ˜

C

λi ˜ Pi 0

Solution recovery

If d⋆ = f ⋆ (strong duality): ˜ x⋆ = arg min

˜ x

L(˜ x, ˜ λ

⋆)

Dual problem

d⋆ = max

˜ λ

d(˜ λ)

slide-35
SLIDE 35

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

QCQP and Lagrangian relaxation

Primal problem

f ⋆ = min

x

˜ x⊤ ˜ Q˜ x, ˜ x ≡ x y

  • ,

s.t. ˜ x⊤ ˜ Pi ˜ x = 0, ∀i ∈ ˜ C

Lagrangian relaxation

L(˜ x, ˜ λ) = γ + ˜ x⊤( ˜ Q +

  • i∈ ˜

C

λi ˜ Pi)˜ x, d(˜ λ) = γ s.t. ˜ Q +

  • i∈ ˜

C

λi ˜ Pi 0

Solution recovery

If d⋆ = f ⋆ (strong duality): ˜ x⋆ ∈ null( ˜ Q +

  • i∈ ˜

C

λ⋆

i ˜

Pi)

Dual problem

d⋆ = max

˜ λ

d(˜ λ)

slide-36
SLIDE 36

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

QCQP and Lagrangian relaxation

Primal problem

f ⋆ = min

x

˜ x⊤ ˜ Q˜ x, ˜ x ≡ x y

  • ,

s.t. ˜ x⊤ ˜ Pi ˜ x = 0, ∀i ∈ ˜ C

Lagrangian relaxation

L(˜ x, ˜ λ) = γ + ˜ x⊤( ˜ Q +

  • i∈ ˜

C

λi ˜ Pi)˜ x, d(˜ λ) = γ s.t. ˜ Q +

  • i∈ ˜

C

λi ˜ Pi 0

Solution recovery

If rank(null( ˜ Q +

i∈ ˜ C λ⋆ i ˜

Pi)) = 1: ˜ x⋆ ∈ null( ˜ Q +

  • i∈ ˜

C

λ⋆

i ˜

Pi)

Dual problem

d⋆ = max

˜ λ

d(˜ λ)

slide-37
SLIDE 37

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

QCQP and Lagrangian relaxation

Primal problem

f ⋆ = min

x

˜ x⊤ ˜ Q˜ x, ˜ x ≡ x y

  • ,

s.t. ˜ x⊤ ˜ Pi ˜ x = 0, ∀i ∈ ˜ C

Lagrangian relaxation

L(˜ x, ˜ λ) = γ + ˜ x⊤( ˜ Q +

  • i∈ ˜

C

λi ˜ Pi)˜ x, d(˜ λ) = γ s.t. ˜ Q +

  • i∈ ˜

C

λi ˜ Pi 0

Solution recovery

If rank(null( ˜ Q +

i∈ ˜ C λ⋆ i ˜

Pi)) = 1: ˜ x⋆ ∈ null( ˜ Q +

  • i∈ ˜

C

λ⋆

i ˜

Pi), y⋆ = 1

Dual problem

d⋆ = max

˜ λ

d(˜ λ)

slide-38
SLIDE 38

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

QCQP and Lagrangian relaxation

Primal problem

f ⋆ = min

x

˜ x⊤ ˜ Q˜ x, ˜ x ≡ x y

  • ,

s.t. ˜ x⊤ ˜ Pi ˜ x = 0, ∀i ∈ ˜ C

Lagrangian relaxation

L(˜ x, ˜ λ) = γ + ˜ x⊤( ˜ Q +

  • i∈ ˜

C

λi ˜ Pi)˜ x, d(˜ λ) = γ s.t. ˜ Q +

  • i∈ ˜

C

λi ˜ Pi 0

Solution recovery

If rank(null( ˜ Q +

i∈ ˜ C λ⋆ i ˜

Pi)) = 1: ˜ x⋆ ∈ null( ˜ Q +

  • i∈ ˜

C

λ⋆

i ˜

Pi), y⋆ = 1

Dual problem

d⋆ = max

˜ λ

d(˜ λ)

