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Its Applications to Rotation Synchronization Yunpeng Shi Joint work - - PowerPoint PPT Presentation

Sep 06, 2023 128 likes โ€ข626 views

Message Passing Least Squares Framework and Its Applications to Rotation Synchronization Yunpeng Shi Joint work with Prof. Gilad Lerman School of Mathematics University of Minnesota Rotation Synchronization 3 2 1 5 4 Given a graph

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SLIDE 1

Message Passing Least Squares Framework and Its Applications to Rotation Synchronization

Yunpeng Shi

Joint work with Prof. Gilad Lerman School of Mathematics University of Minnesota

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SLIDE 2

1 2 4 5 3

Given a graph ๐ป ๐‘œ , ๐น ๐‘œ : = 1,2,3, โ€ฆ , ๐‘œ , ๐น is the set of edges

Rotation Synchronization

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SLIDE 3

๐‘†1

โˆ— = ?

๐‘†2

โˆ— = ?

๐‘†3

โˆ— = ?

๐‘†4

โˆ— = ?

๐‘†5

โˆ— = ?

Each node ๐‘— โˆˆ [๐‘œ] is assigned an unknown ground truth rotation ๐‘†๐‘—

โˆ— โˆˆ ๐‘‡๐‘ƒ(3)

Rotation Synchronization

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SLIDE 4

๐‘†1

โˆ— = ?

๐‘†2

โˆ— = ?

๐‘†3

โˆ— = ?

๐‘†4

โˆ— = ?

๐‘†5

โˆ— = ?

  • Each edge ๐‘—๐‘˜ โˆˆ ๐น is given a possibly noisy and corrupted relative rotation ๐‘†๐‘—๐‘˜
  • The uncorrupted relative rotation for ๐‘—๐‘˜ โˆˆ ๐น is ๐‘†๐‘—๐‘˜

โˆ— = ๐‘†๐‘— โˆ—๐‘†๐‘˜ โˆ—โˆ’1

  • Rotation Synchronization: Estimate ๐‘†๐‘—

โˆ— ๐‘—โˆˆ[๐‘œ] from ๐‘†๐‘—๐‘˜ ๐‘—๐‘˜โˆˆ๐น

  • ๐‘†๐‘—

โˆ—๐‘† ๐‘—โˆˆ[๐‘œ] for any rotation ๐‘† is also a solution ๐‘†12 ๐‘†23 ๐‘†25 ๐‘†14 ๐‘†45 ๐‘†35

Rotation Synchronization

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SLIDE 5

Applications

Camera orientation estimation in 3D reconstruction tasks:

Simultaneous localization and mapping (SLAM) Demonstration by Raรบl Mur-Artal Structure from motion (SfM) Demonstration by Carl Olsson

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SLIDE 6

Adversarial Corruption Model

๐‘†๐‘—๐‘˜ = แ‰ ๐‘†๐‘—๐‘˜

โˆ— : = ๐‘†๐‘— โˆ—๐‘†๐‘˜ โˆ—โˆ’1,

๐‘—๐‘˜ โˆˆ ๐น๐‘• เทจ ๐‘†๐‘—๐‘˜, ๐‘—๐‘˜ โˆˆ ๐น๐‘

(good edges) (bad edges)

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SLIDE 7

Adversarial Corruption Model

๐‘†๐‘—๐‘˜ = แ‰ ๐‘†๐‘—๐‘˜

โˆ— : = ๐‘†๐‘— โˆ—๐‘†๐‘˜ โˆ—โˆ’1,

๐‘—๐‘˜ โˆˆ ๐น๐‘• เทจ ๐‘†๐‘—๐‘˜, ๐‘—๐‘˜ โˆˆ ๐น๐‘ Corruption Level ๐‘ก๐‘—๐‘˜

โˆ— โ‰” ๐‘’(๐‘†๐‘—๐‘˜, ๐‘†๐‘—๐‘˜ โˆ— )

Commonly, ๐‘’ is the geodesic distance on ๐‘‡๐‘ƒ(3)

(good edges) (bad edges)

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SLIDE 8

Least Squares Solvers

minimize

๐‘†๐‘— ๐‘—โˆˆ[๐‘œ]โŠ‚๐‘‡๐‘ƒ(3) เท ๐‘—๐‘˜โˆˆ๐น

๐‘’2(๐‘†๐‘—๐‘†

๐‘˜ โˆ’1, ๐‘†๐‘—๐‘˜)

The most common approximate solution is the Lie algebraic averaging

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SLIDE 9

Robust Solvers: ๐‘š๐‘ž minimization (0 < ๐‘ž โ‰ค 1) minimize

๐‘†๐‘— ๐‘—โˆˆ[๐‘œ]โŠ‚๐‘‡๐‘ƒ(3) เท ๐‘—๐‘˜โˆˆ๐น

๐‘’๐‘ž(๐‘†๐‘—๐‘†

๐‘˜ โˆ’1, ๐‘†๐‘—๐‘˜)

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SLIDE 10

How to minimize ฯƒ๐‘—๐‘˜โˆˆ๐น ๐‘’๐‘ž(๐‘†๐‘—๐‘†๐‘˜

โˆ’1, ๐‘†๐‘—๐‘˜) over ๐‘†๐‘— ๐‘—โˆˆ[๐‘œ] in SO(3)?

