Fractional-Order Adaptive Motion Magnification: Phase-Reliability Weighting for Noise-Constrained Video Amplification

A. Amplitude-based magnification

• Eulerian magnification avoids point tracking by filtering intensity traces at fixed pixels, after decomposing frames into spatial-frequency bands.The band-pass traces are multiplied by a gain and reconstructed.Source: §I · p. 1

I. INTRODUCTION

• Uniform Eulerian gain amplifies sensor noise and genuine motion equally, making textureless regions prone to temporal mottling.Noise across the spectrum means some noise always falls inside the temporal pass-band.Source: §I · p. 1

B. Phase-based magnification

• Phase-based methods, shift structure without changing contrast, but phase becomes unreliable where monogenic amplitude approaches zero.Amplifying that phase can reintroduce noise in low-amplitude regions.Source: §I · p. 1

I. INTRODUCTION

• FrAM combines a fractional-order temporal filter with a per-pixel gain weighted by local monogenic amplitude, reducing amplification where phase is uninformative.Its amplitude-percentile cutoff adapts to band content without manual tuning.Source: §I · p. 1

C. Fractional calculus in image processing

• Applying fractional differentiation along video time extends prior fractional image-processing work, which was almost entirely spatial.Earlier applications include denoising, edge detection, and texture enhancement,.Source: C. Fractional calculus in image… · p. 2

A. The Gr¨unwald–Letnikov operator

• The Gr¨unwald–Letnikov order continuously controls high-frequency emphasis, interpolating from identity at ν = 0 to a first temporal difference at ν = 1.The response magnitude and coefficient memory vary with order; smaller ν implies longer effective memory.Source: A. The Gr¨unwald–Letnikov operator · p. 2, Figure 1 · p. 2
• Gr¨unwald–Letnikov weights decay algebraically, so temporal memory has no sharp cutoff and grows as ν decreases.The offline implementation estimates a truncation length from a relative tolerance; it uses K = 503 at ν = 0.2 and K = 124 at ν = 0.9.Source: A. The Gr¨unwald–Letnikov operator · p. 2

B. Fractional band-pass filter

• A second-order Butterworth band-pass restricts each band to [flo, fhi] before fractional differentiation, preventing direct processing of broadband out-of-band noise.Zero-phase forward–backward filtering avoids group delay.Source: B. Fractional band-pass filter · p. 2
• Unlike prior pipelines, which implicitly fix the temporal operator to identity, FrAM makes spectral shaping continuously tunable through ν.Increasing ν progressively emphasizes the upper part of the pass-band.Source: B. Fractional band-pass filter · p. 3

C. Local amplitude and phase

• For each pyramid band, central differences approximate the two spatial Riesz components used to derive monogenic local amplitude and phase.Amplitude measures local contrast, while phase encodes the spatial position of structure.Source: C. Local amplitude and phase · p. 3

D. Phase-reliability adaptive gain

• The reliability weight uses band amplitude with a 75th-percentile threshold, adapting to band content rather than a hand-tuned global constant.The amplitude is computed from the temporal mean of each band to avoid noise-driven mask fluctuation and flicker.Source: D. Phase-reliability adaptive gain · p. 3
• Reliability-dependent shrinkage recovers nominal gain α for reliable phase, suppresses gain for unreliable phase, and applies a larger penalty there as ν increases.At ν = 0, the gain reduces to the plain weighting αρ.Source: D. Phase-reliability adaptive gain · p. 3

E. Algorithm

• Each non-residual pyramid band is temporally filtered, weighted by a static gain map from its temporal-mean amplitude, and fused with the original band; the low-pass residual bypasses processing.The algorithm returns the collapsed pyramid frames.Source: E. Algorithm · p. 3, Figure 3 · p. 4

A. Why a synthetic sequence

• The 96-frame synthetic sequence combines a 0.2 px, 2 Hz oscillation in a textured half with a flat half and additive Gaussian noise.The 2 Hz oscillation lies at the centre of the experimental [1.5, 2.5] Hz temporal pass-band.Source: A. Why a synthetic sequence · p. 3

B. Metrics

• Amplification is the fitted slope between estimated and ground-truth displacement, while inter-frame differences in the flat region measure temporal noise.The unprocessed-input amplification estimate is 1.01; band SNR measures how much recovered-motion energy falls near the 2 Hz signal frequency.Source: B. Metrics · pp. 3–5