slide-39
SLIDE 39

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

QCQP and Lagrangian relaxation

Primal problem

f ⋆ = min

x

˜ x⊤ ˜ Q˜ x, ˜ x ≡ x y

  • ,

s.t. ˜ x⊤ ˜ Pi ˜ x = 0, ∀i ∈ ˜ C

Lagrangian relaxation

L(˜ x, ˜ λ) = γ + ˜ x⊤( ˜ Q +

  • i∈ ˜

C

λi ˜ Pi)˜ x, d(˜ λ) = γ s.t. ˜ Q +

  • i∈ ˜

C

λi ˜ Pi 0

Solution recovery

If rank(null( ˜ Q +

i∈ ˜ C λ⋆ i ˜

Pi)) = 1: ˜ x⋆ ∈ null( ˜ Q +

  • i∈ ˜

C

λ⋆

i ˜

Pi), y⋆ = 1

Dual problem

d⋆ = max

˜ λ

γ s.t. ˜ Q +

  • i∈ ˜

C

λi ˜ Pi 0

slide-40
SLIDE 40

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

QCQP and Lagrangian relaxation

Primal problem

f ⋆ = min

x

˜ x⊤ ˜ Q˜ x, ˜ x ≡ x y

  • ,

s.t. ˜ x⊤ ˜ Pi ˜ x = 0, ∀i ∈ ˜ C

Lagrangian relaxation

L(˜ x, ˜ λ) = γ + ˜ x⊤( ˜ Q +

  • i∈ ˜

C

λi ˜ Pi)˜ x, d(˜ λ) = γ s.t. ˜ Q +

  • i∈ ˜

C

λi ˜ Pi 0

Solution recovery

If rank(null( ˜ Q +

i∈ ˜ C λ⋆ i ˜

Pi)) = 1: ˜ x⋆ ∈ null( ˜ Q +

  • i∈ ˜

C

λ⋆

i ˜

Pi), y⋆ = 1

Dual problem

d⋆ = max

˜ λ

γ s.t. ˜ Q +

  • i∈ ˜

C

λi ˜ Pi 0

  • Linear objective
slide-41
SLIDE 41

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

QCQP and Lagrangian relaxation

Primal problem

f ⋆ = min

x

˜ x⊤ ˜ Q˜ x, ˜ x ≡ x y

  • ,

s.t. ˜ x⊤ ˜ Pi ˜ x = 0, ∀i ∈ ˜ C

Lagrangian relaxation

L(˜ x, ˜ λ) = γ + ˜ x⊤( ˜ Q +

  • i∈ ˜

C

λi ˜ Pi)˜ x, d(˜ λ) = γ s.t. ˜ Q +

  • i∈ ˜

C

λi ˜ Pi 0

Solution recovery

If rank(null( ˜ Q +

i∈ ˜ C λ⋆ i ˜

Pi)) = 1: ˜ x⋆ ∈ null( ˜ Q +

  • i∈ ˜

C

λ⋆

i ˜

Pi), y⋆ = 1

Dual problem

d⋆ = max

˜ λ

γ s.t. ˜ Q +

  • i∈ ˜

C

λi ˜ Pi 0

  • Linear objective
  • Lin. mat. ineq. (LMI) constraint
slide-42
SLIDE 42

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Semidefinite Programming (SDP)

Primal SDP

f ⋆ = min

X∈Symn

tr(CX) s.t. tr(AiX) = bi, ∀i ∈ C X 0.

Dual SDP

d⋆ = max

y∈Rm b⊤y,

m = #(C) s.t. C +

  • i∈C

yiAi 0

slide-43
SLIDE 43

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Semidefinite Programming (SDP)

Primal SDP

f ⋆ = min

X∈Symn

tr(CX) s.t. tr(AiX) = bi, ∀i ∈ C X 0.

Dual SDP

d⋆ = max

y∈Rm b⊤y,

m = #(C) s.t. C +

  • i∈C

yiAi 0

LP domain: Polytope

slide-44
SLIDE 44

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Semidefinite Programming (SDP)

Primal SDP

f ⋆ = min

X∈Symn

tr(CX) s.t. tr(AiX) = bi, ∀i ∈ C X 0.

