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Diffusion-Encoding Gaussian Field for Joint k-q dMRI Reconstruction

Zhibo Chen, Yajuan Huang, Yu Guan, Qiuyun Fan, Dong Liang, Qiegen Liu

arXiv:2609.02288v1cs.CV

TL;DR

The paper addresses the difficulty of jointly reconstructing spatial structure and diffusion-direction variation when both k-space and q-space are undersampled. It introduces a self-supervised spatial–angular Gaussian field with shared primitives and continuous tensor-residual responses, achieving consistent improvements across missing-direction reconstruction and diffusion measures.

  • Problem

    Existing formulations do not explicitly exploit shared anatomy and direction-dependent signal variation, leaving spatial and angular information indirectly coupled or sequentially processed.

    Source: §I · p. 1

  • Method

    A subject-specific field uses shared overlapping 3D Gaussian primitives with continuous tensor-anchored q-responses, progressively optimized from observed-direction undersampled k-space measurements.

    Source: §I · p. 2, 1) Joint k–q Inverse Problem · p. 3

  • Results

    The method consistently improves missing-direction DWI reconstruction, tensor-derived metrics, and principal diffusion orientation estimation across three HCP shells and multiple acceleration settings.

    Source: 1) DWI Reconstruction and… · p. 6, 1) Ablation on Primitive Response Design · p. 9

  • Takeaways & Limitations

    Jointly refining explicit spatial support and continuous q-responses supports held-out-direction synthesis while preserving diffusion-related quantitative structure.

    Source: 1) Ablation on Primitive Response Design · p. 9, §VI · p. 10

Abstract

from arXiv · show

Diffusion MRI requires repeated k-space acquisitions over multiple diffusion-encoding directions, making acquisition time dependent on both spatial and angular sampling. Existing joint k-q methods either associate directional parameters with fixed voxels or separate spatial reconstruction from angular completion. However, diffusion-weighted images acquired under different directions share the same anatomical organization, while their local signal intensities vary with diffusion encoding. Existing formulations do not fully exploit the complementarity between shared anatomy and direction-dependent signal variation. Consequently, residual spatial errors may be misinterpreted as genuine angular variation and propagated to unobserved directions. We propose a subject-specific spatial-angular Gaussian field for self-supervised joint k-q dMRI reconstruction. Shared 3D Gaussian primitives provide local spatial support, with each primitive carrying a continuous q-conditioned tensor-residual response. The signal at each location is synthesized from multiple overlapping primitive responses, coupling neighboring spatial regions and diffusion directions. The field is progressively optimized from undersampled k-space measurements of observed directions, without fully sampled targets or held-out-direction supervision. Experiments on three HCP diffusion shells under multiple acceleration settings demonstrated consistent improvements in missing-direction DWI reconstruction, tensor-derived metrics, and principal diffusion orientation estimation.

I. INTRODUCTION

Existing joint k–q methods do not explicitly integrate shared anatomy with direction-dependent signal variation during undersampled reconstruction. The proposed self-supervised Gaussian field uses shared spatial support and continuous primitive responses to synthesize unobserved directions from observed k-space measurements.

  • Sparse q-space can cause spatial errors to be misinterpreted as directional variation, affecting FA, MD, and principal diffusion-direction estimates.Source: §I · p. 1
  • Existing methods often couple spatial and angular information indirectly or reconstruct spatial content and angular completion sequentially.Source: §I · p. 1, A. Joint k–q Reconstruction · p. 2
  • The proposed representation shares overlapping 3D Gaussian primitives across directions, with each primitive carrying a continuous q-conditioned response.Source: §I · p. 2
  • Primitive-level responses jointly model shared anatomy and direction-dependent attenuation through local support and response aggregation.Source: §I · p. 2
  • Observed-direction measurements guide adaptive primitive allocation and tensor-anchored continuous-response refinement without fully sampled targets or held-out-direction supervision.Source: §I · p. 2

B. q-Space Modeling

The paper treats dMRI as a continuous spatial–angular signal observed through undersampled k-space measurements. Gaussian primitives provide explicit shared anatomical support, while direction-conditioned responses model q-space variation and enable held-out-direction queries.

