Distance-Based Localization Techniques

• Distance-based localization is the use of distance measurements (direct pairwise, range differences, TDoA, etc.) to estimate positions and poses from partial, noisy data.
• Key methodologies include SDP relaxations, convex optimization, and iterative refinements, with advanced metrics like Earth Mover’s Distance enhancing accuracy.
• Applications span wireless sensor networks, UAV coordination, acoustic source detection, and scan-to-map registration, stressing the importance of metric design to resolve ambiguities.

Searching arXiv for recent and foundational papers on distance-based localization to ground the article. Distance-based localization comprises methods that estimate position, pose, or map-consistent configuration from quantities defined by distance. In the literature, those quantities include direct pairwise ranges, range differences, hop-derived expected distances, distances between signal likelihoods, and continuous distance functions defined on geometric maps. The resulting inverse problems appear in wireless sensor networks, UAV cooperation, acoustic source localization, WiFi and CSI fingerprinting, panoramic visual localization, and scan-to-map registration. Across these settings, the central task is to recover spatial state from partial, noisy, or indirect distance information while accounting for rigid-motion ambiguities, model mismatch, and combinatorial nonuniqueness (Javanmard et al., 2011, Jiang et al., 2017, Schäfermeier et al., 2018, Kim et al., 2023, Millane et al., 2020).

1. Problem formulations and identifiability

A foundational formulation considers $n$ points in $\mathbb{R}^d$ with only a subset $E$ of noisy pairwise squared distances observed: $$\tilde{d}_{ij}^2 = d_{ij}^2 + z_{ij}, \qquad |z_{ij}| \le \Delta, \qquad (i,j)\in E .$$ The semidefinite program proposed for this setting estimates a Gram matrix $Q$ by minimizing $\mathrm{tr}(Q)$ subject to distance-consistency constraints and $Q \succeq 0$, then extracts a rank-$d$ embedding by spectral truncation. For a random geometric graph model, exact recovery up to rigid transformations is obtained in the noiseless case beyond a radius threshold $r \gtrsim (\log n/n)^{1/d}$, while in the noisy case upper and lower bounds on reconstruction error are derived (Javanmard et al., 2011).

In cooperative 3D mobile localization, the unknown is often not the point set itself but the rigid transformation between coordinate frames. For one GPS-equipped UAV and one GPS-denied UAV, the model is

$d[k] = \|R p_1[k] + T - p_0[k]\| ,$

with unknown $\mathbb{R}^d$0 and $\mathbb{R}^d$1. The reported result is that at least $\mathbb{R}^d$2 distance measurements, for generic trajectories, are required to guarantee a unique solution almost always. The proposed solver combines an SDP relaxation with maximum-likelihood refinement by gradient descent on $\mathbb{R}^d$3 (Jiang et al., 2017).

A related reformulation appears in 3D single-source localization with microphones. Instead of optimizing directly over the source coordinates, the Euclidean distance matrix formulation reduces the problem to a single-variable optimization over the source-reference distance $\mathbb{R}^d$4: $\mathbb{R}^d$5 Here the correct $\mathbb{R}^d$6 is identified by the rank-$\mathbb{R}^d$7 structure of the Gram matrix induced by the Euclidean distance matrix. This converts a three-variable search into a one-variable search, and the source position is then recovered from the reconstructed EDM (Brümann et al., 2022).

These formulations show that “distance-based” does not prescribe a single estimator class. It can induce convex relaxation, rigid-alignment, or rank-minimization problems depending on whether the unknowns are coordinates, transformations, or latent distance parameters. This suggests that identifiability in distance-based localization is governed less by sensor modality than by the algebraic structure of the distance constraints.

2. Wireless sensor networks: hop geometry, barycentric coordinates, and RSSI trilateration

In wireless sensor networks, a major line of work uses only distance or range-related information without requiring every node to be directly ranged to anchors. A distributed formulation expresses a node $\mathbb{R}^d$8 in barycentric coordinates with respect to neighbors $\mathbb{R}^d$9: $E$0 For general network configurations without the convex hull assumption, the difficulty is determining the sign pattern of the barycentric coordinates using only pairwise distances. The proposed solution computes unsigned barycentric coordinates from Cayley-Menger determinants, resolves sign ambiguity from distance-based criteria, and guarantees convergence by diagonal preconditioning: $E$1 This removes the convex-hull restriction of DILOC while preserving distributed iterative localization (Diao et al., 2013).

