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SCoPE-Reg: Efficient Rigid Ultrasound Slice-to-Volume Registration via State-Space Correlation and Closed-Form Pose Estimation

Niklas Schwarz, Jens Kleesiek, Moritz Rempe

arXiv:2608.28715v1eess.IVcs.CV

TL;DR

Rigid ultrasound slice-to-volume registration must localize an untracked 2D frame in a 3D reference despite limited anatomy and acoustic ambiguity. SCoPE-Reg uses linear state-space slice–volume interaction to predict dense weighted correspondences, then estimates pose with closed-form weighted Kabsch alignment. It reports lower errors than the strongest baselines on CAMUS and µ-RegPro, with bounded peak error, resolution-independent parameters, and real-time performance at 512^2.

  • Problem

    Rigid ultrasound SVR must estimate a 6-DoF pose from partial, artifact-corrupted, view-dependent 2D observations, while existing methods use costly dense attention or unconstrained direct regression.

  • Method

    SCoPE-Reg combines linear state-space slice–volume fusion and dense coordinate decoding with parameter-free weighted Kabsch pose estimation.

  • Results

    0.73 mm and 2.27 mm mTRE on CAMUS and µ-RegPro improve on EUReg by 41% and 14%, while 6.49 M parameters remain resolution-independent and performance reaches 51 FPS at 512^2.

  • Takeaways & Limitations

    The combination of dense weighted correspondences, bounded coordinate predictions, and closed-form alignment supports rigid ultrasound SVR at native acquisition resolution.

  • Takeaways & Limitations

    Evaluation uses simulated perturbations rather than tracked probe motion, so sequence-level validation on tracked sweeps remains necessary.

Abstract

from arXiv · show

Ultrasound-guided interventions can require localization of an untracked 2D frame within a 3D anatomical reference. Rigid slice-to-volume registration (SVR) estimates this six-degree-of-freedom pose but remains challenging because of limited anatomical context, acoustic artifacts, and view-dependent appearance. Existing methods often use dense cross-attention, whose cost scales with the product of slice and volume token counts, or direct pose regression without explicit correspondence constraints. We introduce SCoPE-Reg, combining state-space slice--volume interaction, dense 3D coordinate prediction, and parameter-free weighted Kabsch estimation. On SVR tasks from CAMUS and $μ$-RegPro, SCoPE-Reg yields mean target registration errors of $0.73$ mm and $2.27$ mm against $1.24$ mm and $2.63$ mm for the state of the art (SOTA), reduces peak error on CAMUS by 56% below SOTA ($12.5\!\to\!5.5$ mm), and registers $100\%$ and $80\%$ of frames within $3$ mm. On CAMUS at $128^2$ it retains the lowest error at increasing pose-perturbation magnitude. It holds $6.49$ M parameters independent of resolution, sustaining $51$ FPS at $512^2$. SCoPE-Reg establishes a SOTA in rigid ultrasound SVR: by coupling correspondence-based accuracy with bounded worst-case error and resolution-independent cost, it becomes viable at native acquisition resolution during intervention, where prior methods trade accuracy, reliability, or frame rate against one another. Supplementary code provided and will be open-sourced upon acceptance.

1. Introduction

Rigid ultrasound slice-to-volume registration must recover a 6-DoF pose from incomplete, artifact-corrupted 2D observations while meeting millimetre accuracy at acquisition frame rates. SCoPE-Reg addresses this with linear state-space interaction, dense correspondence prediction, and closed-form weighted pose estimation.

  • Motivation: Millimetre accuracy at acquisition frame rates is required for image-guided interventions using 2D ultrasound mapped into 3D context.Applications include cardiac fusion, targeted prostate biopsy, hepatic tumor ablation, and margin assessment.
  • Challenges: Partial anatomy, repeated structures, speckle, shadowing, and view-dependent appearance make rigid ultrasound SVR ambiguous and non-convex.These factors also limit the capture range of iterative methods.
  • Existing limitations: Dense cross-attention creates memory bottlenecks, while direct pose regression lacks explicit spatial correspondence constraints and can produce large outliers.Correspondence-based methods instead depend on the reliability and coverage of predicted matches.
  • Proposed approach: SCoPE-Reg replaces quadratic cross-attention with linear state-space fusion and direct pose regression with closed-form weighted Kabsch alignment.Its decoder predicts dense 3D coordinates and evidence weights, learned end-to-end from pose loss without direct correspondence supervision.
  • Reported outcome: 0.73 mm and 2.27 mm mean errors improve on 1.24 mm and 2.63 mm across CAMUS and µ-RegPro, while CAMUS peak error falls by 56%.The method also uses 6.49 M parameters independent of resolution and achieves high throughput with low GPU memory.

