Diffusion Denoiser-Aided Gyrocompass

• The paper introduces a two-stage stationary gyrocompassing architecture that applies a DDPM-style diffusion denoiser followed by an enhanced Bi-LSTM to robustly predict heading from low-SNR MEMS data.
• It addresses the challenge of weak Earth rotation signals by preprocessing noisy triaxial gyro sequences, reducing CRMSE by up to 26% compared to classical methods.
• Empirical results validate that the diffusion denoiser-aided system outperforms both model-based and direct learning methods, enhancing initial alignment in environments lacking magnetic cues.

A diffusion denoiser-aided gyrocompass is a stationary gyrocompassing architecture that estimates initial heading from low-cost MEMS gyroscopes by inserting a diffusion-based denoiser ahead of a learned heading estimator. In the formulation reported in “Diffusion Denoiser-Aided Gyrocompassing” (Ben-Arie et al., 28 Jul 2025), the denoiser operates on raw triaxial gyroscope sequences, and the cleaned sequence is then processed by an enhanced Bi-LSTM to predict heading. The approach addresses the central difficulty of gyrocompassing: the body-frame projection of the Earth’s rotation rate is extremely weak relative to MEMS noise, bias, and drift, so direct model-based azimuth recovery from raw averages is often inaccurate or slow.

1. Problem formulation and physical basis

Gyrocompassing estimates heading from gyroscopes by exploiting the measured projection of the Earth’s rotation rate onto the sensor axes. For a stationary platform, the classical relation is

$\omega^b_{ib} = T^{b}_{n}T^{n}_{e}\omega^e_{ie},$

where $\omega^b_{ib}$ is the body-frame gyroscope measurement, $\omega^e_{ie}$ is the Earth rotation vector, $T^b_n$ is the navigation-to-body rotation matrix, and $T^n_e$ is the Earth-fixed-to-navigation rotation matrix. Heading is then recovered through

$$\text{s}_{\psi} = -\omega_{ib,y}^b \text{c}_{\phi} + \omega_{ib,z}^b \text{s}_{\phi}, \qquad \text{c}_{\psi} = \omega_{ib,x}^b \text{c}_{\theta} + \omega_{ib,y}^b \text{s}_{\phi}\text{s}_{\theta} + \omega_{ib,z}^b \text{c}_{\phi}\text{s}_{\theta},$$

followed by

$$\psi = \mathrm{atan}_2(\text{s}_{\psi},\text{c}_{\psi}).$$

Under leveled conditions this reduces to

$$\psi = \mathrm{atan}_2\left(-\omega^b_{ib,y},\omega^b_{ib,x}\right).$$

This reduction is central: once roll and pitch are obtained from static accelerometer measurements of gravity, gyrocompassing becomes the problem of recovering the weak horizontal Earth-rate vector from the $x$- and $y$-axis gyro channels (Ben-Arie et al., 28 Jul 2025).

The difficulty is set by observability. At latitude $\omega^b_{ib}$0, the maximum horizontal Earth-rate component available to support heading is

$\omega^b_{ib}$1

At $\omega^b_{ib}$2, the maximal sensed heading-relevant signal is only $\omega^b_{ib}$3 deg/s. The paper identifies the resulting failure modes directly: the useful signal is weak, gyroscope bias and random noise can dominate the Earth-rate projection, drift contaminates averaging over time, observability decreases with latitude through $\omega^b_{ib}$4, and no GNSS, vision, RF, or magnetometer aid is assumed. The method is therefore aimed at environments where magnetic heading is unreliable or unavailable, and where low-SNR MEMS gyro data must still support initial azimuth estimation.

2. Two-stage architecture

The reported system is explicitly two-stage. First, a diffusion denoiser is applied to raw stationary triaxial gyroscope data; second, an enhanced Bi-LSTM heading-regression model predicts the heading angle from the denoised sequence. The operational pipeline is stated as: collect raw stationary triaxial gyroscope data; normalize the sequence; apply a trained diffusion denoiser; de-normalize the denoised output; feed the denoised gyroscope sequence into a heading extraction network; predict the heading angle (Ben-Arie et al., 28 Jul 2025).

