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

Alejandro Garnung Menéndez

eess.IVSep 3, 2026

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

Authors: Alejandro Garnung Menéndez

Organizations: University of Oviedo, Gijón Polytechnic School of Engineering, Asturias, Spain

Abstract

Eulerian video amplification boosts sub-pixel motion by band-pass filtering per-pixel intensity traces and applying a uniform gain. That gain ignores local structure, so sensor noise is amplified together with the signal, especially in textureless regions where the monogenic phase is unreliable. We propose FrAM (Fractional-order Adaptive Motion Magnification), a pipeline developed first offline and then as a causal stream. It replaces the constant temporal gain with a Grünwald--Letnikov derivative of fractional order, giving continuous control over high-frequency emphasis, and replaces the uniform spatial gain with a per-pixel weight derived from the local amplitude of the monogenic signal. On a controlled synthetic sequence split into textured and flat halves, FrAM matches the amplification of the Eulerian baseline while keeping flat-region temporal noise at the input level. The reduction holds across an eightfold range of input noise levels. Real videos show improved spatial selectivity and lower background noise in every case. The causal reformulation cuts the per-frame cost by two orders of magnitude, reaching 69,fps at 640 $\times$ 480.

Figures & tables

Fig. 1: Left: magnitude of the Grünwald–Letnikov response ( 4 ), showing the continuous interpolation between the identity ( $\nu=0$ ) and a first derivative ( $\nu=1$ ). Right: decay of the coefficients $w_{k}$ as a function of memory lag $k$ ; smaller $\nu$ implies a longer effective memory. Here $k$ indexes past frames in the GL sum and measures the operator’s temporal reach, not a group-delay or control-loop reaction lag.

Fig. 2: Motivation for the adaptive gain. The input (left) is textured on the left half and flat on the right. Local phase (third panel) is noise in the flat region, where amplitude vanishes; the reliability mask ( 6 ) (right) identifies this without supervision.

Panel of Figure 3 — Fig. 3: End-to-end data flow of FrAM for one pyramid band $\ell$ . The loop over bands runs independently for every scale, and the low-pass residual bypasses it completely. After the temporal stack $S$ the pipeline splits into a temporal/frequency branch that produces the fractionally differentiated signal $\tilde{S}_{t}$ , and a spatial branch computed once from the band’s temporal mean that produces the static gain map $\alpha_{\text{eff}}$ ; the two are fused with the original unfiltered band. Stages with both offline and causal realisations are annotated with both modes (Sec. VII, Appendix A ).

Method	$\alpha$	Amplif.	Noise	Noise	SNR
			(flat)	(textured)	(dB)
Input (unprocessed)	–	1.01 $\times$	0.0565	0.0569	8.86
EVM (Butterworth)	20	18.44 $\times$	0.1137	0.1589	11.99
FrAM ( $\nu=0.0$ )	42	17.03 $\times$	0.0574	0.1525	11.89
FrAM ( $\nu=0.3$ )	64	16.09 $\times$	0.0574	0.1589	11.91
FrAM ( $\nu=0.5$ )	90	14.83 $\times$	0.0575	0.1733	11.74

TABLE I: Comparison with

\alpha

calibrated toward equal effective amplification (EVM

18.44\times

\sigma_{\text{in}}=0.04

. FrAM falls slightly short of the EVM factor because the reliability mask withholds gain in the flat half; lower noise is better. Bold marks the better value of each metric column among the processed methods (noise, SNR); the amplification serves as the calibration target, and the unprocessed input is listed for reference only.

Order	Gain	Amplif.	Noise (flat)	Noise (textured)
$\nu=0.0$	constant	18.44 $\times$	0.1137	0.1589
$\nu=0.0$	adaptive	8.75 $\times$	0.0569	0.0925
$\nu=0.5$	constant	10.05 $\times$	0.0935	0.1188
$\nu=0.5$	adaptive	4.11 $\times$	0.0567	0.0723
$\nu=0.7$	constant	6.70 $\times$	0.0876	0.1072
$\nu=0.7$	adaptive	2.81 $\times$	0.0566	0.0679

TABLE II: Adaptive-gain ablation,

\alpha=20

fixed. Within each order, bold marks the gain with the lower noise in each noise column. At fixed

\alpha

the adaptive gain always lowers noise and always lowers amplification; matching amplification requires recalibrating

\alpha

(Table I ).

