Download - Translation Synchronization via Truncated Least Squaresxrhuang/slides/TranSyncSpotlight_NIPS17.pdf · Translation Synchronization via Truncated Least Squares Xiangru Huang1* Zhenxiao

Transcript
Page 1: Translation Synchronization via Truncated Least Squaresxrhuang/slides/TranSyncSpotlight_NIPS17.pdf · Translation Synchronization via Truncated Least Squares Xiangru Huang1* Zhenxiao

Translation Synchronizationvia Truncated Least Squares

Xiangru Huang1* Zhenxiao Liang2*Chandrajit Bajaj 1 Qixing Huang 1

1Department of Computer ScienceUniversity of Texas at Austin

2Tsinghua University

NIPS, 2017

Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 1 / 20

Page 2: Translation Synchronization via Truncated Least Squaresxrhuang/slides/TranSyncSpotlight_NIPS17.pdf · Translation Synchronization via Truncated Least Squares Xiangru Huang1* Zhenxiao

From a Simple Example

U[µ− σ, µ+ σ]

σ−σ

Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 2 / 20

Page 3: Translation Synchronization via Truncated Least Squaresxrhuang/slides/TranSyncSpotlight_NIPS17.pdf · Translation Synchronization via Truncated Least Squares Xiangru Huang1* Zhenxiao

From a Simple Example

U[µ− σ, µ+ σ]

σ−σ

µ

Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 3 / 20

Page 4: Translation Synchronization via Truncated Least Squaresxrhuang/slides/TranSyncSpotlight_NIPS17.pdf · Translation Synchronization via Truncated Least Squares Xiangru Huang1* Zhenxiao

From a Simple Example

U[µ− σ, µ+ σ]

σ−σ

µ

Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 4 / 20

Page 5: Translation Synchronization via Truncated Least Squaresxrhuang/slides/TranSyncSpotlight_NIPS17.pdf · Translation Synchronization via Truncated Least Squares Xiangru Huang1* Zhenxiao

From a Simple Example

U[µ− σ, µ+ σ]

σ−σ

µ

mean

Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 5 / 20

Page 6: Translation Synchronization via Truncated Least Squaresxrhuang/slides/TranSyncSpotlight_NIPS17.pdf · Translation Synchronization via Truncated Least Squares Xiangru Huang1* Zhenxiao

From a Simple Example

U[µ− σ, µ+ σ]

σ−σ

Outliers

Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 6 / 20

Page 7: Translation Synchronization via Truncated Least Squaresxrhuang/slides/TranSyncSpotlight_NIPS17.pdf · Translation Synchronization via Truncated Least Squares Xiangru Huang1* Zhenxiao

From a Simple Example

U[µ− σ, µ+ σ]

σ−σ

Outliers

Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 7 / 20

Page 8: Translation Synchronization via Truncated Least Squaresxrhuang/slides/TranSyncSpotlight_NIPS17.pdf · Translation Synchronization via Truncated Least Squares Xiangru Huang1* Zhenxiao

From a Simple Example

U[µ− σ, µ+ σ]

σ−σ

Outliers

µ

mean

Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 8 / 20

Page 9: Translation Synchronization via Truncated Least Squaresxrhuang/slides/TranSyncSpotlight_NIPS17.pdf · Translation Synchronization via Truncated Least Squares Xiangru Huang1* Zhenxiao

From a Simple Example

U[µ− σ, µ+ σ]

σ−σ

Outliers

µ

meanmedian

Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 9 / 20

Page 10: Translation Synchronization via Truncated Least Squaresxrhuang/slides/TranSyncSpotlight_NIPS17.pdf · Translation Synchronization via Truncated Least Squares Xiangru Huang1* Zhenxiao

From a Simple Example

U[µ− σ, µ+ σ]

σ−σ

Outliers

µ

meanmedian

1. Delete Sample tj if |tj −mean|>ǫ

Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 10 / 20

Page 11: Translation Synchronization via Truncated Least Squaresxrhuang/slides/TranSyncSpotlight_NIPS17.pdf · Translation Synchronization via Truncated Least Squares Xiangru Huang1* Zhenxiao