Dual SDP

d⋆ = max

y∈Rm b⊤y,

m = #(C) s.t. C +

  • i∈C

yiAi 0

LP domain: Polytope SDP domain: Spectrahedron

slide-45
SLIDE 45

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Semidefinite Programming (SDP)

Primal SDP

f ⋆ = min

X∈Symn

tr(CX) s.t. tr(AiX) = bi, ∀i ∈ C X 0.

Dual SDP

d⋆ = max

y∈Rm b⊤y,

m = #(C) s.t. C +

  • i∈C

yiAi 0

Interior Point Method solvers:

  • SeDuMi
  • SDPT3
  • SDPA
  • Mosek
  • Etc.

SDP domain: Spectrahedron

slide-46
SLIDE 46

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

RCQP problem

Usual Lagrangian relaxation

RCQP problem

f ⋆ = min

R

˜ r⊤ ˜ Q˜ r, ˜ r = vec(R) 1

  • ,

s.t. R ∈ SO(3)

slide-47
SLIDE 47

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

RCQP problem

Usual Lagrangian relaxation

RCQP problem

f ⋆ = min

R

˜ r⊤ ˜ Q˜ r, ˜ r = vec(R) 1

  • ,

s.t. R ∈ SO(3)

Rotation matrix constraints:

R ∈ SO(3) ⇒ {R ∈ R3×3 : R⊤R =I3, det(R) = 1}.

slide-48
SLIDE 48

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

RCQP problem

Usual Lagrangian relaxation

RCQP problem

f ⋆ = min

R

˜ r⊤ ˜ Q˜ r, ˜ r = vec(R) 1

  • ,

s.t. R ∈ SO(3)

Rotation matrix constraints:

R ∈ SO(3) ⇒ {R ∈ R3×3 : R⊤R =I3

  • quadratic

, det(R) = 1

  • cubic

}.

slide-49
SLIDE 49

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

RCQP problem

Usual Lagrangian relaxation

RCQP problem

f ⋆ = min

R

˜ r⊤ ˜ Q˜ r, ˜ r = vec(R) 1

  • ,

s.t. R ∈ SO(3)

Rotation matrix constraints:

R ∈ SO(3) ⇒ {R ∈ R3×3 : R⊤R =I3

  • quadratic

, ✘✘✘✘✘

✘

det(R) = 1

  • cubic

}.

slide-50
SLIDE 50

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

RCQP problem

Usual Lagrangian relaxation

RCQP problem

f ⋆ = min

R

˜ r⊤ ˜ Q˜ r, ˜ r = vec(R) 1

  • ,

s.t. R⊤R = I3

Rotation matrix constraints:

R ∈ SO(3) ⇒ {R ∈ R3×3 : R⊤R =I3

  • quadratic

, ✘✘✘✘✘

✘

det(R) = 1

  • cubic

}.

slide-51
SLIDE 51

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

RCQP problem

Usual Lagrangian relaxation

RCQP problem ≡ QCQP

f ⋆ = min

R

˜ r⊤ ˜ Q˜ r, ˜ r = vec(R) 1

  • ,

s.t. ˜ r⊤ ˜ Pi˜ r = 0, ∀i = 1, . . . , 6

slide-52
SLIDE 52

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

RCQP problem

Usual Lagrangian relaxation

RCQP problem ≡ QCQP

f ⋆ = min

R

˜ r⊤ ˜ Q˜ r, ˜ r = vec(R) 1

  • ,

s.t. ˜ r⊤ ˜ Pi˜ r = 0, ∀i = 1, . . . , 6

Dual problem SDP

?

− → R⋆

d⋆ = max

λ,γ γ, s.t. ˜

Q +

6

  • i=1

λi ˜ Pi + γ ˜ Ph 0

slide-53
SLIDE 53

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

RCQP problem

Usual Lagrangian relaxation

RCQP problem ≡ QCQP

f ⋆ = min

R

˜ r⊤ ˜ Q˜ r, ˜ r = vec(R) 1

  • ,

s.t. ˜ r⊤ ˜ Pi˜ r = 0, ∀i = 1, . . . , 6

Dual problem SDP

?