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SLIDE 11

Iteratively Reweighted Least Squares (IRLS): ๐‘†๐‘—,๐‘ข ๐‘—โˆˆ[๐‘œ] = argmin

๐‘†๐‘— ๐‘—โˆˆ[๐‘œ]โŠ‚๐‘‡๐‘ƒ(3)

ฯƒ๐‘—๐‘˜โˆˆ๐น ๐‘ฅ๐‘—๐‘˜,๐‘ข๐‘’2(๐‘†๐‘—๐‘†๐‘˜

โˆ’1, ๐‘†๐‘—๐‘˜)

๐‘ 

๐‘—๐‘˜,๐‘ข = ๐‘’(๐‘†๐‘—,๐‘ข๐‘†๐‘˜,๐‘ข โˆ’1, ๐‘†๐‘—๐‘˜)

๐‘ฅ๐‘—๐‘˜,๐‘ข+1 = ๐‘ 

๐‘—๐‘˜,๐‘ข ๐‘žโˆ’2

How to minimize ฯƒ๐‘—๐‘˜โˆˆ๐น ๐‘’๐‘ž(๐‘†๐‘—๐‘†๐‘˜

โˆ’1, ๐‘†๐‘—๐‘˜) over ๐‘†๐‘— ๐‘—โˆˆ[๐‘œ] in SO(3)?

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SLIDE 12

Iteratively Reweighted Least Squares (IRLS): ๐‘†๐‘—,๐‘ข ๐‘—โˆˆ[๐‘œ] = argmin

๐‘†๐‘— ๐‘—โˆˆ[๐‘œ]โŠ‚๐‘‡๐‘ƒ(3)

ฯƒ๐‘—๐‘˜โˆˆ๐น ๐‘ฅ๐‘—๐‘˜,๐‘ข๐‘’2(๐‘†๐‘—๐‘†๐‘˜

โˆ’1, ๐‘†๐‘—๐‘˜)

๐‘ 

๐‘—๐‘˜,๐‘ข = ๐‘’(๐‘†๐‘—,๐‘ข๐‘†๐‘˜,๐‘ข โˆ’1, ๐‘†๐‘—๐‘˜)

๐‘ฅ๐‘—๐‘˜,๐‘ข+1 = ๐‘ 

๐‘—๐‘˜,๐‘ข ๐‘žโˆ’2

Ideally, ๐‘ 

๐‘—๐‘˜,๐‘ข โ‰ˆ ๐‘ก๐‘—๐‘˜ โˆ— โ‰” ๐‘’(๐‘†๐‘—๐‘˜, ๐‘†๐‘—๐‘˜ โˆ— ) and ๐‘ฅ๐‘—๐‘˜,๐‘ข+1 โ‰ˆ 1 ๐‘ก๐‘—๐‘˜

โˆ—

2โˆ’๐‘ž

concentrates on ๐น๐‘•

How to minimize ฯƒ๐‘—๐‘˜โˆˆ๐น ๐‘’๐‘ž(๐‘†๐‘—๐‘†๐‘˜

โˆ’1, ๐‘†๐‘—๐‘˜) over ๐‘†๐‘— ๐‘—โˆˆ[๐‘œ] in SO(3)?

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SLIDE 13

Issue 1: Over-Aggressive Reweighting

  • Under severe corruption, ๐‘†๐‘—,๐‘ข โ‰‰ ๐‘†๐‘—

โˆ— and thus ๐‘ ๐‘—๐‘˜,๐‘ข โ‰‰ ๐‘ก๐‘—๐‘˜ โˆ—

  • In certain cases, ๐‘ ๐‘—๐‘˜,๐‘ข โ‰ˆ 0 for ๐‘—๐‘˜ โˆˆ ๐น๐‘, and thus

๐‘ฅ๐‘—๐‘˜,๐‘ข+1 =

1 ๐‘ ๐‘—๐‘˜,๐‘ข 2โˆ’๐‘ž

can be extremely high for ๐‘—๐‘˜ โˆˆ ๐น๐‘

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SLIDE 14

Issue 2: Poor Least Squares Solution

๐‘†๐‘—,๐‘ข ๐‘—โˆˆ[๐‘œ] = argmin

๐‘†๐‘— ๐‘—โˆˆ[๐‘œ]โŠ‚๐‘‡๐‘ƒ(3)

