Phase-Sequential Binomial Self-Compensation

• Phase-Sequential Binomial Self-Compensation (P-BSC) is a temporal error-suppression method for PSP that uses binomial weighting on phase frames to exponentially reduce harmonic errors.
• The method applies Pascal’s triangle coefficients to successive motion-affected phase frames, canceling double-frequency sinusoidal errors without needing external motion tracking.
• P-BSC enables quasi-single-shot dynamic 3D scanning, though higher binomial orders increase computational cost and noise integration.

Searching arXiv for the specified papers and closely related work to ground the article. Phase-Sequential Binomial Self-Compensation (P-BSC) is a temporal error-suppression method for dynamic phase-shifting profilometry (PSP) that operates directly on a sequence of motion-affected wrapped phase frames. In PSP, high-precision 3D scanning is achieved by projecting sinusoidal fringe patterns with known phase shifts and recovering phase per pixel, but the standard formulation assumes that the object remains static during acquisition. In dynamic measurement this assumption is violated, so the phase recovered from multi-frame PSP contains a motion-induced error comprising a DC component and a harmonic component at twice the fringe frequency, the latter appearing as ripple-like artifacts in reconstructed point clouds. P-BSC addresses this problem by summing successive motion-affected phase frames weighted by binomial coefficients, in a pixel-wise and frame-wise loopable manner, so that the harmonic motion error diminishes exponentially as the binomial order increases (Zhang et al., 2024, Zhang et al., 14 Jul 2025).

1. Dynamic PSP error model

In PSP, the projector emits sinusoidal fringe patterns with known phase shifts, and the camera records several images from which phase is demodulated. For an $N$-step equal-phase-shift scheme, the ideal intensity at pixel $(x,y)$ in the $i$-th captured frame is modeled as

$$I_i(x,y) = A(x,y) + B(x,y)\cos\!\left[\phi_0(x,y) - \frac{2\pi i}{N}\right],$$

where $A(x,y)$ is background intensity, $B(x,y)$ is modulation, and $\phi_0(x,y)$ is the underlying object phase at a datum frame (Zhang et al., 14 Jul 2025). Under the static-object assumption, the same surface point is imaged at the same pixel across all $N$ frames, and the wrapped phase at frame index $i$ is obtained by the standard phase-shifting formula

$$\tilde{\phi}_i(x,y) = \tan^{-1}\!\left[ \frac{\displaystyle\sum_{n=0}^{N-1} I_{i+n}(x,y)\,\sin\left(\tfrac{2\pi n}{N}\right)}{\displaystyle\sum_{n=0}^{N-1} I_{i+n}(x,y)\,\cos\left(\tfrac{2\pi n}{N}\right)} \right].$$

For dynamic scenes, the measured intensity is modeled by introducing a small motion-induced temporal phase offset $(x,y)$0. In the $(x,y)$1-shift setting emphasized by the BSC papers, for camera pixel $(x,y)$2,

$(x,y)$3

where $(x,y)$4 is the true phase at frame $(x,y)$5, and $(x,y)$6 is the unknown phase offset due to object motion between the reference and frame $(x,y)$7 (Zhang et al., 2024). The motion sequence $(x,y)$8 is treated pixel-wise, and its discrete differences are defined by

$(x,y)$9

Applying standard PSP demodulation to motion-affected intensities and assuming small motion, so that $i$0 and $i$1, yields a phase error consisting of two components: a DC component, which appears as an overall lag or shift of the reconstructed surface, and a harmonic component proportional to $i$2, which manifests as the visible ripple-like distortion on the reconstructed depth (Zhang et al., 14 Jul 2025). The BSC framework explicitly targets suppression of this harmonic term.

For the important $i$3 case with $i$4 phase shifting, the motion error simplifies to

$i$5

The alternating sign $i$6 in the harmonic term is the structure exploited by binomial self-compensation (Zhang et al., 14 Jul 2025).

2. Definition of P-BSC

P-BSC computes a temporal sequence of motion-affected wrapped phase frames $i$7 using cyclically projected $i$8 phase-shifted patterns and then forms a binomially weighted sum over successive phase frames: $i$9 where $$I_i(x,y) = A(x,y) + B(x,y)\cos\!\left[\phi_0(x,y) - \frac{2\pi i}{N}\right],$$0 is the binomial order, $$I_i(x,y) = A(x,y) + B(x,y)\cos\!\left[\phi_0(x,y) - \frac{2\pi i}{N}\right],$$1 is the raw motion-affected phase, and $$I_i(x,y) = A(x,y) + B(x,y)\cos\!\left[\phi_0(x,y) - \frac{2\pi i}{N}\right],$$2 is the $$I_i(x,y) = A(x,y) + B(x,y)\cos\!\left[\phi_0(x,y) - \frac{2\pi i}{N}\right],$$3-th order compensated phase (Zhang et al., 2024, Zhang et al., 14 Jul 2025).

