Phase-Based Motion Estimation

• Phase-Based Motion Estimation (PME) is a technique that extracts motion information from local phase shifts in band-limited signals, enabling sub-pixel accuracy across various fields.
• It employs complex filter banks and phase unwrapping to reliably compute small displacements, enhancing applications such as radar odometry and optical flow.
• PME offers enhanced motion sensitivity and noise resilience over traditional methods, with innovations integrating adaptive filtering and deep learning for robust performance.

Phase-Based Motion Estimation (PME) is a class of computational techniques for estimating motion by analyzing phase evolution in band-pass filtered representations of time-varying signals. PME exploits the property that small object displacements in spatial or temporal domains manifest as local phase changes in the responses of complex-valued, band-limited filters. This methodology enables robust, sub-pixel (or sub-sample) displacement estimation in contexts including radar odometry, biomedical imaging, motion magnification, and optical flow. Modern PME pipelines integrate complex wavelet or steerable filter banks, phase unwrapping, robust aggregation, and, in some domains, learned post-processing or optimization frameworks.

1. Theoretical Foundations

PME leverages the Fourier shift theorem: a spatial shift in a signal induces a linear phase change in its frequency (or band-pass filtered) representation. Given a signal $I(x, y, t)$—such as an image sequence or radargram—convolution with a complex band-pass filter $\psi$ produces

$$C(x, y, t) = [I(\cdot, \cdot, t) * \psi](x, y) = A(x, y, t) e^{j\phi(x, y, t)}$$

where $A$ is the local amplitude and $\phi$ is the phase. For small translations $\Delta x, \Delta y$, the phase increment satisfies $\Delta\phi \approx k \cdot \Delta x$ with $k$ the carrier spatial frequency of $\psi$.

In 1D settings (such as radar or OCT), the output after mixing or dechirping is of the form $r(t) = A e^{j\phi(t)} + n(t)$, with the phase $\psi$0 encoding distance- or velocity-induced delays. Under constant velocity,

$\psi$1

where $\psi$2 is the Doppler frequency and $\psi$3 is wavelength.

In optical flow, phase correlation in the Fourier domain between frames $\psi$4 and $\psi$5 displaced by a translation $\psi$6 yields a cross-power spectrum with a sharp impulse at $\psi$7.

2. PME Algorithms and Signal Models

Core PME algorithmic steps include:

1. Band-Pass Decomposition: Input signals (image frames, radar returns, or biomedical scans) are decomposed into scale and orientation-specific complex coefficients via Gabor, steerable, or pyramid wavelets (Oshim et al., 2022, Prashnani et al., 2022, Sarrafi et al., 2018).
2. Local Phase Extraction: The phase of each coefficient is extracted, typically via $\psi$8 or equivalent (Oshim et al., 2022).
3. Phase Increment (Motion): For each point (pixel, range-bin, depth sample), temporal or spatial phase differences $\psi$9 are computed, and, after unwrapping, directly mapped to displacements or velocities using the filter’s frequency: for spatial translation, $$C(x, y, t) = [I(\cdot, \cdot, t) * \psi](x, y) = A(x, y, t) e^{j\phi(x, y, t)}$$0; for velocity, $$C(x, y, t) = [I(\cdot, \cdot, t) * \psi](x, y) = A(x, y, t) e^{j\phi(x, y, t)}$$1 ($$C(x, y, t) = [I(\cdot, \cdot, t) * \psi](x, y) = A(x, y, t) e^{j\phi(x, y, t)}$$2 is sampling interval) (Sen et al., 2024, Sarrafi et al., 2018).
4. Motion Magnitude Estimation: For small, local signals, displacement/velocity is proportional to phase increment divided by the effective frequency bandwidth (Prashnani et al., 2022).
5. Phase Unwrapping and Filtering: 1D or 2D phase unwrapping ensures temporal continuity. Temporal filtering can isolate frequency bands of interest (e.g., cardiac or respiratory in biomedical signals, structural vibration modes) (Oshim et al., 2022, Sarrafi et al., 2018).
6. Aggregation, Fusion, and Post-processing: For robustness and noise attenuation, PME pipelines aggregate estimates over consistent spatial/temporal bins, and may employ median or weighted averaging, low-pass, or Kalman-style filters (Sen et al., 2024).