For 3D DV-Hop localization, the probability-based maximum distance estimation (PMDE) and probability-based average distance estimation (PADE) models derive hop-dependent upper bounds and expected distances from the probability distribution of node distances detected by an anchor node. The PMDE model computes

$E$2

and PADE then estimates the expected distance

$E$3

The resulting estimate is embedded, together with the traditional distance loss, into NSGA-II. Reported gains are $E$4 in random 3D WSNs and $E$5 in multimodal 3D WSNs (Wang et al., 2024).

A complementary 2D DV-Hop variant uses distance estimation using multinode (DEMN) and a hop loss. DEMN replaces the classic

$E$6

with an expected distance $E$7 computed from cross-domain information when multiple anchor nodes detect the same unknown node. The hop loss

$E$8

is then jointly optimized with Euclidean distance loss in NSGA-II. The reported average localization accuracy is $E$9 in the randomly distributed network, $$\tilde{d}_{ij}^2 = d_{ij}^2 + z_{ij}, \qquad |z_{ij}| \le \Delta, \qquad (i,j)\in E .$$0 better than DEM-DV-Hop, with DEMN and hop loss contributing $$\tilde{d}_{ij}^2 = d_{ij}^2 + z_{ij}, \qquad |z_{ij}| \le \Delta, \qquad (i,j)\in E .$$1 and $$\tilde{d}_{ij}^2 = d_{ij}^2 + z_{ij}, \qquad |z_{ij}| \le \Delta, \qquad (i,j)\in E .$$2, respectively (Wang et al., 2024).

RSSI-based trilateration remains a distinct branch within WSN localization. The N-times trilateral centroid weighted localization algorithm uses a weighted average of many RSSIs as the current RSSI, selects reliable beacon nodes, computes distances from an empirical formula,

$$\tilde{d}_{ij}^2 = d_{ij}^2 + z_{ij}, \qquad |z_{ij}| \le \Delta, \qquad (i,j)\in E .$$3

and performs $$\tilde{d}_{ij}^2 = d_{ij}^2 + z_{ij}, \qquad |z_{ij}| \le \Delta, \qquad (i,j)\in E .$$4 trilateral centroid localizations over combinations of three beacons. A weighted average of filtered reference coordinates is then used as the final estimate. The reported conclusion is that the method performs better than the trilateral centroid algorithm on STM32W108 IEEE 802.15.4 hardware (Qiu et al., 2013).

Taken together, these works show three recurrent WSN strategies: algebraic localization with signed barycentric coordinates, hop-aware expectation models that replace crude per-hop scaling, and repeated RSSI trilateration with weighting and filtering. Their common purpose is to compensate for the fact that raw connectivity, raw hop counts, or raw RSSI values are not themselves Euclidean distances.

3. Range-only localization, navigation, and TDoA

Distance-only sensing is especially prominent in mobile robotics and acoustic localization. In integrated distance-based docking of UAVs, a single landmark at an arbitrarily unknown position is used to drive simultaneous relative localization and navigation for discrete-time integrators under bounded velocity. The reported architecture couples a nonlinear adaptive estimation scheme with a control scheme, and convergence is proved via the discrete-time LaSalle’s invariance principle. Validation is reported on quadcopters equipped with ultra-wideband ranging sensors and optical flow sensors in a GPS-less environment (Nguyen et al., 2018).

For microphone arrays, the EDM-based 3D source localization method uses known microphone geometry and estimated TDOAs to express all source-microphone distances as functions of the single unknown $$\tilde{d}_{ij}^2 = d_{ij}^2 + z_{ij}, \qquad |z_{ij}| \le \Delta, \qquad (i,j)\in E .$$5, the distance from source to reference microphone. An extension considers the top $$\tilde{d}_{ij}^2 = d_{ij}^2 + z_{ij}, \qquad |z_{ij}| \le \Delta, \qquad (i,j)\in E .$$6 local maxima of the GCC-PHAT function for each microphone pair and jointly optimizes over both $$\tilde{d}_{ij}^2 = d_{ij}^2 + z_{ij}, \qquad |z_{ij}| \le \Delta, \qquad (i,j)\in E .$$7 and the candidate TDOA indices. In a $$\tilde{d}_{ij}^2 = d_{ij}^2 + z_{ij}, \qquad |z_{ij}| \le \Delta, \qquad (i,j)\in E .$$8 m room with 6 microphones, strong reverberation, and babble noise at $$\tilde{d}_{ij}^2 = d_{ij}^2 + z_{ij}, \qquad |z_{ij}| \le \Delta, \qquad (i,j)\in E .$$9 dB SNR, the reported median localization errors are $Q$0 cm, $Q$1 cm, $Q$2 cm, and $Q$3 cm for source distances of $Q$4, $Q$5, $Q$6, and $Q$7 m when using $Q$8, versus $Q$9 cm, $\mathrm{tr}(Q)$0 cm, $\mathrm{tr}(Q)$1 cm, and $\mathrm{tr}(Q)$2 cm for SRP-PHAT (Brümann et al., 2022).