The method is called phase-sequential because the binomial operation is applied in the phase-frame domain: phase is first computed from images, and the resulting phase frames are then combined in time. It is called self-compensation because it uses only the motion-affected phase sequence itself; no external motion model, tracking, estimated motion phase shift, or other intermediate variables are required (Zhang et al., 2024, Zhang et al., 14 Jul 2025).

Operationally, the binomial weights are taken from Pascal’s, or Yang Hui’s, triangle. The recursive neighbor-summation structure produces coefficient rows such as $$I_i(x,y) = A(x,y) + B(x,y)\cos\!\left[\phi_0(x,y) - \frac{2\pi i}{N}\right],$$4, $$I_i(x,y) = A(x,y) + B(x,y)\cos\!\left[\phi_0(x,y) - \frac{2\pi i}{N}\right],$$5, and $$I_i(x,y) = A(x,y) + B(x,y)\cos\!\left[\phi_0(x,y) - \frac{2\pi i}{N}\right],$$6, and the final result is exactly the binomial-weighted combination above (Zhang et al., 2024). This gives P-BSC a direct interpretation as a temporal binomial filter on the phase sequence.

A central distinction introduced in later analysis is between phase-sequential BSC and image-sequential BSC. In phase-sequential BSC, the binomial combination is applied to phase frames; in image-sequential BSC, the same idea is applied to fringe images before phase calculation. This distinction is important because the later image-sequential formulation was proposed specifically to reduce the computational overhead and error accumulation associated with P-BSC, while preserving the same theoretical motion-error suppression capability for the 4-step case (Zhang et al., 14 Jul 2025).

3. Mathematical mechanism and exponential suppression

For 4-step PSP, substitution of the motion-affected phase into the binomial operator yields

$$I_i(x,y) = A(x,y) + B(x,y)\cos\!\left[\phi_0(x,y) - \frac{2\pi i}{N}\right],$$7

where

$$I_i(x,y) = A(x,y) + B(x,y)\cos\!\left[\phi_0(x,y) - \frac{2\pi i}{N}\right],$$8

collects the desired phase component and the DC motion terms (Zhang et al., 14 Jul 2025). In the formulation reported in the 2024 paper, the same structure appears with the residual harmonic term multiplied by $$I_i(x,y) = A(x,y) + B(x,y)\cos\!\left[\phi_0(x,y) - \frac{2\pi i}{N}\right],$$9 and high-order temporal differences of the motion offsets (Zhang et al., 2024).

For the modified 3-step $A(x,y)$0 case, the residual error is more involved: $A(x,y)$1 but it still has the same defining structure: exponential scaling in $A(x,y)$2 and dependence on high-order differences of the motion sequence (Zhang et al., 14 Jul 2025).

The mechanism of suppression has two parts. First, the prefactor $A(x,y)$3 decays exponentially with $A(x,y)$4. Second, under smooth temporal motion, the high-order finite differences $A(x,y)$5 quickly tend toward zero as $A(x,y)$6 grows. If motion is polynomial in time of degree $A(x,y)$7, then $A(x,y)$8 for $A(x,y)$9 (Zhang et al., 14 Jul 2025). This is the formal basis for the statement that summing successive motion-affected phase frames weighted by binomial coefficients causes motion error to diminish exponentially as the binomial order increases (Zhang et al., 2024).

The same papers also provide an intuitive explanation. Adjacent phase frames differ by a $B(x,y)$0 shift in the true phase, which flips the sign of the harmonic coefficients in the motion-error term. Summing successive frames therefore tends to cancel the double-frequency sinusoidal error, and recursively repeating that process produces the binomial coefficients of Yang Hui’s triangle (Zhang et al., 2024). A plausible implication is that P-BSC can be regarded as a temporal polynomial-annihilation filter acting on the motion-induced phase offsets, while preserving the desired phase contribution (Zhang et al., 14 Jul 2025).

