Decoding Motion-Blurred QR Codes: Motion-Parameter Error Tolerance and Model-Based Recovery

I. INTRODUCTION

• The paper measures how far assumed motion parameters can depart from truth before standard decoders fail, because overall decoding rates cannot isolate geometric errors from kernel or noise errors.Related work considered one-dimensional barcodes and kernel perturbations on natural images, but not this two-dimensional QR-code setting under rigid-plane motion.Source: §I · p. 1
• Controlled perturbations across 1,104 synthetic photographs test blur length, direction, speed profile, and motion centre or axis while holding other parameters at truth.A fixed linear inverter and standard decoders process each result, with bisection bracketing the failure boundary.Source: §I · p. 2
• Two compact networks initialise corner trajectories; tolerance-guided search, alternating image-and-path fitting, and regularised linear inversion refine the recovery.A payload is accepted only when at least two of three decoders return the identical string.Source: §I · p. 2
• 6.9% decoding error rate versus 16.4% for the strongest fine-tuned restoration baseline, a 58.0% relative reduction, on 955 handheld photographs.The complete pipeline accepts no wrong strings.Source: §I · p. 2
• A QR-code motion-blur framework links parameter error to payload recovery and specifies when an invertible mapping and convolution kernel permit regularised linear inversion.The framework identifies which information the inversion can recover.Source: §I · pp. 2–3
• Tolerance depends on speed profile, is asymmetric for under- versus overestimated length, and tightens when parameter errors are combined.A four-module sweep with speed rising 5.7-fold loses its payload at 65–66% too short but retains it at 80% too long.Source: Figure 1 · p. 2, §I · p. 3

II. RELATED WORK

• Existing QR and barcode methods use symbol priors or specific targets and generally do not systematically characterize how multi-parameter motion errors affect two-dimensional code decoding.Examples include finder-pattern kernel estimation, region-wise kernels, and barcode kernel-width error bounds.Source: §II · p. 3
• The paper establishes exact conditions for reducing rigid-planar-code blur to convolution under a coordinate change, then quantifies motion-parameter tolerances.Prior work models spatially varying blur with motion trajectories or homographies –,,,.Source: §II · p. 3
• Learned restorers range from direct image-to-image networks –, – to QR-specific methods such as ADNet, while this work predicts geometric parameters for explicit refinement.The paper also requires agreement across three decoders and verifies accepted strings against an independently registered payload; adequate spatial sampling is treated as a precondition.Source: §II · p. 3
• Unlike natural-image studies that perturb kernel parameters and track PSNR or SSIM, this work perturbs physical motion parameters in native units and measures exact-payload decoding.Earlier perturbation studies include kernel-width, length, and orientation changes and rotations up to 60° [26].Source: §II · p. 3

III. FRAMEWORK

• The framework covers blur reducible to convolution after an invertible coordinate change and separates the geometric mapping from the kernel.It applies the framework to short exposures of a rigid planar code whose motion is determined by four corner trajectories.Source: §III · pp. 3–4

A. The exposure operator

• The exposure operator models a sharp reference image transformed by scene motion over exposure time, under brightness constancy and additive sensor noise.The formulation assumes no occlusion, illumination variation, or sensor saturation.Source: A. The exposure operator · p. 4
• A common coordinate mapping makes every point undergo the same displacement; the resulting convolution kernel is the dwell measure of exposure time spent at each displacement.The blur depends on time spent at each displacement, not the temporal order of visits.Source: A. The exposure operator · p. 4
• The common mapping exists only when position mappings at different instants commute and may fail at fixed points; without it, exposure blur need not reduce to one convolution.Rotations and scalings about a fixed centre become translations in log-polar coordinates, whereas a moving rotation centre admits no common mapping.Source: A. The exposure operator · p. 4

B. Where information is lost, and what can be recovered

• The blurred image’s spatial frequencies are multiplied by the kernel transfer function, whose zero-frequency value is one and whose magnitude is at most one.The transfer function describes how the blur affects each frequency.Source: B. Where information is lost, and what… · p. 4
• Regularised least squares acts independently on each frequency in transformed coordinates, neglecting observed-region boundary effects.The inversion uses the adjoint of the blur operator and the identity operator.Source: B. Where information is lost, and what… · p. 4
• Frequencies with |Kµ(ω)|² ≫ λ are recovered with error of order σ/|Kµ(ω)|, whereas frequencies with |Kµ(ω)|² ≪ λ are suppressed.This linear inversion uses neither QR symbolic structure nor a learned image prior.Source: B. Where information is lost, and what… · p. 4
• Payload recovery does not require exact pixel-wise reconstruction: data-region module errors can be corrected up to each block’s Reed–Solomon capacity.Severe blur across finder patterns or format regions prevents localisation or reading.Source: B. Where information is lost, and what… · pp. 4–5