This separation is not merely organizational. The paper argues that a heading estimator trained directly on raw low-cost gyro sequences must implicitly learn denoising and heading extraction simultaneously, despite the Earth-rate signature being close to the sensor noise floor. By contrast, the denoiser-first design assigns noise suppression to a dedicated module and reserves the downstream Bi-LSTM for angular inference. The training strategy is correspondingly sequential rather than end-to-end: the diffusion denoiser is trained first on synthetic simulated gyro data; its weights are then frozen; it is used to preprocess real recorded gyro sequences; and the heading estimator is trained on those denoised real sequences.

The system is limited to stationary gyrocompassing. The paper repeatedly assumes a stationary and leveled platform, and the real recordings are fixed-heading static sequences. It is therefore not a dynamic-motion heading estimator, nor is it presented as a continuous navigation filter. That distinction matters because the method targets initial alignment rather than attitude propagation under motion.

3. Diffusion denoising formulation

The denoising front-end uses a DDPM-style discrete diffusion model rather than an SDE-based score model. The forward process progressively corrupts clean signals with Gaussian noise: $\omega^b_{ib}$5 with marginal

$\omega^b_{ib}$6

where

$\omega^b_{ib}$7

Training uses the standard practical noising equation

$\omega^b_{ib}$8

with $\omega^b_{ib}$9, and the network predicts the added noise $\omega^e_{ie}$0 (Ben-Arie et al., 28 Jul 2025).

The denoiser operates on gyroscope time sequences. Each real or synthetic sequence corresponds to 100 s of triaxial gyro data sampled at 3 Hz after downsampling, producing 300 time steps and 3 channels. Before model input, each $\omega^e_{ie}$1 sequence is flattened into a vector of size $\omega^e_{ie}$2. The denoiser receives the time sequence and the diffusion timestep $\omega^e_{ie}$3. Its architecture is a five-layer bidirectional LSTM with 64 hidden states per layer and timestep embedding dimension 20; the timestep embeddings are injected into the Bi-LSTM layers so that the model can condition its noise estimate on corruption level.

The variance schedule is linear with $\omega^e_{ie}$4, $\omega^e_{ie}$5, and $\omega^e_{ie}$6. The manuscript states that these values were selected based on real data analysis, signal-to-noise ratio evaluations, and empirical adjustments to match real-world conditions. The reverse denoising update is written in the paper in a malformed form, but its intent is the standard DDPM reverse update using the predicted noise; the paper does not define $\omega^e_{ie}$7 beyond its appearance in that equation.

A distinctive feature is the loss. Ordinary MSE was compared against a “spectral filtering MSE,” and ordinary MSE was judged inadequate because it did not regulate predicted noise in a way that matched sensor noise behavior. The adopted objective applies SVD to a matrix $\omega^e_{ie}$8,

$\omega^e_{ie}$9

thresholds the singular values by

$T^b_n$0

reconstructs

$T^b_n$1

and then computes an MSE between filtered prediction and filtered target. The stated interpretation is that this better matches the non-uniform frequency content of real sensor noise.

The deployed reverse process is partial rather than full. Denoising is performed from $T^b_n$2 down to a selected $T^b_n$3, so the number of reverse iterations is $T^b_n$4. The empirically chosen value is $T^b_n$5, meaning only a short reverse segment near the high-noise end is used in the final system. The paper does not fully justify this choice from a generative-modeling standpoint, but reports that it yielded better heading results.

4. Heading estimator, loss, and data regime

The heading estimator extends a prior Bi-LSTM baseline with dropout between the two Bi-LSTM layers, dropout between the Bi-LSTM output and the fully connected layer, and exponential learning-rate decay. The network remains a two-layer bidirectional LSTM with 24 hidden states, followed by a fully connected scalar heading regressor. The chosen dropout ratio is $T^b_n$6. ADAM is used as optimizer with batch size 32, and the learning rate decays exponentially from $T^b_n$7 to $T^b_n$8 (Ben-Arie et al., 28 Jul 2025).