4. Sequential implementation and wrapped-phase accumulation

P-BSC is defined for cyclic $B(x,y)$1 phase-shifting PSP, using either the modified 3-step $B(x,y)$2 scheme or the standard 4-step $B(x,y)$3 scheme (Zhang et al., 14 Jul 2025). For temporal index $B(x,y)$4, the raw wrapped phase is computed as

$B(x,y)$5

for the 3-step $B(x,y)$6 scheme, and

$B(x,y)$7

for the 4-step $B(x,y)$8 scheme (Zhang et al., 14 Jul 2025). Because the patterns are cycled, any sliding window of 3 or 4 consecutive images yields one wrapped phase frame, so the PSP phase extraction itself is frame-wise loopable (Zhang et al., 14 Jul 2025).

To align the temporal phase sequence to a common reference, the constant phase shift is corrected by

$B(x,y)$9

P-BSC then performs binomial accumulation in the phase domain. Because direct addition of wrapped phases is unstable at $\phi_0(x,y)$0 discontinuities, the method uses a custom phase-addition operator $\phi_0(x,y)$1. In the 2024 formulation this operator is defined as

$\phi_0(x,y)$2

which corrects wrap-order discrepancies when $\phi_0(x,y)$3 and reduces to $\phi_0(x,y)$4 otherwise (Zhang et al., 2024). The 2025 mechanism paper presents the same idea as a recursive pyramid accumulation designed to handle wrapping automatically (Zhang et al., 14 Jul 2025).

The recursive update is

$\phi_0(x,y)$5

After $\phi_0(x,y)$6 layers, the final output is the compensated phase $\phi_0(x,y)$7 or, in sliding-window operation, the compensated phase corresponding to the appropriate temporal index (Zhang et al., 2024, Zhang et al., 14 Jul 2025).

This sequential structure makes P-BSC inherently sliding-window and frame-wise loopable. For each image acquisition time, one depth map can be reconstructed using a sliding buffer of $\phi_0(x,y)$8 images; once the initial warm-up has passed, one output phase frame is produced per new input frame (Zhang et al., 2024). The reported latency is $\phi_0(x,y)$9 frames, but the output depth-map frame rate is not reduced below the camera frame rate (Zhang et al., 2024).

5. Computational properties and relation to I-BSC

The later mechanism paper identifies two limitations of P-BSC: high computational overhead and error accumulation (Zhang et al., 14 Jul 2025). P-BSC computes $N$0 phase frames per output, and each phase frame requires one arctangent per pixel. In addition, the binomial pyramid requires arithmetic, comparison, and modular operations whose count grows quadratically with $N$1.

The reported per-pixel operation counts for one output phase are summarized below (Zhang et al., 14 Jul 2025).

Operation	P-BSC	I-BSC
$N$2	$N$3	1
mod	$N$4	0
$N$5	$N$6	$N$7
$N$8	$N$9	$i$0
cmp	$i$1	0

Accordingly, the dominant growth for P-BSC is $i$2 for an image with $i$3 pixels, whereas the image-sequential variant I-BSC reduces this to roughly $i$4 (Zhang et al., 14 Jul 2025). The same paper states that I-BSC reduces the computational complexity by one polynomial order and accelerates the computational frame rate by several to dozen times (Zhang et al., 14 Jul 2025).

The second limitation is error accumulation. The analytical derivation of P-BSC relies on small-motion linearization and on approximating the arctangent. In P-BSC, this approximation is effectively used repeatedly across multiple phase frames, so residual terms accumulate, and practical convergence can be slower than the ideal theoretical formula suggests (Zhang et al., 14 Jul 2025). By contrast, I-BSC performs the binomial weighting in the image domain and computes the arctangent only once, which resolves both limitations in P-BSC according to the 2025 paper (Zhang et al., 14 Jul 2025).

This does not negate the significance of P-BSC. Rather, it situates P-BSC as the phase-domain formulation from which the generalized image-domain variant was developed. The 2025 paper explicitly states that I-BSC is inspired by P-BSC and that the two formulations are theoretically identical, up to sign, in their asymptotic motion-error suppression capability for the 4-step case (Zhang et al., 14 Jul 2025).

6. Experimental behavior, performance, and trade-offs

The experiments reported across the two papers use a dynamic structured-light system with two AVT 1800U-120c cameras, cropped to $i$5, and a TI DLP4500 projector, with cameras synchronized to the projector and the camera frame rate at approximately 90 fps (Zhang et al., 14 Jul 2025). The temporal scheme is cyclic projection of $i$6-shifted high-frequency fringes, together with paraxial stereo phase unwrapping so that only one high-frequency set is needed (Zhang et al., 14 Jul 2025).