2. Related Work

Related work spans classical optimization, learned pose regression, state-space vision models, and closed-form geometric alignment. SCoPE-Reg positions itself as combining scalable cross-dimensional interaction with dense weighted correspondences and geometric pose recovery.

  • Classical registration: Classical slice-to-volume methods pair image-similarity metrics with local or global optimization but remain sensitive to initialization and metric choice.Iterative optimization also limits real-time interventional throughput.
  • Learned rigid SVR: Learned rigid SVR replaces per-case optimization with feedforward inference and varies in pose parameterization and strategies for bridging the 2D–3D domain gap.Examples include geodesic angle–axis regression, differentiable slice sampling, and restricted out-of-plane parameter regression.
  • State-space models: Structured state-space models capture long-range dependencies in linear time, while visual variants differ in bidirectional, four-directional, volumetric, or locality-preserving scans.Mamba introduces input-dependent transitions for selective sequence propagation.
  • Geometric estimation: Rigid point-set alignment estimates translation from weighted centroids and rotation from SVD of the cross-covariance matrix with determinant correction.This provides the geometric basis for closed-form weighted pose recovery.
  • Positioning: SCoPE-Reg unifies efficient interaction, explicit dense correspondence modeling, and geometric constraints by propagating volume coordinates and evidence through state-space fusion.Its dense weighted correspondences differ from sparse-keypoint Kabsch pipelines and from state-space methods confined to individual branches or slice sequences.

3. Method

SCoPE-Reg formulates rigid 2D–3D registration as dense coordinate prediction followed by closed-form weighted alignment. Its state-space decoder scans slice, volume, and coordinate information across four spatial orderings, then reduces evidence across depth.

  • Problem formulation: The method maps each slice pixel coordinate p_i to a target volume coordinate q_i under a six-degree-of-freedom rigid transformation.Rather than directly regressing pose parameters, it predicts dense coordinates and evidence weights before solving the pose.
  • Feature Encoding: Independent 2D and 3D ResNet-8 encoders produce features with an 8× spatial downsampling factor and N = H′W′ in-plane locations.The encoded dimensions are D′ = D/8, H′ = H/8, and W′ = W/8.
  • Coordinate decoding: CWCD fuses encoded slice and volume features with coordinate grids to produce a dense 2D field of 3D coordinates and evidence weights.The decoder applies state-space slice–volume fusion followed by correlation-weighted depth reduction and coordinate normalization.
  • State-space slice–volume fusion: SSVF uses DI-SSM scans to fuse appearance while a parallel coordinate scan accumulates volume locations with interaction-dependent weights.The coordinate stream uses the scan kernel as correspondence weight, so each slice token accumulates matched volume coordinates.
  • Coordinate normalization: Positive evidence accumulation normalizes each predicted coordinate into a convex combination of volume-grid locations.Depth reduction produces the coordinate numerator and strictly positive evidence mass, preserving relative evidence across planes and directions.
  • Directional scanning: Four forward and reverse row- and column-major scans make every slice–volume pair reachable while adding complementary traversal locality.The four orderings reach all in-plane pairs in O(D′NC′S) time, compared with dense attention’s O(D′N^2C′) time and O(D′N^2) affinity matrix.
  • Weighted Kabsch Pose Solver: The decoder yields N source–target correspondences, which WKPS aligns using weighted centroids, cross-covariance, SVD, and determinant correction.Because source points are coplanar, rank(H) ≤ 2; the rank-deficient case remains a unique minimizer for a non-degenerate grid with positive weights.
  • Training: Pose supervision alone trains the system by minimizing mean squared displacement over 14 reference points.The coordinate field and evidence weights receive no direct supervision.

4. Experiments and Results

Across two public benchmarks, SCoPE-Reg improves rigid ultrasound slice-to-volume registration, remains robust to larger pose perturbations, and scales efficiently to higher resolutions. Ablations attribute gains to state-space correspondence modeling and weighted geometric pose recovery, while evaluation remains limited to simulated perturbations.