The output is a scalar heading angle in degrees. The model does not use sine/cosine output parameterization, quaternions, or classification bins. Instead it is trained with a cyclic RMSE loss,

$T^b_n$9

where $T^n_e$0. This loss computes the shortest angular difference and avoids the $T^n_e$1 discontinuity. The same CRMSE is used for validation and test evaluation.

Two datasets are used, with a strict division of labor. The real dataset is used for heading-estimator training and final evaluation. It contains 34 stationary sequences, each at a fixed heading, each 100 s long, for a total duration of 56 min. The input gyroscope is an Emcore MEMS SDC500 IMU, and heading ground truth is supplied by an Inertial Labs MRU-P IMU, whose static heading accuracy is $T^n_e$2. The data were originally sampled at 600 Hz and downsampled to 3 Hz. The split is 24 sequences for training, 4 for validation, and 6 for test. Because the real training set is small, each training sequence is augmented by extrapolating it to 100 different heading angles evenly spaced within a $T^n_e$3 range around the original sequence heading, yielding 2400 training sequences.

The synthetic dataset is used exclusively for denoiser training and evaluation. It is generated from

$T^n_e$4

with $T^n_e$5, i.e. equator, maximizing horizontal Earth-rate observability. It covers heading range $T^n_e$6 to $T^n_e$7 in $T^n_e$8 increments, for 720 sequences of 100 s each and total duration 1200 min, split 60% train, 20% validation, and 20% test. The paper states that synthetic data were used because real clean-denoising supervision was unavailable.

5. Empirical results and ablations

At 100 s gyrocompassing duration, the reported test CRMSE values are $T^n_e$9 for classical model-based gyrocompassing, $$\text{s}_{\psi} = -\omega_{ib,y}^b \text{c}_{\phi} + \omega_{ib,z}^b \text{s}_{\phi}, \qquad \text{c}_{\psi} = \omega_{ib,x}^b \text{c}_{\theta} + \omega_{ib,y}^b \text{s}_{\phi}\text{s}_{\theta} + \omega_{ib,z}^b \text{c}_{\phi}\text{s}_{\theta},$$0 for the baseline data-driven Bi-LSTM, and $$\text{s}_{\psi} = -\omega_{ib,y}^b \text{c}_{\phi} + \omega_{ib,z}^b \text{s}_{\phi}, \qquad \text{c}_{\psi} = \omega_{ib,x}^b \text{c}_{\theta} + \omega_{ib,y}^b \text{s}_{\phi}\text{s}_{\theta} + \omega_{ib,z}^b \text{c}_{\phi}\text{s}_{\theta},$$1 for the diffusion denoiser-aided system (Ben-Arie et al., 28 Jul 2025).

Method	Test CRMSE at 100 s	Relative result
Model-based gyrocompassing	$$\text{s}_{\psi} = -\omega_{ib,y}^b \text{c}_{\phi} + \omega_{ib,z}^b \text{s}_{\phi}, \qquad \text{c}_{\psi} = \omega_{ib,x}^b \text{c}_{\theta} + \omega_{ib,y}^b \text{s}_{\phi}\text{s}_{\theta} + \omega_{ib,z}^b \text{c}_{\phi}\text{s}_{\theta},$$2	baseline
Data-driven gyrocompassing baseline	$$\text{s}_{\psi} = -\omega_{ib,y}^b \text{c}_{\phi} + \omega_{ib,z}^b \text{s}_{\phi}, \qquad \text{c}_{\psi} = \omega_{ib,x}^b \text{c}_{\theta} + \omega_{ib,y}^b \text{s}_{\phi}\text{s}_{\theta} + \omega_{ib,z}^b \text{c}_{\phi}\text{s}_{\theta},$$3	better than model-based
Diffusion denoiser-aided gyrocompassing	$$\text{s}_{\psi} = -\omega_{ib,y}^b \text{c}_{\phi} + \omega_{ib,z}^b \text{s}_{\phi}, \qquad \text{c}_{\psi} = \omega_{ib,x}^b \text{c}_{\theta} + \omega_{ib,y}^b \text{s}_{\phi}\text{s}_{\theta} + \omega_{ib,z}^b \text{c}_{\phi}\text{s}_{\theta},$$4	$$\text{s}_{\psi} = -\omega_{ib,y}^b \text{c}_{\phi} + \omega_{ib,z}^b \text{s}_{\phi}, \qquad \text{c}_{\psi} = \omega_{ib,x}^b \text{c}_{\theta} + \omega_{ib,y}^b \text{s}_{\phi}\text{s}_{\theta} + \omega_{ib,z}^b \text{c}_{\phi}\text{s}_{\theta},$$5 better than model-based; $$\text{s}_{\psi} = -\omega_{ib,y}^b \text{c}_{\phi} + \omega_{ib,z}^b \text{s}_{\phi}, \qquad \text{c}_{\psi} = \omega_{ib,x}^b \text{c}_{\theta} + \omega_{ib,y}^b \text{s}_{\phi}\text{s}_{\theta} + \omega_{ib,z}^b \text{c}_{\phi}\text{s}_{\theta},$$6 better than baseline DL

