Mobile crowdsourcing enables large-scale construction and updating of fingerprint databases for indoor positioning (e.g., Wi-Fi, geomagnetic). Accurate trajectory recovery from such data, however, remains challenging due to cumulative dead-reckoning (DR) errors, susceptibility to local optima, and rapid drift in vehicular scenarios. This paper presents a graph optimization framework that refines crowdsourced trajectories by integrating three types of constraints: (1) adjacent-pose constraints from a smartphone GNSS/MEMS DR filter enhanced with an empirical velocity model to mitigate indoor drift; (2) relative loop closures detected via geomagnetic sequence matching, exploiting the spatial stability of indoor geomagnetic fields; and (3) absolute loop closures obtained from OpenStreetMap (OSM) map matching to enforce global consistency. A joint graph optimization using these constraints produces globally refined trajectories. Experiments in two underground parking garages with multiple smartphones show that, compared with single-constraint approaches, the proposed method achieves clear improvements in trajectory accuracy.
Mobile crowdsourcing is a key technology for dynamic mapping and updating in fingerprint-based
indoor positioning systems, enabling large-scale applications such as Wi-Fi and geomagnetic matching.
The core challenge is obtaining fine-grained and accurate fingerprint location annotations, which
directly determine the positioning accuracy of the fingerprint database [
Previous research has explored various approaches. Wi-Fi-based crowdsourced mapping [
Analysis suggests that exploiting the spatial stability of indoor geomagnetic fields to construct loop closure points, in combination with optimizing crowdsourced DR trajectories, can efectively enhance the accuracy of indoor trajectory recovery. However, two challenges remain: 1. DR-based pose constraints and geomagnetic loop closures are relative spatial constraints, risking local optima in graph optimization. 2. Initial DR trajectories are critical. Pedestrian DR can leverage zero-velocity or step-length constraints, but vehicle-based crowdsourced DR rapidly diverges due to high speeds and lack of efective velocity constraints, undermining pose reliability between consecutive timestamps.
To address these challenges, this paper proposes a crowdsourced trajectory graph optimization framework integrating three constraints. First, a smartphone GNSS/MEMS integrated navigation and DR filter with an empirical velocity model extrapolates outdoor trajectories indoors, producing coarse indoor trajectories and adjacent-pose constraints. Second, an improved dynamic time warping (DTW) algorithm detects geomagnetic loop closures within and between trajectories, establishing relative loop constraints. Third, outdoor map constraints are extracted from OpenStreetMap (OSM) via an HMM-based map-matching algorithm, providing absolute loop closures to enhance global consistency during G2O optimization.
Finally, the joint graph optimization is solved using the Levenberg–Marquardt (LM) algorithm. By incorporating absolute loop closure constraints alongside relative constraints and introducing an empirical vehicle speed model in the DR filter, the method prevents convergence to local optima and mitigates rapid divergence in indoor DR, improving the reliability of pose constraints between consecutive timestamps.
The proposed method constructs a G2O-based graph optimization model with three types of constraints to refine indoor crowdsourced trajectories. The overall workflow, illustrated in Figure 1, comprises the following steps: 1. Adjacent Pose Constraints: A smartphone’s GNSS and MEMS sensors (accelerometer, gyroscope) are integrated in a navigation and DR filter to extrapolate trajectories from outdoor to indoor environments, producing coarse indoor trajectories. Adjacent pose constraints are derived from these trajectories. 2. Geomagnetic Loop Closure Constraints: Geomagnetic measurements are corrected for hard/soft iron interference using quasi-static azimuth compensation. An improved dynamic time warping (DTW) algorithm detects loop closures within and across trajectories, forming geomagnetic loop closure constraints. 3. Map-Matching Constraints: Key points of indoor DR trajectories (e.g., entrances/exits) are aligned with outdoor trajectories and OSM road lines via map matching, providing absolute positional constraints. 4. Backend Graph Optimization: All constraints are integrated into a unified graph. The Levenberg-Marquardt (LM) algorithm iteratively solves for the globally optimal trajectory, yielding refined indoor coordinates.
This framework leverages smartphone sensor data and open-source maps, ensuring wide applicability.
This study extends the smartphone GNSS/MEMS/OBD integrated filtering method [
The filter state vector includes attitude, velocity, position errors, and sensor biases:
= [︀ , , , ∇, ]︀ T , where , , and are attitude, velocity, and position errors, respectively, and ∇, denote accelerometer and gyroscope biases.
