Shapes from Echoes: Uniqueness from Point-to-Plane Distance Matrices

TL;DR

This paper characterizes the uniqueness of localizing points and planes from point-to-plane distances, providing a complete mathematical understanding of when configurations are distinguishable based on acoustic echoes.

Contribution

It offers a complete characterization of the conditions under which point and plane configurations are uniquely identifiable from distance measurements, extending previous work on localization ambiguities.

Findings

Enumerates equivalence classes of configurations with identical distance measurements.

Provides algebraic characterization of transformations leading to indistinguishable configurations.

Focuses on theoretical conditions for uniqueness in 2D and 3D.

Abstract

We study the problem of localizing a configuration of points and planes from the collection of point-to-plane distances. This problem models simultaneous localization and mapping from acoustic echoes as well as the notable "structure from sound" approach to microphone localization with unknown sources. In our earlier work we proposed computational methods for localization from point-to-plane distances and noted that such localization suffers from various ambiguities beyond the usual rigid body motions; in this paper we provide a complete characterization of uniqueness. We enumerate equivalence classes of configurations which lead to the same distance measurements as a function of the number of planes and points, and algebraically characterize the related transformations in both 2D and 3D. Here we only discuss uniqueness; computational tools and heuristics for practical localization from point-to-plane distances using sound will be addressed in a companion paper.

Equations

$$d_{nk}=\frac{1}{2}c{\tau }_{nk},$$

$$d_{nk}=\mathrm{dist}({\mathbf{r}}_n,{\mathcal{P}}_k)=⟨{\mathbf{p}}_k-{\mathbf{r}}_n,{\mathbf{n}}_k⟩=q_k-⟨{\mathbf{r}}_n,{\mathbf{n}}_k⟩,$$

$$\mathbf{D}\stackrel{\text{def}}{=}[d_{nk}{]}_{n,k=1}^{N,K}\in {\mathbb{R}}^{N\times K}$$

$$\mathbf{D}=1{\mathbf{q}}^{\top }-{\mathbf{R}}^{\top }\mathbf{N},$$

$$⟨{\mathbf{r}}_n^0,{\mathbf{n}}_k^0⟩=⟨{\mathbf{r}}_n,{\mathbf{n}}_k⟩,\text{for all}1\le n\le N,1\le k\le K,$$

$$-⟨{\mathbf{r}}_n^0,{\mathbf{n}}_k^0⟩+⟨{\mathbf{p}}_k^0,{\mathbf{n}}_k^0⟩=-⟨{\mathbf{r}}_n,{\mathbf{n}}_k⟩+⟨{\mathbf{p}}_k,{\mathbf{n}}_k⟩.$$

$$\begin{array}{rl}⟨{\mathbf{r}}_n^0,{\mathbf{n}}_k^0⟩-⟨{\mathbf{r}}_n,{\mathbf{n}}_k⟩& =⟨{\mathbf{p}}_k^0,{\mathbf{n}}_k^0⟩-⟨{\mathbf{p}}_k,{\mathbf{n}}_k⟩\\ & =q_k^0-q_k,\end{array}$$

$${\mathbf{n}}_k^0=\left[\begin{array}{c}\cos {\phi }_k^0\\ \sin {\phi }_k^0\end{array}\right]\text{mbox}and{\mathbf{n}}_k=\left[\begin{array}{c}\cos {\phi }_k\\ \sin {\phi }_k\end{array}\right]$$

$${\mathbf{n}}_k^0=\left[\begin{array}{c}\sin {\theta }_k^0\cos {\phi }_k^0\\ \sin {\theta }_k^0\sin {\phi }_k^0\\ \cos {\theta }_k^0\end{array}\right]\text{mbox}and{\mathbf{n}}_k=\left[\begin{array}{c}\sin {\theta }_k\cos {\phi }_k\\ \sin {\theta }_k\sin {\phi }_k\\ \cos {\theta }_k\end{array}\right]$$

$${\left[\begin{array}{ccc}{\mathbf{c}}_{r+1}& \dots & {\mathbf{c}}_{2m}\end{array}\right]}^{\top }=\mathbf{T}{\left[\begin{array}{ccc}{\mathbf{c}}_1& \dots & {\mathbf{c}}_r\end{array}\right]}^{\top },$$