A central reported property is quasi-single-shot operation. Because of cyclic patterns and frame-wise loopable processing, the output depth-map frame rate equals the camera acquisition rate, for example 90 fps, even though each depth map internally uses $i$7 images (Zhang et al., 2024, Zhang et al., 14 Jul 2025). The method is therefore quasi-single-shot rather than true single-shot.

In dynamic experiments, BSC is compared with traditional 4-step PSP, Hilbert-transform compensation (HTC), $i$8-FTP, phase frame difference (PFD), and phase frame sum (PFS). For a periodically waving plate, BSC with $i$9 and 8 images achieves the smallest RMSE over speeds from $$\tilde{\phi}_i(x,y) = \tan^{-1}\!\left[ \frac{\displaystyle\sum_{n=0}^{N-1} I_{i+n}(x,y)\,\sin\left(\tfrac{2\pi n}{N}\right)}{\displaystyle\sum_{n=0}^{N-1} I_{i+n}(x,y)\,\cos\left(\tfrac{2\pi n}{N}\right)} \right].$$0 to 150 mm/s (Zhang et al., 14 Jul 2025). At a specific speed of $$\tilde{\phi}_i(x,y) = \tan^{-1}\!\left[ \frac{\displaystyle\sum_{n=0}^{N-1} I_{i+n}(x,y)\,\sin\left(\tfrac{2\pi n}{N}\right)}{\displaystyle\sum_{n=0}^{N-1} I_{i+n}(x,y)\,\cos\left(\tfrac{2\pi n}{N}\right)} \right].$$1 mm/s, classical 4-step PSP has approximately $$\tilde{\phi}_i(x,y) = \tan^{-1}\!\left[ \frac{\displaystyle\sum_{n=0}^{N-1} I_{i+n}(x,y)\,\sin\left(\tfrac{2\pi n}{N}\right)}{\displaystyle\sum_{n=0}^{N-1} I_{i+n}(x,y)\,\cos\left(\tfrac{2\pi n}{N}\right)} \right].$$2m RMSE, whereas BSC reduces this to approximately $$\tilde{\phi}_i(x,y) = \tan^{-1}\!\left[ \frac{\displaystyle\sum_{n=0}^{N-1} I_{i+n}(x,y)\,\sin\left(\tfrac{2\pi n}{N}\right)}{\displaystyle\sum_{n=0}^{N-1} I_{i+n}(x,y)\,\cos\left(\tfrac{2\pi n}{N}\right)} \right].$$3m (Zhang et al., 2024, Zhang et al., 14 Jul 2025). The histogram of error after BSC is reported as approximately normal or Gaussian, indicating that the non-Gaussian ripple artifacts have been largely removed and the residual error is dominated by sensor or intensity noise (Zhang et al., 2024, Zhang et al., 14 Jul 2025).

Because P-BSC is pixel-wise in the spatial domain, it preserves depth discontinuities better than non-pixel-wise approaches. The reported experiments with static gypsum objects and a moving plate show that $$\tilde{\phi}_i(x,y) = \tan^{-1}\!\left[ \frac{\displaystyle\sum_{n=0}^{N-1} I_{i+n}(x,y)\,\sin\left(\tfrac{2\pi n}{N}\right)}{\displaystyle\sum_{n=0}^{N-1} I_{i+n}(x,y)\,\cos\left(\tfrac{2\pi n}{N}\right)} \right].$$4-FTP and HTC introduce visible edge distortions, whereas BSC maintains sharp edges and lower RMSE near depth jumps (Zhang et al., 2024). Similar observations are reported for moving statues, where BSC preserves fine geometric details more faithfully than HTC (Zhang et al., 14 Jul 2025).