Optimization-based PME variants, e.g., in OCT elastography, combine phase-difference, intensity consistency, and motion continuity in a dynamic programming framework to resolve sub-wavelength and supra-wavelength motion with phase wrap management (Khodadadi et al., 2021).

3. Application Domains

Radar Odometry and Ego-Motion

In mmWave radar ego-velocity estimation (“mmPhase”), PME tracks the return phase from static reflectors across frames, recovering sub-Doppler velocity (well below Doppler-FFT resolution, e.g., $$C(x, y, t) = [I(\cdot, \cdot, t) * \psi](x, y) = A(x, y, t) e^{j\phi(x, y, t)}$$3 cm/s) by exploiting the phase evolution of range-consistent bins. This approach enables centimeter-per-second sensitivity without need for visual, inertial, or wheel odometry (Sen et al., 2024).

Biomedical and Clinical Monitoring

Eulerian PME underpins motion magnification and vital sign extraction from radar or video. Subtle chest and cardiac motions are amplified by phase manipulation in a complex Gabor pyramid, yielding 1D displacement signals for frequency-domain vital sign estimation. This reduces mean absolute error versus FFT baselines (e.g., MAE $$C(x, y, t) = [I(\cdot, \cdot, t) * \psi](x, y) = A(x, y, t) e^{j\phi(x, y, t)}$$4 bpm smaller in respiration, up to 4 bpm in heart rate) (Oshim et al., 2022).

Optical Flow and Video Motion

PME using phase correlation and modern bilateral- or Gabor-filtered approaches enables accurate, sub-pixel motion estimation in video, particularly at motion boundaries or with multiple motions. The Asymmetric Bilateral Phase Correlation (BLPC) method detects and isolates multiple motion peaks, with experiments demonstrating improved accuracy over classical PC (MSE=0.037, PSNR=16.47 dB) and strong real-time performance (Argyriou, 2018).

Medical Imaging—Optical Coherence Elastography

PME achieves sub-wavelength axial sensitivity by tracking phase changes in OCT B-scans. Optimization-based PME unifies phase and intensity cues, enabling robust displacement estimation over a range exceeding half the imaging wavelength, with high axial precision ($$C(x, y, t) = [I(\cdot, \cdot, t) * \psi](x, y) = A(x, y, t) e^{j\phi(x, y, t)}$$5 nm) and effective phase-unwrapping via dynamic programming (Khodadadi et al., 2021).

Structural Health Monitoring and Modal Analysis

PME and motion magnification are employed in non-contact vibration measurement of large structures (e.g., wind turbine blades). PME-derived, phase-magnified videos enable accurate estimation of modal frequencies and mode shapes—matching traditional accelerometer-based experimental modal analysis to within $$C(x, y, t) = [I(\cdot, \cdot, t) * \psi](x, y) = A(x, y, t) e^{j\phi(x, y, t)}$$6 Hz and Modal Assurance Criterion $$C(x, y, t) = [I(\cdot, \cdot, t) * \psi](x, y) = A(x, y, t) e^{j\phi(x, y, t)}$$7 (Sarrafi et al., 2018).

Gait and Human Motion Analysis

For real-time gait phase estimation with IMUs, PME informs the embedding of gait-phase as angular variables in neural architectures, enabling robust identification of gait cycles and events across terrain transitions with RMSE $$C(x, y, t) = [I(\cdot, \cdot, t) * \psi](x, y) = A(x, y, t) e^{j\phi(x, y, t)}$$8–$$C(x, y, t) = [I(\cdot, \cdot, t) * \psi](x, y) = A(x, y, t) e^{j\phi(x, y, t)}$$9\% (Ji et al., 18 Jun 2025).