• FrAM’s gain is scaled toward the baseline amplification using the ratio of baseline amplification to a probe run.The protocol accounts for the pipeline’s linearity in α under a spatially uniform gain.Source: C. Protocol · p. 5
• The reliability mask can leave FrAM’s measured amplification slightly below the baseline because flat pixels receive little gain.Source: C. Protocol · p. 5
• Matching effective amplification makes noise comparisons fairer than comparing methods that simply amplify by different amounts.Source: C. Protocol · p. 5

A. Main result

• Flat-region noise is 0.0574 for FrAM, near the input’s 0.0565 and below Eulerian’s 0.1137; FrAM reaches 17.0× amplification at ν = 0 versus Eulerian’s 18.4×.The residual amplification gap is attributed to withholding gain where there is no structure to amplify.Source: A. Main result · p. 5
• At ν = 0.3, FrAM matches Eulerian’s textured-region noise at 0.1589; higher ν increases that noise, making ν ∈[0, 0.3] the useful operating range.The account links the increase to greater emphasis on high frequencies, where noise dominates.Source: A. Main result · p. 5

B. Ablation

• The ablation attributes input-level flat-region noise to adaptive gain, while fractional order controls the amplification–noise trade-off in textured regions.With constant gain, increasing ν reduces noise but lowers amplification from 18.44× to 6.70×.Source: B. Ablation · p. 5, Table II · p. 5

C. Robustness

• Across an eightfold range of input noise, FrAM roughly halves flat-region noise while retaining approximately 87% of baseline amplification.The figure’s displacement traces show no phase lag for either method under zero-phase filtering.Source: C. Robustness · p. 5, Table III · p. 5, Figure 4 · p. 6

VI. VALIDATION ON REAL VIDEO

• Real-video validation uses four reference sequences covering facial pulse, infant respiration, and structural vibration.The clips are processed as single grayscale channels at reduced resolution, using the first 180 frames.Source: §VI · pp. 5–6

A. Evaluation without ground truth

• Real-video evaluation replaces ground-truth amplification with physiological band energy and pulse rate, noise in static regions, and active-to-static enhancement ratio.Active and static masks are thresholded from the original video and held fixed across methods; unity selectivity indicates indiscriminate amplification.Source: A. Evaluation without ground truth · p. 6

B. Results

• On face, background noise falls 3.0× (0.00302 versus 0.00909), while BER is 0.844 versus 0.825 and spatial selectivity rises from 1.40 to 4.53.Methods are compared at equal active-region enhancement.Source: Table IV · p. 6, B. Results · pp. 6–7, A. Evaluation without ground truth · p. 6
• FrAM raises spatial selectivity on all four sequences and reduces background noise on each, though the noise-reduction margin ranges from 3.0× on face to 1.06× on face2.On subway, background noise falls by 2.6× and selectivity rises from 1.16 to 3.19.Source: B. Results · pp. 6–7, Table IV · p. 6
• The four clips do not support a predictor of the advantage; face2’s low in-band energy is only a plausible explanation for its smaller gain, not a validated applicability rule.Source: B. Results · p. 7

A. Diagnosis of the cost

• The offline sliding-window pipeline takes over 2 seconds per 640×480 frame, with the temporal filter accounting for about 1.95 seconds.Laplacian pyramids take about 90 ms and pyramid collapse about 50 ms for T = 32 frames.Source: A. Diagnosis of the cost · p. 7

B. Causal reformulation

• Reprocessing the window, filtering forward and backward, and shifting K full-volume copies are implementation costs, not requirements of fractional calculus.The causal version uses a state recursion, a circular buffer, and an exponential moving average; it introduces group delay and about 40 transient frames.Source: A. Diagnosis of the cost · p. 7, B. Causal reformulation · p. 7

C. Results

• 166× speed-up at 640×480 reaches 69 fps, with streaming crossing 30 fps at QVGA and VGA.The reported timings are from a single run; the authors regard the order-of-magnitude gap between modes as more dependable than any individual ratio.Source: Table VI · p. 7, C. Results · p. 7
• K = 8 gives the same amplification as K = 31 at roughly 60% of its cost in the reported run.The far tail contributes little to the measured result, despite the operator's long-memory spectral character.Source: C. Results · pp. 7–8, Table VIII · p. 9
• Across four real-video sequences, FrAM injects less noise in the difference maps, most clearly for face and subway.The figure also presents unsupervised region masks, active-region spectra, and temporal noise per region.Source: Figure 5 · p. 8

VIII. LIMITATIONS

• The noise improvement is attributed to adaptive gain, while the spatial phase estimate uses a first-order Riesz approximation and one global ν across bands.A full steerable filter could estimate phase more accurately; per-band order is proposed as a natural extension.Source: §VIII · pp. 8–9
• The background-noise reduction varies from 1.06× to 3.01× across four clips, and the authors cannot determine which scenes benefit most.Input activity contrast does not predict the observed range, and four clips are too few to establish a criterion.Source: §VIII · p. 8

