Phase-Based Motion Estimation Techniques

• Phase-based motion estimation techniques are algorithms that analyze phase shifts in bandlimited signals to detect sub-pixel motions with high sensitivity and precision.
• They employ advanced filter banks such as Gabor and complex steerable pyramids to extract instantaneous phase changes across spatial and temporal domains.
• Robust estimation is achieved through multi-scale decomposition, phase unwrapping, and spatial regularization to mitigate noise and ambiguities in practical settings.

Phase-based motion estimation techniques are a class of algorithms that extract, characterize, or amplify motion by analyzing the phase of signals—typically from bandlimited decompositions of images, videos, radar, or optical sensors. These Eulerian approaches enable highly sensitive measurement of subtle displacements, sub-wavelength movements, or distributed flexural vibrations. The methodology is widely applicable in video-based structural health monitoring, radar vital sign sensing, gait analysis, quantum optomechanics, and general-purpose optical flow, achieving state-of-the-art accuracy and robustness, often at or close to physical or quantum precision limits.

1. Mathematical Foundations: Phase–Motion Coupling

The core principle underlying phase-based motion estimation is that local translations induce proportional phase shifts in complex bandpass filtered representations. Given a signal decomposition $I(x,y,t)$ via a complex bandpass filter (e.g., 2D Gabor, complex steerable pyramid, or 1D Gabor wavelet), the response at position $(u,v)$ and time $t$ can be written: $$C_{\lambda,\theta}(u,v,t) = (I(\cdot,\cdot,t) * g_{\lambda,\theta})(u,v) = A_{\lambda,\theta}(u,v,t)\, e^{i\phi_{\lambda,\theta}(u,v,t)}$$ where $A$ is amplitude and $\phi$ local phase (Sarrafi et al., 2018, Prashnani et al., 2022, Pintea et al., 2016).

A spatial displacement $\Delta x$ yields a phase change

$$\Delta\phi(u,v) \approx \frac{2\pi}{\lambda}\Delta x$$

for a filter of wavelength $\lambda$ and orientation $\theta=0$. For video, the temporal phase difference relates directly to local motion along the filter's axis: $$\Delta x(u,v,t) \approx \frac{\lambda}{2\pi}\left[\phi(u,v,t+\Delta t) - \phi(u,v,t)\right]$$ This paradigm generalizes to radar and OCT, where phase accumulation tracks radial, axial, or tangential displacements with sub-wavelength sensitivity, provided phase ambiguity (wrapping) is controlled (Sen et al., 2024, Khodadadi et al., 2021).

2. Algorithms and Decomposition Frameworks

Local Steerable Filter Banks and Pyramic Analysis

Typical signal decomposition employs multi-scale, multi-orientation filter banks (e.g., Gabor, complex steerable pyramid), implemented either in the spatial or frequency domain:

• 2D Gabor kernels: $g_{\lambda,\theta,\sigma,\gamma}(x,y)$ for 2D video or SHM applications (Sarrafi et al., 2018).
• Complex steerable pyramid: Analytic bandpass subbands indexed by scale and orientation, optimized for capturing local phase changes due to translation/displacement (Prashnani et al., 2022, Pintea et al., 2016).
• 1D Gabor filter banks: Used in radar vital sign sensing, with spatial bandwidths tuned to physiological motion frequencies (Oshim et al., 2022).

Phase Difference Extraction and Motion Estimation

Local phase is computed per filter and per spatial location. The instantaneous motion signal is proportional to the difference (or derivative) of the phase across time. In 1D radar or OCT, the phase–displacement mapping is: $$\Delta z = \frac{\lambda_0}{4\pi n}\Delta\phi \quad (\Delta\phi \in [-\pi, \pi])$$ where $n$ is refractive index and $\lambda_0$ is center wavelength (Khodadadi et al., 2021).

Disambiguation via Smoothing and Optimization

When motion exceeds the filter's local unambiguous range (e.g., $> \lambda/2$ or $2\pi$ phase), phase unwrapping or global optimization is required. Optical flow frameworks may utilize dynamic programming over candidate shift and wrap states, penalizing phase differences, intensity mismatches, and lack of spatial smoothness (Khodadadi et al., 2021). For large video fields, bilateral or multi-scale filtering combined with frequency-domain phase correlation enables robust estimation even at motion boundaries (Argyriou, 2018).

3. Domain-Specific Implementations and Innovations

Application Area	Core Decomposition/Algorithm	Notable Features/Results
Video-based SHM	Gabor/Steerable Filters (Sarrafi et al., 2018)	Full-field vibration, motion magnification, damage detection
Radar Vital Signs	1D Complex Gabor + phase amplification (Oshim et al., 2022)	Respiration/heart rate estimation surpassing FFT baseline
mmWave Ego-Velocity	FMCW phase tracking (Sen et al., 2024)	Sub-Doppler, millimetric-per-second MAE
Human Gait (IMU-based)	Polar embedding, TCN+Transformer (Ji et al., 18 Jun 2025)	$<3\%$ phase RMSE, robust across terrain
OCT Elastography	Phase/intensity DP optimization (Khodadadi et al., 2021)	Sub-wavelength axial tracking under large deformations
Deepfake Detection	CSP + 3D CNN (Prashnani et al., 2022)	High AUC, strong adversarial robustness
Quantum Optomechanics	Homodyne phase tracking, quantum smoothing (Iwasawa et al., 2013)	Estimation beyond coherent-state QCRB

Each application tailors the decomposition and post-processing according to signal-to-noise, ambiguity concerns, and modality-specific artifacts. Structural health monitoring and medical sensing benefit from non-contact, distributed measurement, while quantum-limited estimation exploits phase-squeezed light and optimal smoothing for approaching fundamental precision bounds.