เท

๐‘—๐‘˜โˆˆ๐น

๐‘ฅ๐‘—๐‘˜,๐‘ข๐‘’2(๐‘†๐‘—๐‘†๐‘˜

โˆ’1, ๐‘†๐‘—๐‘˜)

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SLIDE 15

Issue 2: Poor Least Squares Solution

๐‘†๐‘—,๐‘ข ๐‘—โˆˆ[๐‘œ] = argmin

๐‘†๐‘— ๐‘—โˆˆ[๐‘œ]โŠ‚๐‘‡๐‘ƒ(3)

เท

๐‘—๐‘˜โˆˆ๐น

๐‘ฅ๐‘—๐‘˜,๐‘ข๐‘’2(๐‘†๐‘—๐‘†๐‘˜

โˆ’1, ๐‘†๐‘—๐‘˜)

๐œ•๐‘— ๐‘†๐‘— ๐‘†๐‘—,๐‘ขโˆ’1 Riemannian manifold of SO(3) Lie algebraic representation

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SLIDE 16

Issue 2: Poor Least Squares Solution

๐‘†๐‘—,๐‘ข ๐‘—โˆˆ[๐‘œ] = argmin

๐‘†๐‘— ๐‘—โˆˆ[๐‘œ]โŠ‚๐‘‡๐‘ƒ(3)

เท

๐‘—๐‘˜โˆˆ๐น

๐‘ฅ๐‘—๐‘˜,๐‘ข๐‘’2(๐‘†๐‘—๐‘†๐‘˜

โˆ’1, ๐‘†๐‘—๐‘˜)

The Lie Algebraic Averaging uses the approximation ๐‘’ ๐‘†๐‘—๐‘†

๐‘˜ โˆ’1, ๐‘†๐‘—๐‘˜

โ‰ˆ ิก๐œ•๐‘— โˆ’ เธฎ ๐œ•๐‘˜ โˆ’ ๐œ•๐‘—๐‘˜

2

where ๐œ•๐‘— = log( ๐‘†๐‘—,๐‘ขโˆ’1

โˆ’1 ๐‘†๐‘—)

and ๐œ•๐‘—๐‘˜ = log( ๐‘†๐‘—,๐‘ขโˆ’1

โˆ’1 ๐‘†๐‘—๐‘˜๐‘† ๐‘˜,๐‘ขโˆ’1)

The approximation is valid only when ๐‘†๐‘— โ‰ˆ ๐‘†๐‘—

โˆ— and ๐‘†๐‘—๐‘˜ โ‰ˆ ๐‘†๐‘— โˆ—๐‘†๐‘˜ โˆ—โˆ’1

๐œ•๐‘— ๐‘†๐‘— ๐‘†๐‘—,๐‘ขโˆ’1

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SLIDE 17

How to accurately estimate ๐‘ก๐‘—๐‘˜

โˆ— without knowing ๐‘†๐‘— โˆ— and ๐‘†๐‘˜ โˆ— ?

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SLIDE 18

Cycle-Edge Message Passing (CEMP)

  • Goal: Estimate corruption level

๐‘ก๐‘—๐‘˜

โˆ— โ‰” ๐‘’(๐‘†๐‘—๐‘˜, ๐‘†๐‘—๐‘˜ โˆ— )

from 3-cycle inconsistency measure ๐‘’๐‘—๐‘˜๐‘™ โ‰” ๐‘’(๐‘†๐‘—๐‘˜๐‘†๐‘˜๐‘™๐‘†๐‘™๐‘—, ๐ฝ)

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SLIDE 19

Cycle-Edge Message Passing (CEMP)

  • Goal: Estimate corruption level

๐‘ก๐‘—๐‘˜

โˆ— โ‰” ๐‘’(๐‘†๐‘—๐‘˜, ๐‘†๐‘—๐‘˜ โˆ— )

from 3-cycle inconsistency measure ๐‘’๐‘—๐‘˜๐‘™ โ‰” ๐‘’(๐‘†๐‘—๐‘˜๐‘†๐‘˜๐‘™๐‘†๐‘™๐‘—, ๐ฝ)

  • For each ๐‘—๐‘˜ โˆˆ ๐น, sample 50 3-cycles ๐‘—๐‘˜๐‘™ and for each cycle compute ๐‘’๐‘—๐‘˜๐‘™
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SLIDE 20

Cycle-Edge Message Passing (CEMP)

  • Goal: Estimate corruption level

๐‘ก๐‘—๐‘˜

โˆ— โ‰” ๐‘’(๐‘†๐‘—๐‘˜, ๐‘†๐‘—๐‘˜ โˆ— )

from 3-cycle inconsistency measure ๐‘’๐‘—๐‘˜๐‘™ โ‰” ๐‘’(๐‘†๐‘—๐‘˜๐‘†๐‘˜๐‘™๐‘†๐‘™๐‘—, ๐ฝ)