− → R⋆

d⋆ = max

λ,γ γ, s.t. ˜

Q +

6

  • i=1

λi ˜ Pi + γ ˜ Ph 0

But is dual problem tight?

slide-54
SLIDE 54

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

RCQP problem

Usual Lagrangian relaxation

RCQP problem ≡ QCQP

f ⋆ = min

R

˜ r⊤ ˜ Q˜ r, ˜ r = vec(R) 1

  • ,

s.t. ˜ r⊤ ˜ Pi˜ r = 0, ∀i = 1, . . . , 6

Dual problem SDP

?

− → R⋆

d⋆ = max

λ,γ γ, s.t. ˜

Q +

6

  • i=1

λi ˜ Pi + γ ˜ Ph 0

But is dual problem tight?

No, in general d⋆ ≤ f ⋆ [1].

[1] Olsson & Eriksson, "Solving quadratically constrained geometrical problems using Lagrangian duality". ICPR08.

slide-55
SLIDE 55

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Duality strengthening

BUT the dual problem is not intrinsic [2]!

[2] Boyd & Vandenberghe, "Convex optimization". Cambridge University Press (2004).

slide-56
SLIDE 56

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Duality strengthening

BUT the dual problem is not intrinsic [2]!

Strengthening tools

  • compose objective function with increasing function
  • introduce extra (constrained) variables
  • add (linearly independent) redundant constraints

[2] Boyd & Vandenberghe, "Convex optimization". Cambridge University Press (2004).

slide-57
SLIDE 57

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Redundant rotation constraints

Desired properties:

  • Linearly independent
  • Quadratic
slide-58
SLIDE 58

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Redundant rotation constraints

  • Ort. columns R⊤R = I3
slide-59
SLIDE 59

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Redundant rotation constraints

  • Ort. columns R⊤R = I3
  • Ort. rows RR⊤ = I3
slide-60
SLIDE 60

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Redundant rotation constraints

  • Ort. columns R⊤R = I3
  • Ort. rows RR⊤ = I3

R(1) × R(2) = R(3)

slide-61
SLIDE 61

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Redundant rotation constraints

  • Ort. columns R⊤R = I3
  • Ort. rows RR⊤ = I3

R(1) × R(2) = R(3) R(2) × R(3) = R(1)

slide-62
SLIDE 62

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Redundant rotation constraints

  • Ort. columns R⊤R = I3
  • Ort. rows RR⊤ = I3

R(1) × R(2) = R(3) R(2) × R(3) = R(1) R(3) × R(1) = R(2)

slide-63
SLIDE 63

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Redundant rotation constraints

Minimal rotation constraints:

Constraint type Constraint equation # Orthonormal rows RR⊤ = I3 6 Determinant det(R) = +1 1

Redundant rotation constraints:

Constraint type Constraint equation # Orthonormal rows RR⊤ = I3 6 Orthonormal columns R⊤R = I3 6 Handedness R(1) × R(2) = R(3) 3 R(2) × R(3) = R(1) 3 R(3) × R(1) = R(2) 3

slide-64
SLIDE 64

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Redundant rotation constraints

Penalization patterns:

  • Ort. columns
  • Ort. rows

Handedness

slide-65
SLIDE 65

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

RCQP problem

Strengthened Lagrangian relaxation

RCQP problem

f ⋆ = min

R

˜ r⊤ ˜ Q˜ r, ˜ r = vec(R) 1

  • ,

s.t. R ∈ SO(3)

Rotation matrix constraints:

R ∈ SO(3) ⇒ {R ∈ R3×3 : R⊤R =I3, RR⊤ = I3, R(1)×R(2) = R(3), R(2)×R(3) = R(1), R(3)×R(1) = R(2)}.

slide-66
SLIDE 66

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

RCQP problem

Strengthened Lagrangian relaxation

RCQP problem ≡ QCQP

f ⋆ = min

R

˜ r⊤ ˜ Q˜ r, ˜ r = vec(R) 1

  • ,

s.t. ˜ r⊤ ˜ Pi˜ r = 0, ∀i = 1, . . . , 21

slide-67
SLIDE 67

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

RCQP problem

Strengthened Lagrangian relaxation

RCQP problem ≡ QCQP

f ⋆ = min

R

˜ r⊤ ˜ Q˜ r, ˜ r = vec(R) 1

  • ,

s.t. ˜ r⊤ ˜ Pi˜ r = 0, ∀i = 1, . . . , 21

Dual problem SDP

?

