Translation Synchronization via Truncated Least Squares Xiangru - - PowerPoint PPT Presentation

Translation Synchronization via Truncated Least Squares

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

1Department of Computer Science

University of Texas at Austin

2Tsinghua University

NIPS, 2017

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

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

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

From a Simple Example

U[µ − σ, µ + σ] σ −σ

µ

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From a Simple Example

U[µ − σ, µ + σ] σ −σ

µ mean

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From a Simple Example

U[µ − σ, µ + σ] σ −σ Outliers

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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

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

From a Simple Example

U[µ − σ, µ + σ] σ −σ Outliers

µ mean median

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

From a Simple Example

U[µ − σ, µ + σ] σ −σ Outliers

µ mean median

• 1. Delete Sample tj if |tj − mean| > ǫ

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

From a Simple Example

U[µ − σ, µ + σ] σ −σ Outliers

µ mean median

• 1. Delete Sample tj if |tj − mean| > ǫ

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

From a Simple Example

U[µ − σ, µ + σ] σ −σ Outliers

µ mean median

• 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

From a Simple Example

U[µ − σ, µ + σ] σ −σ Outliers

µ mean median

• 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

From a Simple Example

U[µ − σ, µ + σ] σ −σ Outliers

µ mean median

• 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

Translation Synchronization

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

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Translation Synchronization

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

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

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

Exact Recovery

Biased Noise Model (Unbounded Outliers): tij = xgt

i

− xgt

j

+ U[−σ, σ] with probability p Any real number with probability 1 − p (1)

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

Exact Recovery

Biased Noise Model (Unbounded Outliers): tij = xgt

i

− xgt

j

+ U[−σ, σ] with probability p Any real number with probability 1 − p (1) For some constants p, q only depend on graph structure, during

• ptimization we have

x(k) − xgt∞ ≤ qσ + 2pǫck−1 and eventually we’ll reach an ˆ x ˆ x − xgt∞ ≤ 2p + cq c − 4p σ where the RHS is independent of ǫ.

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

Randomized Case

Biased Noise Model: tij =

• xgt

i

− xgt

j

+ U[−σ, σ] with probability p xgt

i

− xgt

j

+ U[−a, b] with probability 1 − p (2)

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

Randomized Case

Biased Noise Model: tij =

• xgt

i

− xgt

j

+ U[−σ, σ] with probability p xgt

i

− xgt

j

+ U[−a, b] with probability 1 − p (2) Theorem There 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

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

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

Normalized Error (Min, 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

Normalized Error (Min, 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

Normalized Error (Min, 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

Normalized Error (Min, Median, Max)

Sparse Irregular

ℓ1 min TranSync

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Joint Alignment of 6K Lidar Scans

(c) Ground Truth (d) Our Method (e) ℓ1 Minimization

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