## 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

- 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

- General comments:
- Input scale is N; all estimations deal with N^2 pairs

- beta is set to 5 cm for TEASER++ in all tests

**Differentiable Hungarian Algorithm**

- TRIMs for s — TLS estimation
- Algorithm: adpative voting, polynomial time, exact

- The solution is trivial: enumerate, sort, and vote

- 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%)