− → R⋆

d⋆ = max

λ,γ γ, s.t. ˜

Q +

21

  • i=1

λi ˜ Pi + γ ˜ Ph 0

slide-68
SLIDE 68

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

RCQP problem

Strengthened Lagrangian relaxation

RCQP problem ≡ QCQP

f ⋆ = min

R

˜ r⊤ ˜ Q˜ r, ˜ r = vec(R) 1

  • ,

s.t. ˜ r⊤ ˜ Pi˜ r = 0, ∀i = 1, . . . , 21

Dual problem SDP

?

− → R⋆

d⋆ = max

λ,γ γ, s.t. ˜

Q +

21

  • i=1

λi ˜ Pi + γ ˜ Ph 0

Is dual problem tight?

slide-69
SLIDE 69

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

RCQP problem

Strengthened Lagrangian relaxation

RCQP problem ≡ QCQP

f ⋆ = min

R

˜ r⊤ ˜ Q˜ r, ˜ r = vec(R) 1

  • ,

s.t. ˜ r⊤ ˜ Pi˜ r = 0, ∀i = 1, . . . , 21

Dual problem SDP

?

− → R⋆

d⋆ = max

λ,γ γ, s.t. ˜

Q +

21

  • i=1

λi ˜ Pi + γ ˜ Ph 0

Is dual problem tight?

Yes, d⋆ = f ⋆, for any problem. Warning: Empirical evidence [3].

[3] Briales & Gonzalez, "Convex Global 3D Registration with Lagrangian Duality". CVPR17.

slide-70
SLIDE 70

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Experiments

Experiment from Olsson and Eriksson [1]

[1] Olsson & Eriksson, "Solving quadratically constrained geometrical problems using Lagrangian duality". ICPR08.

slide-71
SLIDE 71

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Experiments

0.1 0.2 0.3 0.4 0.5

σ (m)

10 20 30 40 50 60 70 80 90 100

% optimal

BnB Ours Olsson

Synthetic problems ( ˆ m = 10)

slide-72
SLIDE 72

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Experiments

7 8 9 10 11 12 13 14 15

ˆ m

10 20 30 40 50 60 70 80 90 100

% optimal

BnB Ours Olsson

Real measurements on model

slide-73
SLIDE 73

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Closure and questions

slide-74
SLIDE 74

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Closure and questions

Conclusions:

  • Generic 3D registration solved globally
slide-75
SLIDE 75

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Closure and questions

Conclusions:

  • Generic 3D registration solved globally
  • Non-convexity of R ∈ SO(3) circumvented via convex (SDP) relaxation
slide-76
SLIDE 76

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Closure and questions

Conclusions:

  • Generic 3D registration solved globally
  • Non-convexity of R ∈ SO(3) circumvented via convex (SDP) relaxation

Code available: http://mapir.isa.uma.es/work/rotlift

Or simply scan the QR code!

slide-77
SLIDE 77

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

Closure and questions

Conclusions:

  • Generic 3D registration solved globally
  • Non-convexity of R ∈ SO(3) circumvented via convex (SDP) relaxation

Code available: http://mapir.isa.uma.es/work/rotlift

Or simply scan the QR code!

Future directions:

  • Theoretical proof of strong duality
  • Faster resolution of SDP problem
  • Optimality verification
  • Multiple global minima
  • Robust registration
slide-78
SLIDE 78

Fast and Global 3D Registration

  • f Points,

Lines and Planes Jesus Briales Problem Formulation

Unification Marginalization

RCQP QCQP SDP Dual of RCQP Experiments Conclusion Summary

References I

  • C. Olsson and A. Eriksson, “Solving quadratically constrained geometrical

problems using lagrangian duality,” in Pattern Recognition, 2008. ICPR

  • 2008. 19th Int. Conf., pp. 1–5, IEEE, 2008.
  • S. Boyd and L. Vandenberghe, Convex optimization.

Cambridge University Press, 2004.

  • J. Briales and J. González-Jiménez, “Convex Global 3D Registration with

Lagrangian Duality,” in Int. Conf. Comput. Vis. Pattern Recognit., jul 2017.