Quantum-ranging formulations reinterpret TDoA itself as a directly measured linear combination of distances. The quantum TDoA protocol measures

$\mathrm{tr}(Q)$3

in a single shot by phase estimation on entangled quantum states. For TDoA, this specializes to $\mathrm{tr}(Q)$4. The localization problem is then relaxed into SDP/SOCP form, and a CRLB is derived for the quantum measurement model. The reported numerical result is over $\mathrm{tr}(Q)$5 lower localization error than classical TDoA on average, with comparable solver time in the reported setup (He et al., 2024).

A recurring misconception is that TDoA is outside distance-based localization because it does not measure absolute range. The literature summarized here treats TDoA as distance-based precisely because it constrains linear combinations of Euclidean distances. A plausible implication is that the essential distinction is not between “range” and “difference of ranges,” but between direct coordinate sensing and distance-structured inference.

4. Distance as statistical discrepancy, transport cost, and learned geometry

In radio localization and mapping, distance often refers not to physical range but to a discrepancy between measurement distributions. For WiFi indoor mapping, a location is represented as an RSSI likelihood,

$\mathrm{tr}(Q)$6

estimated by PMF, normal distribution, or kernel density estimation. Among the compared measures—KL divergence, Jensen-Shannon divergence, Bhattacharyya distance, Hellinger distance, Kolmogorov-Smirnov distance, Earth Mover’s Distance, and absolute difference of means—the Earth Mover’s Distance is reported as the most beneficial for localization. For one-dimensional distributions,

$\mathrm{tr}(Q)$7

and the reported best practice is KDE with AP invisibility modeling and EMD, which preserved room topology in a real-world office scenario (Schäfermeier et al., 2018).

Distance preservation also appears in representation learning. The distance invariant sparse autoencoder compresses 91-dimensional RSS vectors to a 10-dimensional latent space using reconstruction loss, $\mathrm{tr}(Q)$8 sparsity regularization, and a distance invariance term

$\mathrm{tr}(Q)$9

The reported compression ratio is over $Q \succeq 0$0, the average RMSE over all access points is $Q \succeq 0$1 dBm, and the average KL-divergence of localization likelihoods is $Q \succeq 0$2 in the distance-invariant 10D latent space versus $Q \succeq 0$3 in the full 91D input space, outperforming sparse AE and PCA baselines (Miyagusuku et al., 2020).

Channel charting extends this idea to unlabeled CSI geometry. A time-distance-based metric is built from aligned CIR amplitudes,

$Q \succeq 0$4

and global geometry is recovered through geodesic distances on a neighborhood graph. A Siamese neural network is trained so that Euclidean distances in the chart match the geodesic distances. After a linear transformation from chart coordinates to physical coordinates, the reported localization accuracies are $Q \succeq 0$5 m for UWB and $Q \succeq 0$6 m for the 5G setup (Stahlke et al., 2022).

In supervised multi-view crowd localization, the transport cost itself is made distance-aware. The Mahalanobis distance-based multi-view optimal transport loss replaces Euclidean transport cost with

$Q \succeq 0$7

where the covariance matrix encodes the view ray direction and the object-to-camera distance. The reported finding is that the resulting M-MVOT loss outperforms density map-based and Euclidean OT baselines on several multi-view crowd localization datasets (Zhang et al., 2024).

These works enlarge the meaning of distance-based localization from geometric ranging to distributional geometry. This suggests that, in many sensing systems, localization quality depends less on recovering raw Euclidean distances than on constructing a distance surrogate whose topology correlates strongly with physical space.