• A recovery is an exact match when its decoded string equals the registered payload, an accepted wrong string when it is nonempty but differs, and a rejection when it is empty.Source: C. What counts as success, and what… · p. 5
• With the payload unknown during recovery, candidate paths are ranked by known-module error, refinement minimizes forward reprojection residual, and final strings require agreement across multiple decoders.Source: C. What counts as success, and what… · p. 5

• For a rigid planar code, short-exposure camera and object motion is approximated by a twist: fixed-axis rotation combined with fixed-direction translation at time-varying speed.Source: D. Short exposures of a rigid plane · p. 5
• The image-plane trajectory is modeled as homographies generated by a trace-free 3 × 3 matrix with eight dimensionless degrees of freedom, with lateral, axial, roll, and tilt motions as special cases.The homography trajectory departs from the exponential model only at second order in pose change.Source: D. Short exposures of a rigid plane · p. 5
• The parameterization reduces motion unknowns to the generator, a one-dimensional speed profile, and the reference code pose, while temporal quadrature evaluates the explicit linear exposure operator directly.Uniform quadrature weights are exact when progress is proportional to exposure time.Source: D. Short exposures of a rigid plane · p. 5
• Each of four corner trajectories is expanded to second order in progress, with position, half-displacement, and a second-order term capturing curvature or monotonic acceleration or deceleration.Non-monotonic profiles that are slow at both ends and fast in the middle are not considered.Source: D. Short exposures of a rigid plane · p. 5

A. Overall pipeline

• The pipeline predicts corners and trajectories, performs regularised linear inversion, searches for a better path, then alternates refinement and decoding until decoder consensus or rejection.Source: A. Overall pipeline · p. 5

B. Geometry and motion estimation

• A two-pass path network refines the corner window and predicts the final path, while tolerance-guided candidate-path optimisation improves restoration accuracy.Source: B. Geometry and motion estimation · p. 5, Figure 2 · p. 6
• Corner labels encode mid-exposure positions, half-displacements, and second-order terms; a symmetric matrix target handles time-reversal ambiguity by recovering displacement up to sign.Source: B. Geometry and motion estimation · p. 6
• Both networks are trained on synthetic renders of rigid planar codes spanning four error-correction levels, quiet-zone sizes, and four motion categories, with imaging variations applied after exposure.Source: B. Geometry and motion estimation · p. 6

C. Physics-based recovery, refinement and read-out

• The candidate search scans length scale from 0.70 to 1.35, rotation within ±10°, and local length offsets, keeping the nearest grid candidate within 2.5% in length and 2.5° in direction.Each candidate is recovered by conjugate gradients with λ = 10^-3 and ranked by known QR function-pattern classification error.Source: C. Physics-based recovery, refinement… · pp. 6–7
• Three refinement rounds extract a binary code, re-blur it along the current path, and update path, intensity, and illumination parameters to match the observed photograph.Source: C. Physics-based recovery, refinement… · p. 7
• Read-out is attempted on the input and each recovery stage; if fewer than two decoders agree, fallback inversion weights are tried, and processing ends at consensus or rejection.Source: C. Physics-based recovery, refinement… · p. 7

V. EXPERIMENTS

• Controlled simulations quantify tolerance to trajectory error, and handheld photographs test the complete pipeline under practical capture conditions.Source: §V · p. 7

A. Evaluation protocol and evidence scope

• The controlled design crosses four motion families, four amplitudes, three speed profiles, and two card orientations, yielding 1,104 synthetic images, with tolerances measured on 1,065 images decodable under true motion.Each scan perturbs one assumed parameter while holding the others at truth; outcomes distinguish exact matches, wrong strings, and rejections.Source: B. Controlled tolerance measurements · p. 7, A. Evaluation protocol and evidence… · p. 7