$$\left[\begin{array}{c}\cos {\phi }_k\\ \sin {\phi }_k\end{array}\right]=\mathbf{T}\left[\begin{array}{c}\cos {\phi }_k^0\\ \sin {\phi }_k^0\end{array}\right],\text{mbox}where\mathbf{T}=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right].$$

$${\mathbf{N}}^{\top }={\mathbf{T}}^{\prime }{{\mathbf{N}}^0}^{\top },$$

$$\mathbf{T}=\left[\begin{array}{cc}a& b\\ 0& d\end{array}\right].$$

$$(a\cos {\phi }_k^0+b\sin {\phi }_k^0{)}^2+(d\sin {\phi }_k^0{)}^2=\cos {\phi }_k^2+\sin {\phi }_k^2=1,$$

$$A{\cos }^2(2{\phi }_k^0)+B\cos (2{\phi }_k^0)+C=0,$$

$$A$$

$$B$$

$$C$$

$$\mathbf{T}=\left[\begin{array}{cc}\pm 1& 0\\ 0& \pm 1\end{array}\right]\text{mbox}or\mathbf{T}=\left[\begin{array}{cc}\pm 1& 0\\ 0& \mp 1\end{array}\right],$$

$$\cos (2{\phi }_k^0)=\frac{-B\pm \sqrt{B^2-4AC}}{2A}.$$

$${\phi }_k=f({\phi }_k^0,s_k,a,b,d)=\mathrm{atan}\frac{d\sin {\phi }_k^0}{a\cos {\phi }_k^0+b\sin {\phi }_k^0}+s_k\pi ,$$

$$\displaystyle\Big{[}\{\mathcal{P}^{0}_{k}\}_{k=1}^{K}\Big{]}=\Big{\{}\{\mathcal{P}_{k}\}_{k=1}^{K}\,\big{|}\,\varphi_{k}=f(\varphi_{k}^{0},s_{k},a,b,c),$$

$$a,b\in \mathbb{R}\text{mbox}s.t.\text{eqref}eq:2d_{r}ank{2}_{A}BC\text{mbox}satisfied,d\in \mathbb{R},s_k\in \{0,1\},$$

$$q_k=q_k^0\text{ for }1\le k\le K$$

$${\mathbf{v}}_1={\left[\begin{array}{cccc}-a,& -b,& 1,& 0\end{array}\right]}^{\top },$$

$${\mathbf{v}}_2={\left[\begin{array}{cccc}0,& -d,& 0,& 1\end{array}\right]}^{\top },$$

$$\left[\begin{array}{c}{\mathbf{r}}_n^0\\ -{\mathbf{r}}_n\end{array}\right]={\mathbf{v}}_1{\gamma }_1+{\mathbf{v}}_2{\gamma }_2,$$

$${\mathbf{r}}_n^0={\mathbf{T}}^{\top }{\mathbf{r}}_n.$$

$$\left[\begin{array}{c}\sin {\phi }_k^0\\ \cos {\phi }_k\\ \sin {\phi }_k\end{array}\right]=\mathbf{T}\cos {\phi }_k^0,\text{mbox}where\mathbf{T}=\left[\begin{array}{c}a\\ b\\ c\end{array}\right].$$

$$\left[\begin{array}{c}\cos {\phi }_k\\ \sin {\phi }_k\end{array}\right]={\mathbf{T}}^{\prime }\left[\begin{array}{c}\cos {\phi }_k^0\\ \sin {\phi }_k^0\end{array}\right],\text{mbox}where{\mathbf{T}}^{\prime }=\left[\begin{array}{cc}b& 0\\ c& 0\end{array}\right].$$

Full text

I Introduction

Localization methods are traditionally based on geometric information (angles, distances, or both) about known objects, often referred to as landmarks or anchors. Famous examples include global positioning by measuring distances to satellites and navigation at sea by measuring angles of celestial bodies. More recent work on simultaneous localization and mapping (SLAM) addresses the case where the positions of landmarks are also unknown.

In this paper, we address localization from distances to (unknown) planes instead of the more extensively studied localization from distances to points. Concretely, given pairwise distances between a set of points and a set of planes, we wish to localize both the planes and the points. It is clear that a single point does not allow unique localization. As we will show, localization is in general possible with multiple points, though there are surprising exceptions.