  • For each ๐‘—๐‘˜ โˆˆ ๐น, sample 50 3-cycles ๐‘—๐‘˜๐‘™ and for each cycle compute ๐‘’๐‘—๐‘˜๐‘™
  • For fixed ๐‘—๐‘˜ and good 3-cycles w.r.t. ๐‘—๐‘˜ (That is, ๐‘—๐‘™, ๐‘˜๐‘™ โˆˆ ๐น๐‘•)

๐‘’๐‘—๐‘˜๐‘™ = ๐‘’(๐‘†๐‘—๐‘˜๐‘†๐‘˜๐‘™

โˆ— ๐‘†๐‘™๐‘— โˆ— , ๐ฝ)=๐‘’(๐‘†๐‘—๐‘˜๐‘†๐‘˜๐‘™ โˆ— ๐‘†๐‘™๐‘— โˆ— ๐‘†๐‘—๐‘˜ โˆ— , ๐‘†๐‘—๐‘˜ โˆ— )=๐‘’(๐‘†๐‘—๐‘˜, ๐‘†๐‘—๐‘˜ โˆ— )= ๐‘ก๐‘—๐‘˜ โˆ—

๏ธธ

= ๐ฝ by cycle consistency

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SLIDE 21

๐‘’๐‘—๐‘˜๐‘™1 ๐‘’๐‘—๐‘˜๐‘™2 ๐‘’๐‘—๐‘˜๐‘™49 ๐‘’๐‘—๐‘˜๐‘™50

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SLIDE 22

๐‘’๐‘—๐‘˜๐‘™1 ๐‘’๐‘—๐‘˜๐‘™2 = ๐‘ก๐‘—๐‘˜

โˆ—

๐‘’๐‘—๐‘˜๐‘™49 = ๐‘ก๐‘—๐‘˜

โˆ—

๐‘’๐‘—๐‘˜๐‘™50

๐‘ž๐‘—๐‘˜๐‘™1

๐‘ข

๐‘ž๐‘—๐‘˜๐‘™2

๐‘ข

๐‘ž๐‘—๐‘˜๐‘™49

๐‘ข

๐‘ž๐‘—๐‘˜๐‘™50

๐‘ข

๐‘—๐‘˜

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SLIDE 23

๐‘’๐‘—๐‘˜๐‘™1 ๐‘’๐‘—๐‘˜๐‘™2 = ๐‘ก๐‘—๐‘˜

โˆ—

๐‘’๐‘—๐‘˜๐‘™49 = ๐‘ก๐‘—๐‘˜

โˆ—

๐‘’๐‘—๐‘˜๐‘™50

๐‘ž๐‘—๐‘˜๐‘™1

๐‘ข

๐‘ž๐‘—๐‘˜๐‘™2

๐‘ข

๐‘ž๐‘—๐‘˜๐‘™49

๐‘ข

๐‘ž๐‘—๐‘˜๐‘™50

๐‘ข

๐‘ก๐‘—๐‘˜

โˆ— โ‰ˆ ๐‘ก๐‘—๐‘˜,๐‘ข+1: = 1 ๐‘Ž๐‘—๐‘˜

๐‘ข ฯƒ๐‘™ ๐‘ž๐‘—๐‘˜๐‘™

๐‘ข ๐‘’๐‘—๐‘˜๐‘™

๐‘—๐‘˜

๐‘Ž๐‘—๐‘˜

๐‘ข = ฯƒ๐‘™ ๐‘ž๐‘—๐‘˜๐‘™ ๐‘ข ๐‘’๐‘—๐‘˜๐‘™

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SLIDE 24

๐‘ก๐‘—๐‘™,๐‘ข ๐‘ก

๐‘˜๐‘™,๐‘ข

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SLIDE 25

๐‘“โˆ’๐›พ๐‘ขโˆ™ ๐‘ก๐‘—๐‘™,๐‘ข ๐‘ž๐‘˜๐‘™,๐‘ข ๐‘ž๐‘—๐‘™,๐‘ข ๐‘ก

๐‘˜๐‘™,๐‘ข

๐‘“โˆ’๐›พ๐‘ขโˆ™

  • Prob. that ๐‘—๐‘™ โˆˆ ๐น๐‘• given ๐‘ก๐‘—๐‘™,๐‘ข
  • Prob. that j๐‘™ โˆˆ ๐น๐‘• given ๐‘ก

๐‘˜๐‘™,๐‘ข

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SLIDE 26

๐‘—๐‘˜๐‘™ ๐‘ž๐‘—๐‘˜๐‘™

๐‘ข

= ๐‘“โˆ’๐›พ๐‘ข(๐‘ก๐‘—๐‘™,๐‘ข+๐‘ก๐‘˜๐‘™,๐‘ข)