B. Controlled tolerance measurements

• At larger motion amplitudes, constant-speed length tolerance falls to 8–11%, versus 19–54% at speed ratio 2.1 and at least 80% for overestimation at 5.7.At speed ratio 5.7, underestimation tolerance is 61–80%.Source: Figure 3 · p. 8, Figure 3a
• For 2-module translation at speed ratio 2.1, length tolerance is 54% for underestimation versus 29% for overestimation.The figure also shows that underestimation generally remains more tolerated, especially under large tilt.Source: Figure 3 · p. 8, Figure 3a
• At T2, a 10% length overestimate narrows direction tolerance from 13° to 8°; combined boundary offsets leave only 0–13% of combinations readable under constant speed.Accelerated motion tolerates assumed speed ratios up to the scan cap of 39 on most images, while lower limits remain near 0.91 and 1.27 for ratios 2.1 and 5.7.Source: B. Controlled tolerance measurements · p. 7, Figure 4 · p. 8
• With true motion, decoding succeeds under added shot noise at 13–30% of full-brightness standard deviation; repeated noise draws change length tolerance by about 2%, versus 19% across scenes.Upright and 45°-rotated cards have similar median tolerances.Source: B. Controlled tolerance measurements · pp. 7, 9

C. Synthetic evaluation of learned estimation

• Across 4,560 in-plane translation and roll images, initial length ratios span 0.84–1.12 at the 1st–99th percentiles, with extremes of 0.60 and 1.35.The initial direction-error median is 1.2° and its 99th percentile is 4.9°; a search over 0.70–1.35 and ±10° covers 99.9% of these cases with 22 inversions.Source: C. Synthetic evaluation of learned… · p. 9
• The complete pipeline is evaluated on 100 held-out synthetic images rendered with perspective distortion, defocus, sensor noise, and JPEG compression against a ground-truth-motion oracle.The comparison reports decoding error rates and trajectory errors.Source: C. Synthetic evaluation of learned… · p. 9
• 83 of 100 synthetic images decode with ground-truth quadratic paths; adding search and fitting lowers error from 27% to 23% while mid-exposure corner error falls from 0.231 to 0.143 modules.Median trajectory-shape error rises from 0.090 to 0.184 modules despite the decoding gain.Source: C. Synthetic evaluation of learned… · p. 9
• The authors suggest that failures even under recorded motion may reflect limitations of the quadratic path approximation or unmodelled defocus.This is presented as a likely explanation for the 17 unrecovered images.Source: C. Synthetic evaluation of learned… · p. 9

D. Handheld photographs and comparison

• The benchmark comprises 955 photographs captured by five photographers across four phone models and four motion categories, with reference payloads verified from separate still captures.Seven learned baselines were fine-tuned on synthetic pairs with training payloads disjoint from the photographed cards; two classical methods required no training.Source: D. Handheld photographs and comparison · p. 9
• Decoding error falls from 16.4% for NAFNet-32 to 6.9%, a 58.0% relative reduction, on the 955 photographs.The dataset spans four motion categories and four phone models.Source: D. Handheld photographs and comparison · p. 9, Table II · p. 10
• Our method achieves the highest recovery rate in every motion category and overall; its lowest rate is 87.4% on approach or retreat, accounting for 30 of 66 rejections.At processed resolution, photographs below 12 pixels per module account for 63 of 66 rejections. Across four phone models, recovery rates range from 88.9% to 94.5%.Source: D. Handheld photographs and comparison · p. 10
• Adding corner localisation, trajectory estimation and inversion recovers 588 photographs; search adds 98 and alternating fitting 53, bringing final error to 6.9%.Search and fitting run only on photographs rejected by earlier stages.Source: D. Handheld photographs and comparison · p. 10, Table III · p. 10
• The complete pipeline takes a median 5.8 s per photograph and 33.8 s at the 90th percentile; search and fitting consume 9% and 38% of processing time.Source: D. Handheld photographs and comparison · p. 10

VI. LIMITATIONS AND FUTURE WORK

• The current formulation covers planar rigid targets with quadratic corner trajectories and uniform temporal weights.Future extensions include non-planar deformation, partial occlusion and coupled optical defocus.Source: §VI · p. 10
• The pipeline takes a median 5.8 s per photograph, mainly because candidate search and fitting repeatedly perform inversion and decoding.The authors identify early path pruning or amortized refinement as possible routes toward real-time mobile deployment.Source: §VI · p. 10

VII. CONCLUSION

• Motion-parameter tolerances contract as blur amplitude grows, are tightest under uniform speed, relax under acceleration, and become stricter when errors combine.Length tolerance also differs markedly between underestimation and overestimation.Source: §VII · pp. 10–11
• On 955 handheld photographs, the pipeline lowers decoding error from 16.4% for the strongest fine-tuned network to 6.9%, a 58.0% relative reduction.Source: §VII · p. 11
• Neither whole-trajectory geometric accuracy nor visual edge sharpness guarantees payload decoding, so the paper argues for evaluating deblurring around payload recovery.Source: §VII · p. 11
• The 955-photograph benchmark has verified reference payloads across four motion categories and four phone models, and is released with code.Source: §VII · p. 11