4. Practical Implementations and Comparative Performance

PME has consistently demonstrated improved motion sensitivity and robustness over traditional intensity-based methods and FFT-only estimates:

• Radar ego-motion (mmPhase): Mean absolute error in velocity reduced by $A$0 over Doppler-FFT, especially at low speeds where Doppler-FFT fails (Sen et al., 2024).
• Vital sign estimation (radar/clinical video): PME reduces error and sharpens spectral peaks for respiration/heart rate over FFT techniques—improving mean error by 1–4 bpm across settings (Oshim et al., 2022).
• Optical Flow: BLPC-based PME achieves subpixel accuracy and edge precision, outperforming classical phase correlation, especially at motion boundaries. Reported for 4K datasets and standard benchmarks (MSE, PSNR, AE) (Argyriou, 2018).
• OCT Elastography: PME outperforms both pure phase-difference and block-matching, delivering robust tracking over both sub- and supra-wavelength regimes (Khodadadi et al., 2021).
• Structural Health Monitoring: PME estimates of frequencies and deflection shapes match accelerometer ground truth; damage-induced changes in mode shapes are readily detected (Sarrafi et al., 2018).

A summary comparison table (condensed):

Application Domain	PME Advantage	Benchmark Error
Radar Odometry	Sub-cm/s velocity; $A$1 MAE	MAE: 0.02–0.08 m/s
Clinical Vital Sign	$A$2–$A$3 bpm lower error	MAE: Resp. 1.90 vs 2.65
Optical Flow, BLPC	Subpixel boundary accuracy	MSE: 0.037 (BLPC)
OCT Elastography	Sub-100 nm axial sensitivity	NMAE lowest across range
SHM (Wind blade)	Frequency error $A$4\%, MAC$A$5	$A$6 Hz
Human Gait (IMU)	RMSE $A$7–$A$8\%	$A$93.3\% (gait-phase)

5. Limitations and Challenges

PME performance depends on several factors:

• Phase Unwrapping: Large or rapid motion may exceed $\phi$0 radians per interval, leading to phase-wrapping errors; unwrapping can fail under high noise (Oshim et al., 2022, Khodadadi et al., 2021).
• Filter Design: Choice of spatial/temporal filter scale, orientation, and bandwidth critically affects sensitivity and artifact rejection (Oshim et al., 2022, Sarrafi et al., 2018).
• Clutter/Multipath: In radar, range bin consistency is challenged in cluttered or multipath-heavy environments, possibly introducing ambiguities (Sen et al., 2024).
• Computational Cost: Multi-scale, multi-orientation convolutions and phase arithmetic impose heavy CPU/GPU/FPGA loads for real-time applications (Oshim et al., 2022, Argyriou, 2018).
• Motion Types: Classical phase correlation is limited to translation; extensions are needed for rotation, scale, and more complex deformations (Argyriou, 2018).
• Speckle and Noise: In OCT, phase decorrelation in low SNR regions can impair PME, motivating joint optimization and regularization (Khodadadi et al., 2021).

6. Extensions and Future Directions

Research avenues include:

• Physics-Informed and Deep Learning: Augmenting PME phase-to-motion mappings with physics-informed neural networks and end-to-end learned feature stacking (Sen et al., 2024, Oshim et al., 2022).
• Multi-Dimensional Velocity Estimation: Fusing multi-azimuth (radar) or multi-direction (video) channels to recover full 2D/3D velocity vectors (Sen et al., 2024).
• Adaptive and Robust Filtering: Dynamic adaptation of scale, amplification, and phase-handling parameters based on context (e.g., subject posture, scene clutter, terrain) (Oshim et al., 2022, Ji et al., 18 Jun 2025).
• Nonlinear and Multimodal Fusion: Integration of PME outputs with conventional motion features (e.g., micro-Doppler, IMU) and explicit modeling of non-rigid deformations (Oshim et al., 2022, Ji et al., 18 Jun 2025).
• Structural and Biomedical Sensing: Scaling PME to large structures with distributed, automated video inference; parallelization and regularization for high-throughput medical imaging (Sarrafi et al., 2018, Khodadadi et al., 2021).

Emerging PME research continues to target high-sensitivity, robust motion estimation under weak SNR, domain shift, and complex multiphysics settings across radar, healthcare, and vision domains.