4. Comparison to Lagrangian (Feature/Tracking-Based) Schemes

The phase-based (Eulerian) approach contrasts sharply with classic optical flow (Lagrangian) models, which rely on point or patch correspondences and solve for explicit displacement fields by minimizing intensity differences under continuity/smoothness constraints (Pintea et al., 2016):

• Eulerian strengths: No explicit tracking or matching, always dense per-pixel phase, robust to small and distributed deformations, high sensitivity to tiny (sub-pixel, sub-wavelength) motion.
• Limitations: For large or discontinuous motions, phase wrapping can induce ambiguity; without global optimization, outliers may proliferate, and the method may fail on abrupt occlusion or large-scale non-affine transformations.
• Lagrangian methods: Excel at modeling large displacements or tracking over long time frames but suffer in low-texture regions or at the sub-pixel scale.

Hybrid methods optimize over phase and intensity, or invoke spatial regularization, bilateral filtering, or multi-resolution analysis to bridge the gap (Argyriou, 2018, Khodadadi et al., 2021).

5. Practical Challenges and Boundary Conditions

Phase Wrapping and Smoothing

Phase-based estimation is intrinsically limited by $2\pi$ ambiguity. Strategies include:

• Global dynamic programming to simultaneously solve for displacements and wrap counts (Khodadadi et al., 2021).
• Robust spatial regularizers or bilateral filtering for edge preservation and outlier rejection (Argyriou, 2018).
• Phase amplification for signal magnification, with trade-offs between gain and stability—excessive gain induces destructive interference or ringing (Oshim et al., 2022).

Noise, Multipath, and Hardware Considerations

• Multipath/Specular Reflection: Temporal and spatial bin consistency checks filter out noisy or spurious phase traces (mmWave/radar) (Sen et al., 2024).
• Bandwidth and SNR: Phase-based methods excel at mid-band frequencies (bandpass filtering rejects DC/low-frequency drift and adversarial high-frequency noise) but degrade when local amplitude is low, requiring gain adjustment or filter reparameterization (Prashnani et al., 2022, Oshim et al., 2022).
• Quantum Limitations: In optical detection/quantum regimes, detection inefficiency and finite squeezing bandwidth set ultimate accuracy; optimal smoothing (Wiener or Kalman) approaches quantum Cramér–Rao bounds, and phase-squeezed states enable estimation beyond coherent-state limits (Iwasawa et al., 2013).

6. Canonical Use Cases and Performance Metrics

Substantial empirical validation exists across domains:

• Vibration-based SHM: PME + magnification discriminates modal frequencies and detects damage in wind turbine blades, with non-contact, high-density spatial coverage (Sarrafi et al., 2018).
• Vital sign monitoring: Phase-based radar magnification reduces HR/RR MAE by $20-40\%$ relative to temporal FFT in lab and clinical settings (e.g., MAE for HR in sleep clinic: FFT=7.98, phase-based RF=4.28 bpm) (Oshim et al., 2022).
• Autonomous vehicle odometry: Phase-resolved mmWave radar tracking yields MAE $\sim$1 cm/s, outperforming Doppler-FFT and IMU baselines at sub-Doppler velocities (Sen et al., 2024).
• Gait phase: Transformer-based implicit phase modeling achieves RMSE $2.7\%$ (level) and $3.2\%$ (transitions), outperforming oscillators and LSTM/CNN models, running real-time ($<4$ ms per window) (Ji et al., 18 Jun 2025).
• Quantum optomechanics: Squeezed-state phase estimation approaches/achieves QCRB, with experimental MSE $\sim$15\% below coherent-state limit for position and momentum (Iwasawa et al., 2013).

7. Extensions, Open Problems, and Future Directions

Research efforts are progressing toward:

• Learned filters: End-to-end fused complex-valued filter banks for personalized or posture-adaptive phase decomposition (Oshim et al., 2022).
• Hybrid fusion: Joint optimization over phase, intensity, and regularized flow fields, especially for cases with mixed motion regimes or tissue elasticity (OCT elastography, OCE) (Khodadadi et al., 2021).
• Large motion & occlusion: Multi-hypothesis, occlusion-aware variants, adaptive high-order models (Fourier–Mellin, affine) to extend phase-based methods to scale/rotation/in-plane deformation (Argyriou, 2018).
• Adversarial/deepfake detection: Phase-based motion features for temporal dynamics in faces, robust to adversarial attacks and cross-dataset artifacts (Prashnani et al., 2022).
• Quantum measurement: Real-time phase tracking and smoothing for quantum-limited sensing, with broader implications for gravitational wave detection and nanoscale optomechanics (Iwasawa et al., 2013).

Continued advances in algorithmic robustness, computational efficiency, and fusion with learned representations are expected to further broaden the impact of phase-based motion estimation across scientific, engineering, and clinical domains.