Localization from point-to-plane distances models many practical problems. Our motivation comes from indoor localization with sound. Imagine a mobile device equipped with a single omnidirectional source and a single omnidirectional receiver that measures its distance to the surrounding reflectors, for example by emitting acoustic pulses and receiving echoes. The times of flight of the first-order echoes recorded by the device correspond to point-to-plane distances. They could be used to pinpoint its location given the positions of the walls, but the problem is harder and more interesting when we do not know where the walls are. A similar principle is used by bats to echolocate, although we do not assume having any directional information. Another problem that can be cast in this mold is the well-known “structure from sound” [1], where the task is to localize a set of microphones from phase differences induced by a set of unknown far field sources.

Prior work on localization from point-to-plane distances has so far been mostly computational [2, 3]. Although several papers point out problems with uniqueness [4, 5], a complete study was up to now absent. The most notable result is presented in [6], which shows that one can reconstruct a room from the first-order echoes from one omnidirectional loudspeaker to four non-planar microphones, placed together on a drone with generic position and orientation.

In this work, we focus on uniqueness of reconstruction from point-to-plane distance matrices (PPDMs). Unlike in the case of localization from points, where with sufficiently many points the only possible ambiguity is that of translation, rotation, and reflection [7], our analysis shows that localization from PPDMs suffers from additional ambiguities that correspond to certain continuous deformations of the points–planes system.

I-A Related work

The PPDM problem is related to the more standard multidimensional unfolding [8]: localization of a set of points from distances to a set of point landmarks. There are several variations of this problem that correspond to different assumptions of what is known: 1) given distances to known landmarks, localize unknown points (i.e., estimate the unknown trajectory), 2) given distances to known points, reconstruct unknown landmarks (i.e., map the unknown environment), 3) estimate both unknown landmarks and unknown points from their pairwise distances.

The first scenario is solved by simple multilateration [9]. The second scenario is a topic of active research in signal processing and room acoustics, where it is known as “hearing the shape of a room” [10, 11, 12]. Much of that work assumes that the geometry of the microphone array is known. If that is the case, since the source is fixed, the landmarks are modeled by points that correspond to virtual sources.

When neither the landmarks nor the points are known, we get an instance of SLAM. In general SLAM, the task is to simultaneously build some representation of the map of the environment and estimate the trajectory. Different flavors of SLAM involve different sensing modalities; prior work has considered visual [13, 14, 15, 16], range-only [17, 18, 19], and acoustic SLAM [20, 21, 22, 23], as well as solutions based on multiple sensor modalities [24, 25, 26]. Localization from PPDMs corresponds to range-only SLAM, though conventional approaches to SLAM rely on some noisy estimate of the trajectory, which is more information than we assume in scenario (3) above.

Methods for SLAM from reflections of sound or radio waves [10, 27, 28, 29, 30] usually assume a fixed source or a fixed receiver, so that the echoes correspond to virtual beacons that provide range measurements. This information in turn allows to localize both sets using tools such as multidimensional unfolding [8]. More recent works [31, 32, 27, 33] show how to exploit multipath reflections. An appeal of our collocated setup is that it does not require any preinstalled infrastructure [34].

I-B Our contributions

We have previously shown that range-only SLAM can be addressed effectively using Euclidean distance matrices (EDM) [35]. Here we show how our new problem can be similarly cast as localization from PPDMs. This completes and extends our work on the 2D case [36]. Unlike in standard SLAM, we do not assume any motion model; the waypoints can be scattered arbitrarily.

We study uniqueness of reconstruction of point–plane configurations from their pairwise distances. We derive conditions under which the localization is unique, and provide a complete characterization of non-uniqueness by enumerating the equivalence classes of configurations that lead to same PPDMs. Since we are motivated by SLAM, we refer to point–plane configurations as rooms and trajectories. The conclusions, however, are general, and can be applied to any of the discussed applications.

Finally, while PPDMs provide a good basic model for SLAM from echoes with a collocated source and receiver, the full SLAM problem presents a number of additional challenges. Problems of associating echoes to walls, dealing with missing echoes, and telling first-order from higher-order echoes will be addressed in a companion paper in preparation. Here we assume having a full PPDM as defined in Section II.