The principal trade-off is in the choice of binomial order $$\tilde{\phi}_i(x,y) = \tan^{-1}\!\left[ \frac{\displaystyle\sum_{n=0}^{N-1} I_{i+n}(x,y)\,\sin\left(\tfrac{2\pi n}{N}\right)}{\displaystyle\sum_{n=0}^{N-1} I_{i+n}(x,y)\,\cos\left(\tfrac{2\pi n}{N}\right)} \right].$$5. Larger $$\tilde{\phi}_i(x,y) = \tan^{-1}\!\left[ \frac{\displaystyle\sum_{n=0}^{N-1} I_{i+n}(x,y)\,\sin\left(\tfrac{2\pi n}{N}\right)}{\displaystyle\sum_{n=0}^{N-1} I_{i+n}(x,y)\,\cos\left(\tfrac{2\pi n}{N}\right)} \right].$$6 reduces motion-induced harmonic error exponentially, but it also increases the effective temporal window, the computational cost, and the integration of intensity noise (Zhang et al., 2024, Zhang et al., 14 Jul 2025). The 2024 paper states that the minimum $$\tilde{\phi}_i(x,y) = \tan^{-1}\!\left[ \frac{\displaystyle\sum_{n=0}^{N-1} I_{i+n}(x,y)\,\sin\left(\tfrac{2\pi n}{N}\right)}{\displaystyle\sum_{n=0}^{N-1} I_{i+n}(x,y)\,\cos\left(\tfrac{2\pi n}{N}\right)} \right].$$7 for 4-step PSP is 1, corresponding to 5 images, while 8 images are a good compromise (Zhang et al., 2024). The 2025 paper gives the corresponding recommendation as $$\tilde{\phi}_i(x,y) = \tan^{-1}\!\left[ \frac{\displaystyle\sum_{n=0}^{N-1} I_{i+n}(x,y)\,\sin\left(\tfrac{2\pi n}{N}\right)}{\displaystyle\sum_{n=0}^{N-1} I_{i+n}(x,y)\,\cos\left(\tfrac{2\pi n}{N}\right)} \right].$$8, or 8 patterns, for dynamic scanning (Zhang et al., 14 Jul 2025). These statements are consistent in practice, since 4-step PSP with $$\tilde{\phi}_i(x,y) = \tan^{-1}\!\left[ \frac{\displaystyle\sum_{n=0}^{N-1} I_{i+n}(x,y)\,\sin\left(\tfrac{2\pi n}{N}\right)}{\displaystyle\sum_{n=0}^{N-1} I_{i+n}(x,y)\,\cos\left(\tfrac{2\pi n}{N}\right)} \right].$$9 uses 8 images total in the reported implementations.

7. Scope, constraints, and interpretation

The derivations assume small motion, expressed through $(x,y)$00 and $(x,y)$01, so the inter-frame displacement must remain moderate (Zhang et al., 2024, Zhang et al., 14 Jul 2025). The reported effective speed range is on the order of $(x,y)$02 mm/s at 90 fps for the described setup, although this depends on frame rate, binomial order, and signal-to-noise ratio (Zhang et al., 14 Jul 2025). Very fast or highly non-smooth motions reduce effectiveness because the finite differences are no longer small and wrap handling becomes more difficult (Zhang et al., 2024).

The method is derived specifically for $(x,y)$03 phase-shift PSP. The 2024 paper notes that the method in this form is specific to equal-step $(x,y)$04 shifting, and that extension to arbitrary phase steps may require more elaborate formulations (Zhang et al., 2024). The later mechanism paper likewise focuses on cyclic $(x,y)$05 3-step and 4-step schemes (Zhang et al., 14 Jul 2025).

The data also note a texture-related constraint: the motion model assumes low-frequency textures on the object. High-frequency surface textures can break the intensity model and lead to artifacts (Zhang et al., 2024). Nonlinearity in projector or camera response is another practical limitation. The 2025 paper reports that nonlinearity degrades both P-BSC and I-BSC, especially 3-step P-BSC, and introduces Full-chain Nonlinearity Rectification (FNR), including OPWM and camera response correction, to improve convergence under nonlinear response (Zhang et al., 14 Jul 2025). If nonlinearity is not corrected, 4-step P-BSC or I-BSC is preferred, and 3-step P-BSC is discouraged (Zhang et al., 14 Jul 2025).

A common misconception is to treat P-BSC as a true single-shot method. The papers do not make that claim. They describe the method as quasi-single-shot because one depth map is produced per acquired frame after warm-up, but each output still depends on a temporal window of multiple images (Zhang et al., 2024, Zhang et al., 14 Jul 2025). Another misconception is that P-BSC requires explicit motion estimation; in fact, its defining feature is that motion-induced phase errors are compensated directly from the phase sequence itself, without tracking, phase-shift estimation, or other intermediate variables (Zhang et al., 2024, Zhang et al., 14 Jul 2025).

Within dynamic 3D measurement, P-BSC is positioned as a plug-and-play enhancement to standard $(x,y)$06 PSP pipelines. It uses no customized patterns, preserves the pixel-wise property of PSP, and offers an analytic mechanism for exponential suppression of motion-induced ripple artifacts through binomial combinations and high-order temporal differences (Zhang et al., 2024, Zhang et al., 14 Jul 2025). A plausible implication is that its importance is both methodological and conceptual: it reframes motion compensation in dynamic PSP as a purely temporal, per-pixel filtering problem in the phase domain, from which later image-domain generalizations can be derived.