๐‘“โˆ’๐›พ๐‘ขโˆ™ ๐‘ก๐‘—๐‘™,๐‘ข ๐‘ž๐‘˜๐‘™,๐‘ข ๐‘ž๐‘—๐‘™,๐‘ข ๐‘ก

๐‘˜๐‘™,๐‘ข

๐‘“โˆ’๐›พ๐‘ขโˆ™

  • Prob. that ๐‘—๐‘™ โˆˆ ๐น๐‘• given ๐‘ก๐‘—๐‘™,๐‘ข
  • Prob. that j๐‘™ โˆˆ ๐น๐‘• given ๐‘ก

๐‘˜๐‘™,๐‘ข

  • Prob. that ๐‘—๐‘˜๐‘™ is good given {๐‘ก๐‘๐‘,๐‘ข: ๐‘๐‘ โˆˆ ๐น}
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SLIDE 27

๐‘ž๐‘—๐‘˜๐‘™

๐‘ข

= ๐‘“โˆ’๐›พ๐‘ข(๐‘ก๐‘—๐‘™,๐‘ข+๐‘ก๐‘˜๐‘™,๐‘ข) = ๐‘„๐‘ (๐‘’๐‘—๐‘˜๐‘™ = ๐‘ก๐‘—๐‘˜

โˆ— |{๐‘ก๐‘๐‘,๐‘ข: ๐‘๐‘ โˆˆ ๐น})

The conditional probability that ๐‘—๐‘˜๐‘™ is good (๐‘’๐‘—๐‘˜๐‘™ = ๐‘ก๐‘—๐‘˜

โˆ— ):

The estimate of the corruption level:

๐‘ก๐‘—๐‘˜,๐‘ข+1: =

1 ๐‘Ž๐‘—๐‘˜

๐‘ข ฯƒ๐‘™ ๐‘ž๐‘—๐‘˜๐‘™

๐‘ข ๐‘’๐‘—๐‘˜๐‘™ = ๐”ฝ(๐‘ก๐‘—๐‘˜ โˆ— |{๐‘ก๐‘๐‘,๐‘ข: ๐‘๐‘ โˆˆ ๐น})

Cycle-Edge Message Passing (CEMP)

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SLIDE 28

Theory

If f

  • the maximal ratio of corrupted cycles per edge <

1 5

  • ๐›พ๐‘ข increases exponentially with a sufficiently small rate,

the hen

  • for all ๐‘—๐‘˜ โˆˆ ๐น, ๐‘ก๐‘—๐‘˜,๐‘ข computed by CEMP linearly and uniformly

converges to ๐‘ก๐‘—๐‘˜

โˆ— .

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SLIDE 29

Init Initialization

Run CEMP for ๐‘ˆ iterations Build a weighted graph with edge weights ๐‘ก๐‘—๐‘˜,๐‘ˆ ๐‘—๐‘˜โˆˆ๐น Find the minimal spanning tree using Primโ€™s Algorithm Initi nitializ ize ๐‘†๐‘—,0 by fixing ๐‘†1,0 = ๐ฝ and ๐‘†๐‘—,0 = ๐‘†๐‘—๐‘˜ ๐‘†๐‘˜,0 Initi nitializ ize weights ๐‘ฅ๐‘—๐‘˜,0 = ๐บ(๐‘ก๐‘—๐‘˜,๐‘ˆ) For ๐‘š๐‘ž minimization, ๐บ ๐‘ฆ = ๐‘ฆ๐‘žโˆ’2

Message Passing Least Squares (MPLS)

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SLIDE 30

Message Passing Least Squares (MPLS)

where ๐œ•๐‘— = log( ๐‘†๐‘—,๐‘ขโˆ’1

โˆ’1 ๐‘†๐‘—) and ๐œ•๐‘—๐‘˜ = log( ๐‘†๐‘—,๐‘ขโˆ’1 โˆ’1 ๐‘†๐‘—๐‘˜๐‘†๐‘˜,๐‘ขโˆ’1).

After solving ๐œ•๐‘—,๐‘ข, update ๐‘†๐‘—,๐‘ข = ๐‘†๐‘—,๐‘ขโˆ’1exp(๐œ•๐‘—,๐‘ข). Residual ๐‘ ๐‘—๐‘˜,๐‘ข: = เธฎ๐œ•๐‘—,๐‘ข โˆ’ เธฎ ๐œ•๐‘˜,๐‘ข โˆ’ ๐œ•๐‘—๐‘˜