  • Diffusion-direction sampling affects tensor fitting, fiber orientation estimation, and FA and MD, so accelerated reconstruction must address spatial and directional sampling together.Source: B. q-Space Modeling · p. 2
  • Prior q-space methods include deep learning, recurrent models, spatio-angular convolutions, and continuous implicit representations for angular modeling.Source: B. q-Space Modeling · p. 2, C. Gaussian Splatting · p. 2
  • 3D Gaussian Splatting represents fields with explicit learnable Gaussian primitives optimized through differentiable splatting.Source: C. Gaussian Splatting · p. 2
  • The method assigns shared Gaussian spatial attributes to anatomy and direction-conditioned primitive responses to q-space signal variation.Source: C. Gaussian Splatting · p. 3, §III · p. 3
  • For one diffusion-weighting shell, observed-direction k-space measurements use direction-dependent masks, in-plane Fourier transforms, and noise terms.Source: 1) Joint k–q Inverse Problem · p. 3
  • The learned continuous field remains consistent with observed k-space samples and is queried at unobserved directions, which are excluded from optimization.Source: 1) Joint k–q Inverse Problem · p. 3

2) Shared Gaussian Support and Primitive-Centered Aggregation:

The field uses shared Gaussian primitives as overlapping spatial support, with each primitive carrying a continuous direction-dependent response. Tensor anchoring supplies physically constrained attenuation while residual angular terms add flexibility.

Shared Gaussian Support and Primitive-Centered Aggregation:

Tensor-Anchored Continuous Diffusion Response:

B. Progressive Field Construction and Optimization

The field is constructed progressively, beginning with a q-independent Gaussian scaffold initialized from observed-data-derived structural references. Subsequent optimization introduces directional responses while using observed-direction reconstruction, k-space consistency, and residuals to guide allocation.

Progressive construction

  • Optimization proceeds from a q-independent scaffold to discrete observed-direction responses and finally a continuous tensor-anchored field.This progression addresses the poorly conditioned joint optimization of Gaussian geometry and continuous q-responses from sparse measurements.Source: B. Progressive Field Construction and… · p. 3

Scaffold initialization

  • The initial scaffold fits a q-independent Gaussian field whose amplitudes are direction-independent and initializes Gaussian centers, spatial extents, and amplitudes.Tensor or angular residuals are introduced only after this structural initialization.Source: 1) Anatomy-Aware Scaffold Initialization · pp. 3–4

Optimization

  • A shared Gaussian scaffold with continuous tensor–residual responses is optimized using observed-direction reconstruction and sampled k-space consistency.Fitting residuals guide adaptive primitive allocation within the framework.Source: Figure 2 · p. 4

Scaffold initialization

  • The direction-averaged zero-filled volume provides an observed-data-derived structural reference rather than a fully sampled supervision target.Unmeasured k-space coefficients are set to zero before inverse Fourier transformation for each observed direction.Source: 1) Anatomy-Aware Scaffold Initialization · pp. 3–4

2) Observed-Response Calibration and Adaptive Allocation:

Observed-direction amplitudes temporarily expose spatial and directional deficiencies before continuous q-response fitting. An allocation score then concentrates Gaussian capacity through voxel birth and primitive splitting, followed by recalibration and continuous response projection.