Observation equations incorporate multiple sources: GNSS when available, otherwise MEMS-based motion constraints. Indoors, empirical velocity est guides state updates through motion constraints. The state vector is sequentially updated via Kalman filtering.
Adjacent position and attitude constraints are derived from the indoor dead reckoning trajectory to ensure local continuity and overall trajectory consistency in subsequent optimization.
In vehicle-mounted scenarios, smartphone magnetometers are afected by non-orthogonal errors, scale factor errors, zero-bias errors, and vehicle-induced magnetic interference. Calibration is thus essential prior to geomagnetic loop closure detection. A simplified linear model is adopted, and ellipsoid fitting tests indicate that zero-bias errors dominate. The model is expressed as
e = m − , where m is the raw measurement, is the zero-bias vector, and e is the calibrated measurement.
Using the vehicle heading from GNSS and the horizontal geomagnetic components, the zero-bias
parameters are estimated under a quasi-static geomagnetic field and known local declination [
To solve the problem of non-uniform geomagnetic field sequence matching caused by changes in
vehicle speed, this paper uses an improved head-tail unconstrained DTW algorithm [
The head-tail unconstrained DTW algorithm compensates for nonlinear temporal variations and speed-induced scale deviations, enabling flexible subsegment matching.
For crowdsourced trajectories, geomagnetic sequence loop closure matching uses this method, with ifnal points obtained via distance-weighted averaging.
The first two types of constraints provide only relative positional information, which can reduce relative trajectory error but may still allow overall drift. To improve absolute spatial accuracy, OpenStreetMap (OSM) is introduced to supply absolute positional references. Map-matched points from outdoor trajectory segments form a third type of constraint, jointly optimized with relative constraints to ensure global consistency.
OSM road segments are represented as straight-line polylines, while real vehicle trajectories may
include curved turns. Straight segments are identified via curve detection, and an improved HMM-based
map-matching algorithm [
The vehicle trajectory is denoted as = {1, . . . , }, with each containing 2D coordinates and heading. Reference points = {1, . . . , } are interpolated from OSM segments, each with The observation probability between a trajectory point and reference point is modeled as ( | ) = exp − 222 )︁ , ︁( where is the observation at time , is the Euclidean distance, and reflects expected positioning error.
The state transition probability between consecutive reference points and is ( | ) = ⎧ ⎪⎨exp − 222 ︂) , if on the same road ︂( ︁( ⎪⎩exp − ︁) , otherwise assigning higher probability to transitions along the same road.
Finally, the Viterbi algorithm computes the most likely sequence of matched points: [, ] = max (︀ [, − 1] · ( | )︀) · ( | ), where [, ] denotes the maximum likelihood at reference point at time . The resulting matched points are then used as constraints in the graph-based optimization framework.
The backend graph optimization refines trajectory estimates by jointly incorporating multiple constraints. A trajectory with poses at the -th iteration is denoted as = {︁1 , 2 , . . . , }︁ , (4) (5) (6) (7) where each pose = [︀ ,
, ]︀ ⊤ consists of 2D coordinates and a heading angle.
The global nonlinear least-squares objective over all trajectories is formulated as where denotes the set of all poses, , the residual between vertices and , Ω the information matrix weighting constraint confidence, and the set of adjacent-pose, geomagnetic loop-closure, and map-matching constraints.
The adjacent-pose constraint enforces smooth motion between consecutive poses, with residual where the superscript 0 denotes observations and ˆ· denotes estimates at iteration .
Geomagnetic loop-closure constraints include intra-trajectory loops, and inter-trajectory loops between trajectories and , () =
∑︁ ⊤, Ω , , (,)∈ ⎡∆ 0,+1 − ∆ ˆ,+1 ⎤ adj ,+1 = ⎢∆ 0,+1 − ∆ ˆ,+1 ⎦⎥ , ⎣ ∆ 0,+1 − ∆ ˆ,+1
⎡− ∆ ˆ, ⎤ lo,op = ⎣− ∆ ˆ, ⎦ ,
− ∆ ˆ, mag = [︃− ∆ ˆ, .
]︃ − ∆ ˆ, o sm = [︃− ∆ ˆ,osm .