2 โ‰ˆ ๐‘’ ๐‘†๐‘—,๐‘ข๐‘†๐‘˜,๐‘ข โˆ’1, ๐‘†๐‘—๐‘˜ โ‰ˆ ๐‘ก๐‘—๐‘˜ โˆ—

๐œ•๐‘—,๐‘ข ๐‘†๐‘—,๐‘ข ๐‘†๐‘—,๐‘ขโˆ’1

๐œ•๐‘—,๐‘ข ๐‘—โˆˆ[๐‘œ] = argmin

๐œ•๐‘—โˆˆโ„3 เท ๐‘—๐‘˜โˆˆ๐น

๐‘ฅ๐‘—๐‘˜,๐‘ขิก๐œ•๐‘— โˆ’ เธฎ ๐œ•๐‘˜ โˆ’ ๐œ•๐‘—๐‘˜

2 2

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SLIDE 31

Reweighting

  • Ideally, ๐‘ฅ๐‘—๐‘˜,๐‘ข = ๐บ ๐‘ก๐‘—๐‘˜

โˆ— , where ๐บ ๐‘ฆ = ๐‘ฆ๐‘žโˆ’2

  • We set ๐‘ฅ๐‘—๐‘˜,๐‘ข = ๐บ ๐‘๐‘—๐‘˜,๐‘ข , where ๐‘๐‘—๐‘˜,๐‘ข is a better approximation of ๐‘ก๐‘—๐‘˜

โˆ— than ๐‘ก๐‘—๐‘˜,๐‘ข

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SLIDE 32

IRLS: ๐‘ก๐‘—๐‘˜

โˆ— โ‰ˆ ๐‘ ๐‘—๐‘˜,๐‘ข โ‰ˆ ๐‘’ ๐‘†๐‘—,๐‘ข๐‘†๐‘˜,๐‘ข โˆ’1, ๐‘†๐‘—๐‘˜

and ๐‘ฅ๐‘—๐‘˜,๐‘ข+1 = ๐บ ๐‘ ๐‘—๐‘˜,๐‘ข CEMP: ๐‘ก๐‘—๐‘˜

โˆ— โ‰ˆ ๐‘ก๐‘—๐‘˜,๐‘ข: = 1 ๐‘Ž๐‘—๐‘˜

๐‘ข ฯƒ๐‘™ ๐‘ž๐‘—๐‘˜๐‘™

๐‘ขโˆ’1๐‘’๐‘—๐‘˜๐‘™ and ๐‘ž๐‘—๐‘˜๐‘™ ๐‘ขโˆ’1 = ๐‘“โˆ’๐›พ๐‘ข(๐‘ก๐‘—๐‘™,๐‘ขโˆ’1+๐‘ก๐‘˜๐‘™,๐‘ขโˆ’1)

MPLS: ๐‘ก๐‘—๐‘˜

โˆ— โ‰ˆ ๐‘๐‘—๐‘˜,๐‘ข: = ๐›ฝ๐‘ข โ„Ž๐‘—๐‘˜,๐‘ข + (1 โˆ’ ๐›ฝ๐‘ข) ๐‘ ๐‘—๐‘˜,๐‘ข

and ๐‘ฅ๐‘—๐‘˜,๐‘ข+1 = ๐บ ๐‘๐‘—๐‘˜,๐‘ข where โ„Ž๐‘—๐‘˜,๐‘ข: =

1 ๐‘Ž๐‘—๐‘˜

๐‘ข ฯƒ๐‘™ ๐‘Ÿ๐‘—๐‘˜๐‘™

๐‘ข ๐‘’๐‘—๐‘˜๐‘™ and ๐‘Ÿ๐‘—๐‘˜๐‘™ ๐‘ข

= ๐‘“โˆ’๐›พ๐‘ˆ(๐‘ ๐‘—๐‘™,๐‘ข+๐‘ ๐‘˜๐‘™,๐‘ข)

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SLIDE 33

IRLS: ๐‘ก๐‘—๐‘˜

โˆ— โ‰ˆ ๐‘ ๐‘—๐‘˜,๐‘ข โ‰ˆ ๐‘’ ๐‘†๐‘—,๐‘ข๐‘†๐‘˜,๐‘ข โˆ’1, ๐‘†๐‘—๐‘˜

and ๐‘ฅ๐‘—๐‘˜,๐‘ข+1 = ๐บ ๐‘ ๐‘—๐‘˜,๐‘ข CEMP: ๐‘ก๐‘—๐‘˜

โˆ— โ‰ˆ ๐‘ก๐‘—๐‘˜,๐‘ข: = 1 ๐‘Ž๐‘—๐‘˜

๐‘ข ฯƒ๐‘™ ๐‘ž๐‘—๐‘˜๐‘™

๐‘ขโˆ’1๐‘’๐‘—๐‘˜๐‘™ and ๐‘ž๐‘—๐‘˜๐‘™ ๐‘ขโˆ’1 = ๐‘“โˆ’๐›พ๐‘ข(๐‘ก๐‘—๐‘™,๐‘ขโˆ’1+๐‘ก๐‘˜๐‘™,๐‘ขโˆ’1)