II Problem setup

Suppose that a mobile device carrying an omnidirectional source and an omnidirectional receiver traverses a trajectory described by $N$ waypoints $\{{\mathbf{r}_{n}}\}_{n=1}^{N}$. At every waypoint, the source produces a pulse, and the receiver registers the echoes. Since the source and receiver are collocated, the distance $d_{nk}$ between the $n$th waypoint and $k$th wall is given by

[TABLE]

where $c$ is the speed of sound and $\tau_{nk}$ is the propagation time of the first-order echo. We can thus find the distances between waypoints and walls by measuring the times of arrival of first-order echoes.

To describe a room, we consider $K$ walls $\{{{\cal P}_{k}}\}_{k=1}^{K}$ (lines in $2$D and planes in $3$D) defined by their unit normals $\mathbf{n}_{k}\in\mathbb{R}^{m}$ and any point $\mathbf{p}_{k}\in\mathbb{R}^{m}$ on the wall, where $m\in\left\{2,3\right\}$. For any $\mathbf{x}\in\mathcal{P}_{k}$ we have $\left\langle\mathbf{n}_{k},\mathbf{x}\right\rangle=q_{k}$, where $q_{k}=\left\langle\mathbf{n}_{k},\mathbf{p}_{k}\right\rangle$ is the distance of the wall from the origin.

Given the distances between walls and waypoints,

[TABLE]

for $n=1,...,N$ and $k=1,...,K$, we define

[TABLE]

to be the point-to-plane distance matrix (PPDM); we always assume $N\geq K$.

By setting $\mathbf{q}\overset{\text{def}}{=}[q_{1},\ldots,q_{K}]^{\top}$, $\mathbf{R}\overset{\text{def}}{=}[\mathbf{r}_{1},\ldots,\mathbf{r}_{N}]$, and $\mathbf{N}\overset{\text{def}}{=}[\mathbf{n}_{1},\ldots,\mathbf{n}_{K}]$, we can express a PPDM as

[TABLE]

where $\mathbf{q}$ is the vector of distances between the planes and the origin, columns of $\mathbf{R}\in\mathbb{R}^{m\times N}$ are the waypoint coordinates, and columns of $\mathbf{N}\in\mathbb{R}^{m\times K}$ outward looking normal vectors of the planes. Letting $\mathbf{P}\overset{\text{def}}{=}\begin{bmatrix}\mathbf{p}_{1}&\ldots&\mathbf{p}_{K}\end{bmatrix}$, the vector $\mathbf{q}$ can be written as $\mathbf{q}=\mathop{\mathrm{diag}}(\mathbf{P}^{\top}\mathbf{N})$, where $\mathop{\mathrm{diag}}(\mathbf{M})$ denotes the vector formed from the diagonal of $\mathbf{M}$.

A pair of planes and waypoints defines a room–trajectory configuration ${\cal R}=\left(\left\{{\cal P}_{k}\right\}_{k=1}^{K},\left\{\mathbf{r}_{n}\right\}_{n=1}^{N}\right)$, and the corresponding PPDM $\mathbf{D}({\cal R})$. In realistic convex configurations, all entries of the PPDM (4) are non-negative. However, in our relaxed definition of a room, the waypoints can lie on either side of a wall, so we allow for signed distances.

Our central question is whether a given PPDM $\mathbf{D}({\cal R})$ specifies a unique room-trajectory configuration ${\cal R}$, or, equivalently, whether the map $\mathcal{R}\mapsto\mathbf{D}(\mathcal{R})$ is injective. It is clear that rotated, translated, and reflected versions of $\mathcal{R}$ all give the same $\mathbf{D}$, so we consider them to be the same configuration (we consider the equivalence class of all room–trajectory configurations modulo rigid motions and reflections).

We formalize the uniqueness question as follows:

Problem 1**.**

Are there distinct room–trajectory configurations ${\cal R}^{1}=\left(\left\{{\cal P}^{1}_{k}\right\}_{k=1}^{K},\left\{\mathbf{r}_{n}^{1}\right\}_{n=1}^{N}\right)$ and ${\cal R}^{2}=\left(\left\{{\cal P}^{2}_{k}\right\}_{k=1}^{K},\left\{\mathbf{r}^{2}_{n}\right\}_{n=1}^{N}\right)$ which are not rotated, translated, and reflected versions of each other, such that $\mathbf{D}({\cal R}^{1})=\mathbf{D}({\cal R}^{2})$?

III Uniqueness of the reconstruction

Perhaps surprisingly, there are many examples of rooms from Problem 1. The main tool in identifying the sought equivalence classes is the following lemma.