  • Registration performance: SCoPE-Reg reaches 0.73 ± 0.04 mm mTRE on CAMUS and 2.27 ± 0.08 mm on µ-RegPro, improving over EUReg by 41% and 14%.Paired frame-level tests give padj < 10^-4, and SCoPE-Reg has the lowest fold-to-fold standard deviation.
  • Registration performance: 100.0% of CAMUS frames and 80.4% of µ-RegPro frames fall within 3 mm, compared with 97.2% and 66.2% for EUReg.The µ-RegPro result improves success despite EUReg retaining lower translation error.
  • Error tails: 5.49 mm peak error on CAMUS is lower than EUReg’s 12.47 mm, while p95 error falls to 1.37 mm from 2.01 mm.On µ-RegPro, p95 and peak errors are 4.22 mm and 5.95 mm versus EUReg’s 6.77 mm peak error.
  • Robustness and scalability: SCoPE-Reg maintains the lowest CAMUS mTRE as perturbations increase, degrading from 0.73 ± 0.04 mm at pm = 10 to 3.67 ± 0.14 mm at pm = 25.Fold standard deviation stays within ±0.14 mm, whereas EUReg reaches 5.54 ± 1.10 mm at pm = 25.
  • Robustness and scalability: At 512^2, SCoPE-Reg retains 6.49 M parameters, uses 678 MiB memory, and reaches 51 FPS, while baseline costs grow steeply with resolution.Its GFLOPs and memory scale linearly with pixel count; it reaches 77/51 FPS at 384^2/512^2 versus EUReg’s 56/32 FPS.
  • Ablations: Replacing cross-attention with SSVF lowers CAMUS mTRE from 1.91 to 0.68 mm at comparable throughput, while evidence-weighted Kabsch reduces peak error to 5.49 mm.CWCD supplies the coordinate field required by the solver but costs throughput without improving the mean.
  • Limitations: Evaluation uses simulated pose perturbations rather than tracked probe motion, leaving sequence-level validation on tracked sweeps as the next step.On µ-RegPro, one fifth of frames exceed 3 mm.

5. Conclusion

SCoPE-Reg constrains dense coordinate prediction to the encoded volume grid and recovers rigid pose with a closed-form weighted Kabsch solver. Its evaluation protocol uses shared preprocessing and perturbation settings across benchmarks, while implementation details document calibrated training configurations for unavailable references.

  • Conclusion: SCoPE-Reg predicts dense correspondences and evidence weights, then recovers pose with a closed-form weighted Kabsch solver.The coordinate field and weights receive no direct supervision.
  • Conclusion: Every predicted coordinate remains inside the convex hull of the encoded voxel grid because the scan uses non-negative gates, a shared kernel, and no passthrough term.The stabilizing ε preserves this property when the volume coordinate grid is centred.
  • Conclusion: Summing numerators and normalizers separately preserves evidence-dependent weighting across scan directions and depth planes.This makes the evidence mass interpretable as total matching confidence and supports 2D-to-3D localization.
  • Conclusion: The method-specific training budgets and schedules were empirically calibrated where reference configurations were unavailable.FVR-Net and CU-Reg received benchmark-specific optimization schedules under the stated implementation protocol.
  • Conclusion: The evaluation uses published case-level cross-validation protocols, with a harder in-plane perturbation setting on µ-RegPro than its reference protocol.All methods share data loaders, pose conventions, perturbation seeds, and metrics.

C. Statistical Significance and Error Distributions

Because volumetric ultrasound registration errors are heavy-tailed and non-Gaussian, frame-level paired testing is used to expose localized failure modes beyond fold-level summaries.

  • Statistical significance and error distributions: Frame-level paired hypothesis testing evaluates registration errors across 4000 CAMUS frames and 584 µ-RegPro frames.The analysis is motivated by heavy-tailed, non-Gaussian mTRE distributions that can hide localized failures in fold-level means and standard deviations.

Methodology

The methodology pairs identical perturbation seeds across models and tests frame-level error differences with corrected non-parametric statistics and bootstrap confidence intervals.

  • Methodology: Identical initial perturbation seeds are assigned to every evaluation frame across competing architectures.For each baseline, the analysis computes paired frame-level error differences.
  • Methodology: Two-sided paired Wilcoxon signed-rank tests evaluate whether paired error differences are symmetric about zero.P-values are adjusted across four baseline comparisons per dataset using Holm–Bonferroni correction.
  • Methodology: Standardized Wilcoxon effect magnitudes r are classified as small, medium, or large using thresholds of 0.1, 0.3, and 0.5.The reported effect-size categories follow the stated Co-hen classification.
  • Methodology: Non-parametric 95% confidence intervals for mean reduction are estimated from 1000 bootstrap resamples.The bootstrap intervals complement the paired hypothesis tests and effect magnitudes.

Discussion of Statistical Findings

SCoPE-Reg shows statistically significant error reductions across direct regression baselines and improves distributional robustness relative to EUReg on both benchmarks. The findings include narrower error ranges, lower medians, and reduced peak drift.