]︃ − ∆ ˆ,osm (8) (9) (10) (11) (12) (13) (14) The complete objective function for ≥ 2 trajectories is
() = ∑︁ a,djΩ a,dj + ∑︁ lo,opΩlooplo,op + ∑︁ magΩmagmag + ∑︁ o smΩosmo sm. For single-trajectory cases ( = 1), the inter-trajectory geomagnetic term is omitted.
The optimal trajectory is obtained by minimizing () with the Levenberg–Marquardt algorithm
[
This optimization yields globally consistent trajectories by jointly balancing all constraint types.
For map-matching, residuals are defined with respect to OSM reference points as
The experiments were conducted in two underground parking structures and adjacent urban-canyon roadways in Haidian District, Beijing. Figure 2(a) shows the street view at the entrance of Scenario 1, lfanked by high-rise buildings that create a typical urban-canyon environment with severe multipath and non-line-of-sight (NLOS) GNSS efects. Figure 2(b) presents a remote-sensing image of Scenario 2, highlighting the underground parking structure and its surrounding roadways where the experiments were carried out. (a) Scenario 1: urban canyon at parking entrance
Source: Baidu Map (https://map.baidu.com/) (b) Scenario 2: underground parking and roadways
Source: Baidu Map (https://map.baidu.com/) (c) Data acquisition interface and mounting setup (d) Crowdsourced smartphone models
Test data were collected using two smartphones vivo X100 and iQOO Z9, both supporting dualfrequency L1/L5 GNSS and tri-axial MEMS IMUs. Figure 2(c) shows the multi-sensor data acquisition software interface and the mounting configuration of the devices. Each device was rigidly mounted in a strapdown setup on the front interior of the vehicle, below the windshield, with the Y-axis aligned to the forward direction, the X-axis to the vehicle right-hand side, and the Z-axis upward, thereby fixing the relation between the phone body frame and the vehicle frame.
Before each test, the vehicles remained stationary for 30 s to collect static IMU data, enabling ofline estimation of gyroscope and accelerometer biases as well as magnetometer calibration.
In addition to these test devices, the crowdsourced database was constructed from nine diferent smartphones, including Huawei Mate60Pro, Huawei Mate20, Xiaomi14, and Honor Magic6, providing diverse sensor data for trajectory optimization (Figure 2(d)).
This paper addresses the challenge of precise indoor trajectory recovery from mobile crowdsourced data by proposing a graph optimization framework leveraging geomagnetic and map-based loop closure constraints. The methodology proceeds as follows: 1. A smartphone GNSS/MEMS integrated navigation and dead-reckoning (DR) filter, enhanced with an empirical velocity model, extrapolates trajectories from outdoor to indoor environments. This generates coarse indoor trajectories and establishes adjacent-pose constraints. 2. Exploiting the spatial stability of indoor geomagnetic fields, an improved dynamic time warping (DTW) algorithm detects loop closures within and across trajectories, forming relative loopclosure constraints. 3. Outdoor map constraints are obtained from OpenStreetMap (OSM) via a hidden Markov model (HMM)-based map-matching algorithm. These absolute loop closures enhance global consistency during G2O optimization. 4. A joint graph optimization model integrates adjacent-pose, geomagnetic loop, and map-matching constraints, which is solved using the Levenberg-Marquardt (LM) algorithm to yield globally optimized trajectories.
The main contributions are: 1. While existing geomagnetic loop closure-based methods rely solely on relative spatial constraints and risk convergence to local optima, the proposed approach incorporates absolute spatial references—such as entrance/exit points and OSM map-matching points—into the G2O loss function, improving optimization robustness. 2. To mitigate rapid divergence of indoor DR trajectories due to lack of external velocity references, the empirical velocity model in the GNSS/MEMS filter provides reliable extrapolation from outdoor to indoor trajectories. This generates adjacent-pose constraints that preserve consistent pose relationships between consecutive trajectory points, ensuring continuous indoor trajectory recovery for crowdsourced data.
(a) Scenario 1: Map matching (b) Scenario 2: Map matching (c) Scenario 1: Geomagnetic loop closure (d) Scenario 2: Geomagnetic loop closure (e) Scenario 1: Joint graph optimization (f) Scenario 2: Joint graph optimization During the preparation of this work, the author(s) used the following generative AI tools: • ChatGPT-5 (OpenAI): Used for grammar and spelling checks.
• Kimi-1.5 (Moonshot AI): For generating LaTeX versions of equations.
After using these tools, the author(s) reviewed, verified, and edited the content as needed, and take(s) full responsibility for the publication’s content.