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# TEASER++

## Remark

• Registration as a Simultaneous Pose and Correspondence (SPC) problem, in contrast to correspondence-based problems

## Quaternion

• Quaternion multiplication can be represented in matrix-vector form, just as cross product
• Quaternion inversion is linear
• Rotating a vector is non-linear

## Problem formulation

• Upper bound can be understood as a 3-sigma bound for inlier. No assumption for outliers.

## Decoupling

### Translation Invariant Measurements (TIMs)

• Insight: relative positions are translation-invariant
• Number of TIMs: N(N-1)/2
• Equivalent with graph representation
• Each vertex is a correspondence
• Each edge is a pair of correspondences

### Translation and Rotation Invariant Measurements (TRIMs)

• Insight: norm of relative positions are rotation-invariant
• By dividing ||a_{ij}|| from both sides, we get an expression only depending on s

## TEASER

• Input scale is N; all estimations deal with N^2 pairs
• beta is set to 5 cm for TEASER++ in all tests

• Heat methods
• Pairwise features? CRF?
1. TRIMs for s — TLS estimation
• Algorithm: adpative voting, polynomial time, exact
• The solution is trivial: enumerate, sort, and vote
1. s + TIMs for R
• Algorithm: Robust rotation search via tight semidefinite relaxation, polynomial time, exact

is equivalent with

can be rewritten with

with

Now q_k = q for inliers and -q for outliers. It is equivalent to Quadratically-Constrained Quadratic Program:

Q depends on TIMs. Reformat Z with x,

this turns to

3. d + R for t

• Compute t's 3 dimensions independently

### Boost performance

• Theorem: Edges corresponding to inlier TIMs form a clique in E0, and there is at least one maximal clique in E0 that contains all the inliers.
• Sparse max-clique can be fast in practice
• An edge (i, j) (and the corresponding TIM) is an inlier if both i and j are correct correspondences
• In practice:
• w max-clique: slow, high performance (90%)
• w/o max-clique: fast, low performance (50%)