  • Statistical significance: padj < 10^-4: SCoPE-Reg achieves statistically significant, large-magnitude error reductions over all direct regression baselines across both benchmarks.On CAMUS, mean reductions are +4.17 mm over FVR-Net, +1.73 mm over CU-Reg, and +5.77 mm over DreamReg; µ-RegPro gains are also large.
  • EUReg comparison: +0.51 mm on CAMUS and +0.36 mm on µ-RegPro: SCoPE-Reg significantly outperforms EUReg on mean error reduction.The corresponding adjusted p-values are below 10^-4 on both benchmarks.
  • Error distributions: CAMUS IQR = [0.493, 0.884] mm versus EUReg’s [0.683, 1.260] mm, indicating a narrower frame-level error distribution.The comparison is reported from the frame-level distribution analysis.
  • Error distributions: CAMUS maximum error decreases to 5.49 mm from EUReg’s 12.47 mm, while µ-RegPro peak drift decreases to 5.95 mm from 6.77 mm.On µ-RegPro, SCoPE-Reg also has the lower median error: 2.057 mm versus 2.488 mm.

D. Qualitative Results: Edge Cases

SCoPE-Reg remains accurate across representative error percentiles on CAMUS and µ-RegPro, including difficult worst-case frames. Its advantage is especially pronounced on CAMUS, where competing methods lose pose in the worst case.

  • Best and Median Cases: 0.10 mm: SCoPE-Reg reduces the best CAMUS frame’s initial 7.55 mm offset, with black error-map coverage over 91% of the frame.CU-Reg and EUReg reach 0.38 mm and 0.53 mm, while FVR-Net and DreamReg remain at 6.85 mm and 5.03 mm.
  • Best and Median Cases: 0.66 mm and 2.06 mm: SCoPE-Reg median frames remain structurally correct on CAMUS and µ-RegPro despite minor residual rotation.Direct regression baselines show 3.7–6.1 mm errors in these cases.
  • p95 and Worst Cases: 5.49 mm: SCoPE-Reg’s worst CAMUS frame stays below EUReg at 12.47 mm and CU-Reg at 12.83 mm, whereas on µ-RegPro its 5.95 mm worst frame ranks third.Across µ-RegPro, SCoPE-Reg leads at mean, p95, and maximum error: 2.27, 4.22, and 5.95 mm.

E. Implementation of the Ablation Variants

The ablation variants are controlled comparisons: all share the same major training and pose components, while fusion and pose readout are the only changed factors.

  • Controlled Ablation Design: All Tab. 3 arms share the same encoders, pose parameterization, loss, optimizer, and data pipeline.
  • Controlled Ablation Design: The variants differ only in their fusion operator.
  • Controlled Ablation Design: The variants also differ in their pose readout.

Direct pose regression heads

Direct pose-regression variants pool a descriptor and use shared translation and rotation heads, contrasting with CWCD’s coordinate-field representation. Pooling keeps their parameter counts independent of grid dimensions.

  • Head Architecture: All head-based arms pool a descriptor f ∈ R^B×C with C = 256 and share the subsequent heads.Cross-attention and SSVF first produce fused volume features, whereas CWCD encodes its coordinate field with a 2D CNN.
  • Pose Regression Heads: Two MLPs map the pooled descriptor to translation with d = 3 and rotation with d = 6.The 6D rotation output is mapped to SO(3).
  • Parameter and Memory Effects: 0.20 M parameters: the heads add this amount, while replacing them with the parameter-free solver reduces memory from 80.8 to 79.3 MiB.
  • Training and Complexity: 6.61 M, 6.81 M, and 7.07 M: cross-attention, SSVF, and CWCD head-based arms differ by less than 3% in parameter count.All variants train for 500 epochs on CAMUS at 128^2; the cross-attention arm uses lr = 10^-4, versus 5 × 10^-5 for state-space variants.

F. Supplementary Ablations

Supplementary ablations test tri-plane scanning and dense-field regularizers, finding that neither improves performance on CAMUS. Tri-plane scanning adds computational cost because CAMUS’s third axis is cardiac time rather than spatial depth.

  • Regularizer and Scan Definitions: Dense displacement regularizers act on ϕ_i′ = q_i′ − p_i′, while tri-plane scanning adds coronal and sagittal scans into Y_f.
  • Scan and Regularization Ablations: 0.14 mm: tri-plane scanning increases error while adding 24% memory and reducing throughput by 27%.The ablation interprets CAMUS’s third axis as cardiac time rather than spatial depth.
  • Scan and Regularization Ablations: 0.82 mm and 0.79 mm: the physics loss and flow regularizer degrade performance by competing with evidence weights.
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