These figures establish the paper’s central empirical claim: diffusion-based preprocessing improves heading estimation beyond both the analytic Earth-rate solution applied directly to raw data and a prior learning-based approach without explicit diffusion denoising. The absolute gains are $$\text{s}_{\psi} = -\omega_{ib,y}^b \text{c}_{\phi} + \omega_{ib,z}^b \text{s}_{\phi}, \qquad \text{c}_{\psi} = \omega_{ib,x}^b \text{c}_{\theta} + \omega_{ib,y}^b \text{s}_{\phi}\text{s}_{\theta} + \omega_{ib,z}^b \text{c}_{\phi}\text{s}_{\theta},$$7 relative to model-based gyrocompassing and $$\text{s}_{\psi} = -\omega_{ib,y}^b \text{c}_{\phi} + \omega_{ib,z}^b \text{s}_{\phi}, \qquad \text{c}_{\psi} = \omega_{ib,x}^b \text{c}_{\theta} + \omega_{ib,y}^b \text{s}_{\phi}\text{s}_{\theta} + \omega_{ib,z}^b \text{c}_{\phi}\text{s}_{\theta},$$8 relative to the baseline learning model.

The ablations focus on preprocessing and denoising depth rather than on architectural alternatives. Two normalization strategies were tested. Time sample normalization yielded training loss $$\text{s}_{\psi} = -\omega_{ib,y}^b \text{c}_{\phi} + \omega_{ib,z}^b \text{s}_{\phi}, \qquad \text{c}_{\psi} = \omega_{ib,x}^b \text{c}_{\theta} + \omega_{ib,y}^b \text{s}_{\phi}\text{s}_{\theta} + \omega_{ib,z}^b \text{c}_{\phi}\text{s}_{\theta},$$9 and validation loss $$\psi = \mathrm{atan}_2(\text{s}_{\psi},\text{c}_{\psi}).$$0, whereas time sequence normalization yielded training loss $$\psi = \mathrm{atan}_2(\text{s}_{\psi},\text{c}_{\psi}).$$1 and validation loss $$\psi = \mathrm{atan}_2(\text{s}_{\psi},\text{c}_{\psi}).$$2; time-sequence normalization was therefore selected. The paper also reports that the denoising process changes the scale of the raw sensor data, with stronger distortion as more reverse denoising iterations are applied. Directly passing denoised data into the heading model often caused low accuracy or divergence, which motivated the normalize–denoise–de-normalize pipeline.

The choice of $$\psi = \mathrm{atan}_2(\text{s}_{\psi},\text{c}_{\psi}).$$3 was also swept. Values from 980 down to 100 were examined. Best validation performance appeared in the range 900–980, with validation CRMSE around $$\psi = \mathrm{atan}_2(\text{s}_{\psi},\text{c}_{\psi}).$$4–$$\psi = \mathrm{atan}_2(\text{s}_{\psi},\text{c}_{\psi}).$$5, while lower values often produced validation loss above $$\psi = \mathrm{atan}_2(\text{s}_{\psi},\text{c}_{\psi}).$$6. The final setting $$\psi = \mathrm{atan}_2(\text{s}_{\psi},\text{c}_{\psi}).$$7 was selected based on performance and better train/validation/test convergence behavior. Importantly, denoiser quality is evaluated indirectly through downstream heading performance rather than through a standalone real-data denoising metric such as reconstruction SNR.