MPLS: ๐‘ก๐‘—๐‘˜

โˆ— โ‰ˆ ๐‘๐‘—๐‘˜,๐‘ข: = ๐›ฝ๐‘ข โ„Ž๐‘—๐‘˜,๐‘ข + (1 โˆ’ ๐›ฝ๐‘ข) ๐‘ ๐‘—๐‘˜,๐‘ข

and ๐‘ฅ๐‘—๐‘˜,๐‘ข+1 = ๐บ ๐‘๐‘—๐‘˜,๐‘ข where โ„Ž๐‘—๐‘˜,๐‘ข: =

1 ๐‘Ž๐‘—๐‘˜

๐‘ข ฯƒ๐‘™ ๐‘Ÿ๐‘—๐‘˜๐‘™

๐‘ข ๐‘’๐‘—๐‘˜๐‘™ and ๐‘Ÿ๐‘—๐‘˜๐‘™ ๐‘ข

= ๐‘“โˆ’๐›พ๐‘ˆ(๐‘ ๐‘—๐‘™,๐‘ข+๐‘ ๐‘˜๐‘™,๐‘ข) โ„Ž๐‘—๐‘˜,๐‘ข = ๐”ฝ(๐‘ก๐‘—๐‘˜

โˆ— |{๐‘ ๐‘๐‘,๐‘ข: ๐‘๐‘ โˆˆ ๐น})

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SLIDE 34

MPLS: ๐‘๐‘—๐‘˜,๐‘ขโˆ’1 โ†’ ๐‘ฅ๐‘—๐‘˜,๐‘ข ๐‘ฅ๐‘—๐‘˜,๐‘ข: weights

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SLIDE 35

MPLS: ๐‘๐‘—๐‘˜,๐‘ขโˆ’1 โ†’ ๐‘ฅ๐‘—๐‘˜,๐‘ข โ†’ ๐‘†๐‘—๐‘˜,๐‘ข ๐‘ฅ๐‘—๐‘˜,๐‘ข: weights ๐‘†๐‘—๐‘˜,๐‘ข : estimated relative rotations ๐‘†๐‘—,๐‘ข๐‘†

๐‘˜,๐‘ข โˆ’1

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SLIDE 36

MPLS: ๐‘๐‘—๐‘˜,๐‘ขโˆ’1 โ†’ ๐‘ฅ๐‘—๐‘˜,๐‘ข โ†’ ๐‘†๐‘—๐‘˜,๐‘ข โ†’ ๐‘ 

๐‘—๐‘˜,๐‘ข

๐‘ฅ๐‘—๐‘˜,๐‘ข: weights ๐‘†๐‘—๐‘˜,๐‘ข : estimated relative rotations ๐‘†๐‘—,๐‘ข๐‘†

๐‘˜,๐‘ข โˆ’1

๐‘ 

๐‘—๐‘˜,๐‘ข: residuals

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SLIDE 37

MPLS: ๐‘๐‘—๐‘˜,๐‘ขโˆ’1 โ†’ ๐‘ฅ๐‘—๐‘˜,๐‘ข โ†’ ๐‘†๐‘—๐‘˜,๐‘ข โ†’ ๐‘ 

๐‘—๐‘˜,๐‘ข โ†’ โ„Ž๐‘—๐‘˜,๐‘ข

๐‘ฅ๐‘—๐‘˜,๐‘ข: weights ๐‘†๐‘—๐‘˜,๐‘ข : estimated relative rotations ๐‘†๐‘—,๐‘ข๐‘†

๐‘˜,๐‘ข โˆ’1

๐‘ 

๐‘—๐‘˜,๐‘ข: residuals

โ„Ž๐‘—๐‘˜,๐‘ข: analogue of ๐‘ก๐‘—๐‘˜,๐‘ข (Note that ๐‘ก๐‘—๐‘˜,๐‘ข =CEMP(๐‘ก๐‘—๐‘˜,๐‘ขโˆ’1) and โ„Ž๐‘—๐‘˜,๐‘ข =CEMP(๐‘ 

๐‘—๐‘˜,๐‘ข))

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SLIDE 38

MPLS: ๐‘๐‘—๐‘˜,๐‘ขโˆ’1 โ†’ ๐‘ฅ๐‘—๐‘˜,๐‘ข โ†’ ๐‘†๐‘—๐‘˜,๐‘ข โ†’ ๐‘ 

๐‘—๐‘˜,๐‘ข โ†’ โ„Ž๐‘—๐‘˜,๐‘ข โ†’ ๐‘๐‘—๐‘˜,๐‘ข

๐‘ฅ๐‘—๐‘˜,๐‘ข: weights ๐‘†๐‘—๐‘˜,๐‘ข : estimated relative rotations ๐‘†๐‘—,๐‘ข๐‘†

๐‘˜,๐‘ข โˆ’1

๐‘ 

๐‘—๐‘˜,๐‘ข: residuals

โ„Ž๐‘—๐‘˜,๐‘ข: analogue of ๐‘ก๐‘—๐‘˜,๐‘ข (Note that ๐‘ก๐‘—๐‘˜,๐‘ข =CEMP(๐‘ก๐‘—๐‘˜,๐‘ขโˆ’1) and โ„Ž๐‘—๐‘˜,๐‘ข =CEMP(๐‘ 