C. Results

• Flat-region noise falls from 0.1605 to 0.1091 in streaming, a 1.47× reduction versus 2× offline.The causal mask is noisier because it lacks the full-window average; the paper cautions that its other FrAM figures are offline results, not streaming numbers.Source: C. Results · p. 7, Table VII · p. 9, §VIII · p. 9

IX. CONCLUSION

• FrAM combines a fractional-order Grunwald–Letnikov temporal filter with gain weighted by local phase reliability.The reliability weighting is independent of the rest of the pipeline and could be applied to existing phase-based magnification methods.Source: §IX · p. 9
• On synthetic video, FrAM approaches Eulerian amplification while holding textureless-region noise at input level, where the baseline doubles it, across an eightfold input-noise range.On real sequences, background noise falls by 3.0× on the primary facial-pulse clip and 2.6× on the structural-vibration clip; in-band energy and pulse rate are preserved, and spatial selectivity improves on every sequence.Source: §IX · p. 9
• The causal formulation cuts per-frame cost 166×, reaching 69 fps in grayscale and 54 fps in colour at 640×480, while retaining a 1.47× noise reduction over baseline.Truncating fractional memory to K = 8 costs no measurable amplification in the reported results.Source: §IX · p. 9
• The implementation uses NumPy, SciPy, OpenCV, and Matplotlib, and the paper's code and result videos are publicly available.The real-time colour stream magnifies only the Y channel after conversion to YCrCb.Source: REPRODUCIBILITY · pp. 9–10

APPENDIX A IMPLEMENTATION OF THE CAUSAL FORMULATION

• The appendix details implementation of the causal formulation; the operator remains the one defined in the main method.It addresses realization rather than a change to the method.Source: Appendix A · p. 10

A. Band-pass as a state recursion

• The band-pass is realized as a biquad because the lowpass-to-bandpass substitution doubles the prototype order.For a first-order prototype, the discrete transfer function has two poles and two zeros, and therefore three numerator and three denominator coefficients.Source: A. Band-pass as a state recursion · p. 10
• A state recursion retaining two past inputs and outputs reproduces a forward pass of lfilter to floating-point precision, with maximum deviation 6 × 10^-16.The causal mode differs from offline filtfilt, which filters forward and backward to achieve zero phase; streaming therefore introduces group delay.Source: A. Band-pass as a state recursion · p. 10

B. Fractional sum over a circular buffer

• The truncated fractional sum needs the current band-pass output and its K most recent predecessors, each paired with its corresponding weight.A naive sliding-window implementation shifts all K + 1 frames at every step.Source: B. Fractional sum over a circular buffer · p. 10
• A circular buffer stores K + 1 frames in fixed slots and uses wrap-around indexing to locate each past frame without copying data between slots.The pointer is advanced modulo K + 1 after each new frame is written.Source: B. Fractional sum over a circular buffer · p. 10
• The buffer's weighted sum pairs the newest image with w0 and older images with successive weights, replacing K shifts of the full volume.At 640×480 with K = 31, the incremental weighted sum costs a few milliseconds per frame.Source: B. Fractional sum over a circular buffer · p. 11

C. Incremental reliability mask

• The causal mask replaces unavailable temporal-mean amplitude with an EMA, refreshes and smooths its percentile threshold every eighth frame, and uses Sobel quadrature smoothing to stabilize noisier instantaneous bands.The transverse smoothing attenuates noise before amplitude formation; mask computation costs sub-millisecond per frame at 640×480.Source: C. Incremental reliability mask · p. 11

D. Numerical precision and colour

• 0.31 ms per frame for float32 versus 1.78 ms for float64, with no perceptible output difference; the reported gain is bandwidth, not arithmetic.All state and arithmetic use single precision.Source: D. Numerical precision and colour · p. 11
• The colour variant processes luminance only, leaving chrominance untouched apart from optional saturation; 18.5 ms for colour versus 14.6 ms for grayscale falls within run-to-run timing noise.The pipeline is applied to the luminance channel because magnification acts on geometry rather than colour.Source: D. Numerical precision and colour · p. 11

E. Parameter invalidation

• Scalar amplification and adaptation controls can change between frames without rebuilding state, while changes to filter or buffer-shape parameters require rebuilding it.The viewer keys state on a parameter tuple, so interactive changes to the scalar group are free.Source: E. Parameter invalidation · p. 11