6. Relation to adjacent work, limitations, and scope

The diffusion denoiser-aided formulation sits within a broader transition from analytic gyrocompassing toward learned extraction of Earth-rate structure from long stationary gyro sequences. “Towards Learning-Based Gyrocompassing” (Engelsman et al., 2023) established that a direct Bi-LSTM regressor could outperform simple averaging on stationary MEMS data, reporting median heading-error improvements across 10 s, 20 s, 30 s, 60 s, and 240 s windows, and reducing the duration required to reach $$\psi = \mathrm{atan}_2(\text{s}_{\psi},\text{c}_{\psi}).$$8 median error from 203.1 s to 13.8 s. The diffusion-denoiser-aided system preserves that stationary-learning setting but changes the division of labor: instead of asking a single sequence model to perform implicit denoising and heading extraction, it inserts an explicit DDPM-style denoiser before the heading regressor.

A separate but related trajectory appears in “Underwater MEMS Gyrocompassing: A Virtual Testing Ground” (Engelsman et al., 2024). That work does not use diffusion models, score matching, or an explicit probabilistic denoiser; rather, it uses a supervised disturbance-robust estimator trained on real stationary gyro measurements augmented with synthetic underwater rotational dynamics. Its relevance lies in showing that learned processing of disturbed inertial data can improve heading estimation when the Earth-rate signal is submerged beneath structured, low-frequency nuisance motion. This suggests that diffusion-denoiser-aided gyrocompassing can be interpreted not as an isolated architectural choice, but as one point within a broader class of learned signal-refinement strategies for Earth-rate extraction.

A common misconception is to equate any learned gyrocompassing model with diffusion-denoiser-aided gyrocompassing. The literature described here distinguishes three categories. First, there is classical model-based gyrocompassing, which directly applies the Earth-rate equations to measured gyroscope outputs. Second, there are direct learning-based regressors, such as the prior Bi-LSTM approach, which map raw sequences to heading without explicit generative denoising. Third, there is the diffusion denoiser-aided variant, which explicitly uses a DDPM-style noise-prediction model as a preprocessing front-end. The underwater work belongs to the second category despite its signal-refinement language; the 2025 method belongs to the third.

The limitations of the diffusion-denoiser-aided method are clearly bounded by the reported experiments. The platform is assumed stationary and leveled; the denoiser depends on synthetic training data; normalization and de-normalization are essential; performance is sensitive to $$\psi = \mathrm{atan}_2(\text{s}_{\psi},\text{c}_{\psi}).$$9; the system is not jointly optimized end-to-end; there is no demonstration under motion; and robustness across multiple IMU grades, multiple real sensors, different latitudes, or dynamic motion profiles is not established. Reproducibility is only partial: the paper gives substantial architectural and dataset detail, but does not clearly specify the exact epoch count for either model, the SVD threshold $$\psi = \mathrm{atan}_2\left(-\omega^b_{ib,y},\omega^b_{ib,x}\right).$$0, the schedule or definition of $$\psi = \mathrm{atan}_2\left(-\omega^b_{ib,y},\omega^b_{ib,x}\right).$$1 in the reverse step, the hardware used, training time, whether flattening is reversed before LSTM processing, the exact real-data augmentation mechanism, or whether the synthetic noise includes realistic bias instability and random walk. A plausible implication is that the method is best understood as a proof of effectiveness for initialization-centric stationary gyrocompassing rather than as a complete deployment recipe.

Within those bounds, the topic denotes a specific technical synthesis: preserve the classical Earth-rate geometry, acknowledge that low-cost MEMS data are dominated by noise processes that simple averaging does not handle well, and use an explicit diffusion denoiser to expose heading-relevant structure before angular regression. In the reported experiments, that synthesis reduces 100-second test CRMSE to $$\psi = \mathrm{atan}_2\left(-\omega^b_{ib,y},\omega^b_{ib,x}\right).$$2, improving over both raw analytic gyrocompassing and a prior direct learning baseline (Ben-Arie et al., 28 Jul 2025).