๐‘—๐‘˜,๐‘ข))

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SLIDE 39

IRLS: ๐‘ 

๐‘—๐‘˜,๐‘ขโˆ’1 โ†’ ๐‘ฅ๐‘—๐‘˜,๐‘ข โ†’ ๐‘†๐‘—๐‘˜,๐‘ข โ†’ ๐‘  ๐‘—๐‘˜,๐‘ข

MPLS: ๐‘๐‘—๐‘˜,๐‘ขโˆ’1 โ†’ ๐‘ฅ๐‘—๐‘˜,๐‘ข โ†’ ๐‘†๐‘—๐‘˜,๐‘ข โ†’ ๐‘ 

๐‘—๐‘˜,๐‘ข โ†’ โ„Ž๐‘—๐‘˜,๐‘ข โ†’ ๐‘๐‘—๐‘˜,๐‘ข

CEMP: ๐‘ก๐‘—๐‘˜,๐‘ขโˆ’1 โ†’ ๐‘ก๐‘—๐‘˜,๐‘ข ๐‘ฅ๐‘—๐‘˜,๐‘ข: weights ๐‘†๐‘—๐‘˜,๐‘ข : estimated relative rotations ๐‘†๐‘—,๐‘ข๐‘†๐‘˜,๐‘ข

โˆ’1

๐‘ 

๐‘—๐‘˜,๐‘ข: residuals

โ„Ž๐‘—๐‘˜,๐‘ข: analogue of ๐‘ก๐‘—๐‘˜,๐‘ข (Note that ๐‘ก๐‘—๐‘˜,๐‘ข =CEMP(๐‘ก๐‘—๐‘˜,๐‘ขโˆ’1) and โ„Ž๐‘—๐‘˜,๐‘ข =CEMP(๐‘ 

๐‘—๐‘˜,๐‘ข))

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SLIDE 40

Experiments

๐‘ˆ = 5, ๐›พ๐‘ข = 2๐‘ข for ๐‘ข = 0, โ€ฆ , 5. For ๐‘ข > 0 ๐›ฝ๐‘ข = 1 ๐‘ข + 1 , ๐บ ๐‘ฆ = ๐‘ฆโˆ’3

2 โˆ™ 1(๐‘ฆ < ๐œ๐‘ข)

๐‘š1/2 minimization Additional Thresholding

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SLIDE 41

๐‘†12 โ‰ˆ ๐‘†12

โˆ—

๐‘†23 โ‰ˆ ๐‘†23

โˆ—

๐‘†35~Haar(SO(3)) ๐‘†14~Haar(SO(3)) ๐‘†25 โ‰ˆ ๐‘†25

โˆ—

๐‘†45 โ‰ˆ ๐‘†45

โˆ—

1 2

Uniform Corruption Model

3 5 4

Corrupted with Prob. q

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SLIDE 42

๐‘•12 โ‰ˆ ๐‘•12

โˆ—

๐‘•23 โ‰ˆ ๐‘•23

โˆ—

๐‘•35~Haar(G) ๐‘•14~Haar(G) ๐‘•25 โ‰ˆ ๐‘•25

โˆ—

๐‘•45 โ‰ˆ ๐‘•45

โˆ—

1 2 3 5 4

Corrupted with Prob. q

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SLIDE 43

Uniform Corruption

q: prob. of corruption ๐œ: noise level

slide-44
SLIDE 44

๐‘†12 โ‰ˆ ๐‘†12

โˆ—

๐‘†23 = เทจ ๐‘†2 เทจ ๐‘†3

โˆ’1

๐‘†35 = เทจ ๐‘†3 เทจ ๐‘†5

โˆ’1

เทจ ๐‘†๐‘—~Haar(SO(3)) for ๐‘— โˆˆ [๐‘œ] ๐‘†25 = เทจ ๐‘†2 เทจ ๐‘†5

โˆ’1

๐‘†45 โ‰ˆ ๐‘†45

โˆ—

1 2

Self-consistent Corruption Model

3 5 4

Corrupted with Prob. q ๐‘†14 = เทจ ๐‘†1 เทจ ๐‘†4

โˆ’1

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SLIDE 45

Self-Consistent Corruption

q: prob. of corruption ๐œ: noise level

slide-46
SLIDE 46
slide-47
SLIDE 47
slide-48
SLIDE 48

Conclusion

  • We proposed the MPLS framework for robustly solving rotation synchronization
  • Our reweighting strategy is more reliable under high corruption and noise
  • Future directions:

theory for MPLS (exact recovery and convergence); more applications; adaptive reweighting parameters/optimal reweighting functions

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SLIDE 49

Thank you!