measure op• Data kinds: video → table
• Call: import fullseye as fs; fs.ledger.phase_displacement(video, f_lo, f_hi, fps, scales: 'int' = 4, orientations: 'int' = 4) -> 'dict' (to call the implementation directly, import motionmag; motionmag.phase_displacement(video, f_lo, f_hi, fps, scales: 'int' = 4, orientations: 'int' = 4) -> 'dict'; from the registry, opsmotionmag.get("phase_displacement"))
Sub-pixel displacement field from local phase -> dict.
The quantitative sibling of motion_magnify: nothing is amplified and nothing is re-rendered, the displacement itself is returned in pixels.
For each oriented sub-band, the temporal phase deviation dphi(t) (taken against the band's temporal mean as the wrapped angle in (-pi, pi] — never unwrapped in time, for the reason given in the design note above _AMP_FLOOR — then band-passed) obeys dphi = -(kx*dx + ky*dy), where (kx, ky) is the band's local spatial frequency in radians per pixel — computed exactly as Im(conj(z) grad z)/|z|^2 with a spectral derivative, not from the band's nominal centre. Each band contributes one linear constraint on the same two unknowns, so the bands are combined per pixel by weighted least squares with weights |z|^2 (a band with no contrast gets no vote).
Returns {"dx": (T, H, W), "dy": (T, H, W), "weight": (H, W), "valid": (H, W) bool, "rank": (H, W) int8, "fps": ..., "band_hz": ..., "wrap_limit_px": ..., "reference_coherence": ...}. dx/dy follow flow: dx is column motion, dy row motion, positive towards increasing index, and both are zero-mean along time because the band-pass removed DC.
rank says how much of the 2-D displacement the data could constrain at each pixel: 2 = both components, 1 = the aperture problem (every band with contrast there shares one orientation, so only the component along it is observable — that component is returned and the unobservable direction is exactly 0), 0 = no contrast at all, both zero. Measured on a purely horizontal grating moving 0.3 px horizontally, every pixel is rank 1 and the answer is dx = 0.3000000000000001, dy = 0.0 exactly.
Accuracy and where it stops (measured, 64x64x64 clip at 32 fps, 8 px horizontal grating, 4 Hz bin-centred, noiseless, defaults; the error is on the peak of the recovered waveform):
============== ============== ============== ===================
true d (px) k*d (rad) measured (px) relative error
============== ============== ============== ===================
0.001 0.0008 0.00100000 8.7e-15
0.010 0.0079 0.01000000 3.1e-15
0.100 0.0785 0.10000000 1.8e-15
0.500 0.3927 0.50000000 6.7e-16
1.000 0.7854 1.00000000 4.4e-16
2.000 1.5708 2.00000000 6.7e-16
3.000 2.3562 3.00000000 5.9e-16
3.050 2.3954 3.05000000 2.9e-16
3.100 2.4347 1.72842712 4.4e-01 <- broken
4.000 3.1416 2.10461396 4.7e-01 <- broken
6.000 4.7124 2.35441757 6.1e-01 <- broken
============== ============== ============== ===================
Exact to rounding — and then it stops, abruptly, between 3.05 and 3.10 px. That boundary is closed-form, and it is *not* the naive phase-wrap bound. The temporal phase reference is the band's temporal mean z_mean, and for a sinusoidal displacement of amplitude A that mean equals c * J0(k*A) — a Bessel function. At k*A = 2.4048, the first zero of J0, the reference amplitude passes through zero and its phase flips by pi; every deviation measured against it is then wrong by pi. For the 8 px grating that is A = 2.4048/(2*pi/8) = 3.0619 px, which is exactly where the table breaks. reference_coherence in the return is |z_mean| / mean_t|z|, i.e. |J0(k*A)| blended over the bands, and it falls monotonically from 1.00000 at 0.001 px to 0.50252 at the 3.05 px edge — a runtime warning that the reference is going degenerate.
Beyond that sits the harder bound |k*d| < pi (half a band wavelength, reported as wrap_limit_px from the measured local frequencies), which no phase-based method can pass with a single band.
Under noise the accuracy is set by the noise, not by the method: on the same clip with A = 0.5 px, additive sigma = 0.001 / 0.01 / 0.05 gives relative errors 2.2e-04 / 1.8e-03 / 1.9e-03.
• motion_magnification family guide
• Sample-data catalog (download URLs / licences) — 2-D uses skimage.data (BSD/public domain) plus synthetic images; 3-D lists download URLs for real data sources (Stanford, PDS, …).
• Operator provenance and references — the sources of the research/methods this op family came from.
• The canonical algorithm (author, year) and its uses are named in the family usage guide above.
• motion_magnification — py -3.11 examples/motion_magnification.py
• poc_beam_modal_video — py -3.11 examples/poc_beam_modal_video.py
• poc_motion_magnification — py -3.11 examples/poc_motion_magnification.py
• quaternion_monogenic — py -3.11 examples/quaternion_monogenic.py
table as input)measure)*Provenance: motionmag.py — MOTIONMAG operator registry. This per-op note is generated by tools/opdocs.py md (do not hand-edit).*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.