From a Simple Example

U[µ− σ, µ+ σ]

σ−σ

Outliers

µ

meanmedian

1. Delete Sample tj if |tj −mean|>ǫ

Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 11 / 20

Page 12: Translation Synchronization via Truncated Least Squaresxrhuang/slides/TranSyncSpotlight_NIPS17.pdf · Translation Synchronization via Truncated Least Squares Xiangru Huang1* Zhenxiao

From a Simple Example

U[µ− σ, µ+ σ]

σ−σ

Outliers

µ

meanmedian

1. Delete Sample tj if |tj −mean|>ǫc1

2. Recompute mean and Shrink threshold

Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 12 / 20

Page 13: Translation Synchronization via Truncated Least Squaresxrhuang/slides/TranSyncSpotlight_NIPS17.pdf · Translation Synchronization via Truncated Least Squares Xiangru Huang1* Zhenxiao

From a Simple Example

U[µ− σ, µ+ σ]

σ−σ

Outliers

µ

meanmedian

1. Delete Sample tj if |tj −mean|>ǫc2

2. Recompute mean and Shrink threshold

Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 13 / 20

Page 14: Translation Synchronization via Truncated Least Squaresxrhuang/slides/TranSyncSpotlight_NIPS17.pdf · Translation Synchronization via Truncated Least Squares Xiangru Huang1* Zhenxiao

From a Simple Example

U[µ− σ, µ+ σ]

σ−σ

Outliers

µ

meanmedian

1. Delete Sample tj if |tj −mean|>ǫc3

2. Recompute mean and Shrink threshold

Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 14 / 20

Page 15: Translation Synchronization via Truncated Least Squaresxrhuang/slides/TranSyncSpotlight_NIPS17.pdf · Translation Synchronization via Truncated Least Squares Xiangru Huang1* Zhenxiao

Translation Synchronization

Ground Truth {xi}Relative measurements tij = xi − xj + noise ∀i , j ∈ E

Algorithm: iteratively update x and E .Can be applied to Pairwise Ranking, Joint Alignment of point clouds,and etc.

Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 15 / 20

Page 16: Translation Synchronization via Truncated Least Squaresxrhuang/slides/TranSyncSpotlight_NIPS17.pdf · Translation Synchronization via Truncated Least Squares Xiangru Huang1* Zhenxiao

Translation Synchronization

Ground Truth {xi}Relative measurements tij = xi − xj + noise ∀i , j ∈ EAlgorithm: iteratively update x and E .

Can be applied to Pairwise Ranking, Joint Alignment of point clouds,and etc.

Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 15 / 20

Page 17: Translation Synchronization via Truncated Least Squaresxrhuang/slides/TranSyncSpotlight_NIPS17.pdf · Translation Synchronization via Truncated Least Squares Xiangru Huang1* Zhenxiao

Translation Synchronization

Ground Truth {xi}Relative measurements tij = xi − xj + noise ∀i , j ∈ EAlgorithm: iteratively update x and E .Can be applied to Pairwise Ranking, Joint Alignment of point clouds,and etc.

Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 15 / 20

Page 18: Translation Synchronization via Truncated Least Squaresxrhuang/slides/TranSyncSpotlight_NIPS17.pdf · Translation Synchronization via Truncated Least Squares Xiangru Huang1* Zhenxiao

Exact Recovery

Biased Noise Model (Unbounded Outliers):

tij =

{xgti − xgtj + U[−σ, σ] with probability p

Any real number with probability 1− p(1)

For some constants p, q only depend on graph structure, duringoptimization we have

‖x (k) − xgt‖∞ ≤ qσ + 2pεck−1

and eventually we’ll reach an x̂

‖x̂ − xgt‖∞ ≤2p + cq

c − 4pσ

where the RHS is independent of ε.

Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 16 / 20

Page 19: Translation Synchronization via Truncated Least Squaresxrhuang/slides/TranSyncSpotlight_NIPS17.pdf · Translation Synchronization via Truncated Least Squares Xiangru Huang1* Zhenxiao

Exact Recovery

Biased Noise Model (Unbounded Outliers):

tij =

{xgti − xgtj + U[−σ, σ] with probability p

Any real number with probability 1− p(1)

For some constants p, q only depend on graph structure, duringoptimization we have

‖x (k) − xgt‖∞ ≤ qσ + 2pεck−1

and eventually we’ll reach an x̂

‖x̂ − xgt‖∞ ≤2p + cq

c − 4pσ

where the RHS is independent of ε.

Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 16 / 20

Page 20: Translation Synchronization via Truncated Least Squaresxrhuang/slides/TranSyncSpotlight_NIPS17.pdf · Translation Synchronization via Truncated Least Squares Xiangru Huang1* Zhenxiao

Randomized Case

Biased Noise Model:

tij =

{xgti − xgtj + U[−σ, σ] with probability p

xgti − xgtj + U[−a, b] with probability 1− p(2)

TheoremThere exists a constant c so that if p > c/

√log(n), then w.h.p,

‖x (k) − xgt‖∞ ≤ (1− p/2)k(b − a),

∀ k = 0, · · · , [− log(b + a

2σ)/log(1− p/2)].

Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 17 / 20

Page 21: Translation Synchronization via Truncated Least Squaresxrhuang/slides/TranSyncSpotlight_NIPS17.pdf · Translation Synchronization via Truncated Least Squares Xiangru Huang1* Zhenxiao

Randomized Case

Biased Noise Model:

tij =

{xgti − xgtj + U[−σ, σ] with probability p

xgti − xgtj + U[−a, b] with probability 1− p(2)

TheoremThere exists a constant c so that if p > c/

√log(n), then w.h.p,

‖x (k) − xgt‖∞ ≤ (1− p/2)k(b − a),

∀ k = 0, · · · , [− log(b + a

2σ)/log(1− p/2)].

Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 17 / 20

Page 22: Translation Synchronization via Truncated Least Squaresxrhuang/slides/TranSyncSpotlight_NIPS17.pdf · Translation Synchronization via Truncated Least Squares Xiangru Huang1* Zhenxiao

Experiments on Synthetic Graphs{Dense, Sparse } × { Regular, Irregular }

{Dense, Sparse} × {Regular, Irregular}

(a) Regular (b) Irregular

Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 18 / 20

Page 23: Translation Synchronization via Truncated Least Squaresxrhuang/slides/TranSyncSpotlight_NIPS17.pdf · Translation Synchronization via Truncated Least Squares Xiangru Huang1* Zhenxiao

Experiments on Synthetic Graphs{Dense, Sparse } × { Regular, Irregular }

0.4, 0.01 0.4, 0.04 0.8, 0.01 0.8, 0.04{p, σ

}0.0

0.2

0.4

0.6

0.8

1.0

Norm

alized Error (M

in, Median, Max) Dense Regular

ℓ1 min

TranSync

0.4, 0.01 0.4, 0.04 0.8, 0.01 0.8, 0.04{p, σ

}0.0

0.2

0.4

0.6

0.8

1.0

Norm

alized Error (M

in, Median, Max) Dense Irregular

ℓ1 min

TranSync

0.8, 0.01 0.8, 0.04 1.0, 0.01 1.0, 0.04{p, σ

}0.0

0.2

0.4

0.6

0.8

1.0

Norm

alized Error (M

in, Median, Max) Sparse Regular

ℓ1 min

TranSync

0.8, 0.01 0.8, 0.04 1.0, 0.01 1.0, 0.04{p, σ

}0.0

0.2

0.4

0.6

0.8

1.0

Norm

alized Error (M

in, Median, Max) Sparse Irregular

ℓ1 min

TranSync

Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 19 / 20

Page 24: Translation Synchronization via Truncated Least Squaresxrhuang/slides/TranSyncSpotlight_NIPS17.pdf · Translation Synchronization via Truncated Least Squares Xiangru Huang1* Zhenxiao

Joint Alignment of 6K Lidar Scans

(c) Ground Truth (d) Our Method (e) `1 Minimization

Xiangru Huang*, Zhenxiao Liang* , Chandrajit Bajaj , Qixing Huang Translation Synchronization via Truncated Least Squares 20 / 20