Provably Approximated ICP

• URL: http://arxiv.org/abs/2101.03588v1
• Date: Sun, 10 Jan 2021 18:09:29 GMT
• Title: Provably Approximated ICP
• Authors: Ibrahim Jubran, Alaa Maalouf, Ron Kimmel, Dan Feldman
• Abstract summary: We prove that there emphalways exists a "witness" set of $3$ pairs in $P times Q$ that, via novel alignment algorithm, defines a constant factor approximation to this global optimum. Our approximation constants are, in practice, close to $1$, and up to x$10$ times smaller than state-of-the-art algorithms.
• Score: 40.349822671753024
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The goal of the \emph{alignment problem} is to align a (given) point cloud $P = \{p_1,\cdots,p_n\}$ to another (observed) point cloud $Q = \{q_1,\cdots,q_n\}$. That is, to compute a rotation matrix $R \in \mathbb{R}^{3 \times 3}$ and a translation vector $t \in \mathbb{R}^{3}$ that minimize the sum of paired distances $\sum_{i=1}^n D(Rp_i-t,q_i)$ for some distance function $D$. A harder version is the \emph{registration problem}, where the correspondence is unknown, and the minimum is also over all possible correspondence functions from $P$ to $Q$. Heuristics such as the Iterative Closest Point (ICP) algorithm and its variants were suggested for these problems, but none yield a provable non-trivial approximation for the global optimum. We prove that there \emph{always} exists a "witness" set of $3$ pairs in $P \times Q$ that, via novel alignment algorithm, defines a constant factor approximation (in the worst case) to this global optimum. We then provide algorithms that recover this witness set and yield the first provable constant factor approximation for the: (i) alignment problem in $O(n)$ expected time, and (ii) registration problem in polynomial time. Such small witness sets exist for many variants including points in $d$-dimensional space, outlier-resistant cost functions, and different correspondence types. Extensive experimental results on real and synthetic datasets show that our approximation constants are, in practice, close to $1$, and up to x$10$ times smaller than state-of-the-art algorithms.

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