$\sigma_{\text{in}}$	EVM		FrAM
	Amplif.	Noise (flat)	Amplif.	Noise (flat)
0.01	19.15 $\times$	0.0284	16.81 $\times$	0.0142
0.02	18.98 $\times$	0.0569	16.64 $\times$	0.0284
0.04	18.44 $\times$	0.1137	16.09 $\times$	0.0574
0.08	16.64 $\times$	0.2275	14.36 $\times$	0.1207

TABLE III: Robustness to input noise, at matched amplification,

\nu=0.3

. Bold marks the lower flat-region noise of the two methods on each row; the amplification columns record the calibration target, not a quantity to maximise.

Fig. 4: Comparison at equal effective amplification. Top: spatio-temporal slices; right of the dashed line the region is flat, and the baseline has filled it with spurious texture that FrAM leaves untouched. Bottom left: injected noise per column, dropping sharply at the boundary for FrAM. Bottom centre: both methods track the true oscillation without phase lag. Bottom right: the trade-off controlled by $\nu$ .

Video	Method	BER	Noise	Spatial	Rate
			(bg)	select.	(bpm)
face	input	0.136	0.00233	1.00	50
face	EVM	0.825	0.00909	1.40	50
face	FrAM	0.844	0.00302	4.53	50
face2	EVM	0.332	0.00702	1.14	50
face2	FrAM	0.244	0.00662	1.31	70

TABLE IV: Real video, at equal enhancement in the active region. Within each video, bold marks the better of EVM and FrAM on each metric (higher BER, lower background noise, higher spatial selectivity); EVM retains slightly higher BER on three of the four clips. The unprocessed input is shown once as a reference, and the pulse rate is a consistency check rather than a quantity to maximise.

Video	Activity	Noise (bg)	Noise (bg)	Gain
	contrast	EVM	FrAM
subway	6.04	0.02026	0.00780	$2.60\times$
face2	3.77	0.00702	0.00662	$1.06\times$
face	1.98	0.00909	0.00302	$3.01\times$
baby	1.07	0.00468	0.00246	$1.90\times$

TABLE V: Activity contrast of the input against the observed advantage. Bold marks the lower background noise of the two methods on each row; the gain column is the ratio of the two noise columns. The ordering is not monotonic: contrast does not predict the margin.

Fig. 5: Real-video validation. Columns: unsupervised region masks; temporal spectrum of the active region with the target band shaded; temporal noise per region; and the difference map of injected noise (red where FrAM injects less). Red dominates on all four sequences, most clearly for face and subway .

Resolution	Offline	Streaming	Speed-up	fps
$320\times 240$	382 ms	6.3 ms	$60\times$	158
$640\times 480$	2430 ms	14.6 ms	$\mathbf{166\times}$	69
$1280\times 720$	7772 ms	62.6 ms	$124\times$	16
$1920\times 1080$	17937 ms	108.8 ms	$165\times$	9

TABLE VI: Per-frame cost: offline versus causal streaming. Streaming is cheaper at every resolution tested; bold highlights the

640\times 480

row, the reference configuration quoted elsewhere in this paper.

Method	$\alpha$	Amplif.	Noise (flat)	Noise (text.)
Input	–	1.01 $\times$	0.0565	0.0569
EVM (streaming)	20	20.85 $\times$	0.1605	0.2002
FrAM ( $\nu=0.0$ )	33	20.08 $\times$	0.1091	0.1827
FrAM ( $\nu=0.15$ )	39	19.78 $\times$	0.1149	0.1887
FrAM ( $\nu=0.3$ )	47	19.36 $\times$	0.1250	0.2026
FrAM ( $\nu=0.5$ )	64	18.47 $\times$	0.1480	0.2375

TABLE VII: Streaming, at equal effective amplification. Bold marks the lowest noise of each column among the processed methods; the unprocessed input is listed for reference only.

$K$	Amplif.	Noise (flat)	ms/frame
4	8.17 $\times$	0.0779	7.8
8	8.94 $\times$	0.0783	11.2
16	8.78 $\times$	0.0781	12.1
24	8.93 $\times$	0.0782	15.0
31	8.82 $\times$	0.0781	17.9

TABLE VIII: Truncating the fractional memory (

640\times 480

). Amplification and flat-region noise vary little for

K\geq 8

, while cost grows with

K

; bold highlights the truncation

K=8

used by the interactive viewer.

Fig. 6: Causal reformulation. Left: per-frame cost, log scale, with the $30$ fps threshold marked. Centre: truncating the fractional memory costs little amplification while cost grows with $K$ . Right: at equal amplification in streaming, the advantage over the baseline is retained.

Noise Robustness

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