5. Geometric distance functions, implicit maps, and direct registration

A separate family of methods defines localization directly against distance functions on maps. For panoramic localization, LDL introduces 2D and 3D line distance functions on the sphere: $Q \succeq 0$8 Candidate poses are scored by a robust inlier-count loss over sampled spherical directions, optionally decomposed by the three principal line directions. The reported result is robust localization under illumination shifts, object layout changes, and large-scale scenes, with pose search terminating within a matter of milliseconds (Kim et al., 2023).

Signed distance functions provide a volumetric alternative to pointcloud descriptors. In Freetures, keypoints are detected from the determinant of the Hessian of the SDF,

$Q \succeq 0$9

local reference frames are assigned from the structure tensor, and descriptors are augmented with mean SDF value and curvature class. The method exploits both surfaces and free space rather than only the zero level set. Reported average improvements are approximately $d$0 on an RGB-D dataset and approximately $d$1 on a LiDAR-based dataset relative to handcrafted surfaces-only descriptors (Millane et al., 2020).

Continuous Euclidean distance fields have also been used for direct 6-DoF localization. G-EDF-Loc models the Euclidean Distance Field with a Block-Sparse Gaussian Mixture Model,

$d$2

with analytic gradients and $d$3 continuity across blocks via Smoothstep blending. Scan-to-map registration minimizes robustified squared field values of transformed scan points. The reported system is CPU-based, real-time, and resilient under severe odometry degradation or in the complete absence of IMU priors (Maese et al., 6 Apr 2026).

Depth measurements in polygonal workspaces produce yet another geometric inverse problem. For a workspace $d$4, pose $d$5, and depth function $d$6, the task is to compute the full preimage

$d$7

The reported output-sensitive data structure answers single-measurement queries in $d$8 time, where $d$9 is the number of vertices and maximal arcs of low degree algebraic curves in the answer, and yields analogous results for two antipodal measurements with $r \gtrsim (\log n/n)^{1/d}$0 query time for the simpler structures (Ugav et al., 2022).

Here the operative “distance” is a map-defined field rather than an inter-object measurement. A plausible implication is that distance-based localization can be viewed as direct optimization or inversion in a metricized environment representation, not only as triangulation from anchors.

6. Ambiguity, error models, and robustness mechanisms

Distance-based localization is especially sensitive to ambiguity generated by symmetry, sparsity, and bounded error. In micro-UAV localization, AFALA addresses flip ambiguities under bounded distance measurement errors and constrained flying motions by introducing a bi-boundary communication model derived from the unit disk graph. For bilateration and trilateration, unique localization criteria are stated in terms of whether candidate regions are consistent with communication or distance constraints. The reported result is excellent localization performance in terms of average estimated error and strong suppression of flip ambiguities in simulation (Guo et al., 2018).

In graph-based localization, ambiguity also arises from incomplete distance measurements and rigid-motion invariance. The SDP analysis of incomplete noisy distances makes this explicit through centered Gram-matrix error bounds and exact recovery only up to rigid transformations (Javanmard et al., 2011). In DV-Hop localization, ambiguity appears as multiple solutions produced by Euclidean distance loss alone; the hop loss in DEMN is specifically introduced to select a suitable solution from multiple solutions obtained by the Euclidean distance loss (Wang et al., 2024).

Noise models vary substantially across modalities. The PADE framework models hop-dependent detection distance probabilistically and reports competitive confidence interval widths in addition to improved average localization accuracy (Wang et al., 2024). The EDM-based acoustic method explicitly addresses reverberation by selecting among multiple TDOA candidates, which approximately halves the number of large $r \gtrsim (\log n/n)^{1/d}$1 cm localization errors compared with using a single candidate (Brümann et al., 2022). Quantum TDoA reduces classical error accumulation by measuring a distance difference directly rather than subtracting two noisy ranges, and the reported average gain exceeds $r \gtrsim (\log n/n)^{1/d}$2 (He et al., 2024). In direct scan-to-map localization, robustness is tied to the regularity of the distance field itself: G-EDF-Loc emphasizes analytic gradients, $r \gtrsim (\log n/n)^{1/d}$3 continuity, and resilience in the absence of IMU priors (Maese et al., 6 Apr 2026).

The broader pattern is that robustness is rarely achieved by a single estimator. It is usually obtained by redesigning the distance object itself: signed barycentric coordinates rather than unsigned areas, expected hop distances rather than mean hop length, direct TDoA phase estimation rather than subtraction of two ranges, or continuous distance fields rather than discontinuous occupancy maps. This suggests that progress in distance-based localization often depends as much on metric design as on optimization.