Observed-response calibration

  • Independent positive observed-direction amplitudes reveal insufficient support near boundaries, fine structures, and regions with complex directional variation.These amplitudes are nonparametric samples supported by the current scaffold and do not define held-out-direction signals.Source: 2) Observed-Response Calibration and… · p. 4

Adaptive allocation

  • The allocation score combines spatial discrepancy, back-projected k-space residuals, and angular heterogeneity computed exclusively from observed directions.The brain mask restricts the score to supported tissue.Source: 2) Observed-Response Calibration and… · p. 4
  • Voxel birth adds primitives at high-score locations, while primitive splitting refines existing high-score primitives with smaller spatial extents.Observed-direction amplitudes are recalibrated after allocation, and the process repeats for several rounds.Source: 2) Observed-Response Calibration and… · p. 4, Figure 3 · p. 4

Continuous response projection

  • Calibrated discrete directional amplitudes are projected onto a continuous tensor–residual response family for evaluation at arbitrary directions.Initialization uses robust amplitude statistics, isotropic diffusivity, and a regularized low-order angular fit before nonlinear optimization.Source: 3) Observed-to-Continuous Response… · pp. 4–5

4) Self-Supervised Learning Objectives:

Self-supervision uses objectives matched to each progressive representation level while relying only on observed measurements. The final field combines measurement consistency with zero-filled anchoring and regularization before Fourier encoding.

Objective design

  • All progressive levels use observed-data-only supervision, with objectives matched to the complexity of each representation.The rendered signal is restricted to the brain support before Fourier encoding.Source: 4) Self-Supervised Learning Objectives · p. 5

Scaffold objective

  • Scaffold initialization minimizes a masked image-domain discrepancy to the zero-filled reconstruction with weak spatial regularization.This objective establishes the initial spatial representation from observed data.Source: 4) Self-Supervised Learning Objectives · p. 5

Response calibration

  • Observed-response calibration combines sampled k-space consistency with an auxiliary zero-filled anchor.This couples acquired Fourier samples with the observed-direction structural reference.Source: 4) Self-Supervised Learning Objectives · p. 5

Final field objective

  • The final continuous-field objective uses zero-filled anchoring, local image-variation regularization, primitive-response transfer, tensor regularization, and angular-correction regularization.The complete procedure progresses from scaffold initialization through calibration and allocation to continuous-response projection and joint refinement.Source: 4) Self-Supervised Learning Objectives · p. 5

IV. EXPERIMENTS

The method was evaluated on retrospectively undersampled HCP diffusion MRI across three shells and multiple joint spatial–angular acceleration settings. Evaluation used held-out directions for angular reconstruction after consistent preprocessing within the brain mask.

Data and preprocessing

  • All methods used the same normalization, brain-mask crop, four-voxel margin, and within-mask reconstruction and evaluation procedure.DWIs were normalized by the mean b = 0 image.Source: 1) Dataset · p. 5

Reconstruction procedure

  • The progressive reconstruction algorithm initializes shared Gaussian primitives, calibrates observed responses, allocates and recalibrates primitives, then jointly refines the continuous field.Its output is the spatial–angular field bS(x, g).Source: 1) Dataset · p. 5

Acceleration settings

  • Spatial undersampling used direction-dependent variable-density one-dimensional Cartesian masks with a fully sampled central fraction of 0.08.Experiments covered Rk ∈ {3, 4} and Nobs ∈ {30, 15, 10}, corresponding to Rq ∈ {3, 6, 9}.Source: 1) Dataset · p. 5

2) Comparison Methods:

The study compares the proposed method with sequential, joint model-based, and Gaussian-based reconstruction baselines under matched acquisition and evaluation conditions.

  • Six representative baselines cover sequential spatial–angular reconstruction, joint k–q model-based reconstruction, and Gaussian-based decoupled reconstruction.Source: 2) Comparison Methods · p. 5
  • All methods use identical crops, brain masks, observed and held-out directions, and k-space undersampling masks.Source: 2) Comparison Methods · p. 5
  • Held-out directions are excluded from subject-specific optimization and reserved only for retrospective evaluation.Source: 2) Comparison Methods · p. 6
  • Evaluation measures missing-direction DWI PSNR and SSIM, tensor-derived FA and MD PSNR and SSIM, and local orientation angular error.Source: 3) Evaluation Metrics · p. 6
  • Tensor-glyph visualizations additionally assess local anisotropy and orientation consistency.Source: 3) Evaluation Metrics · p. 6

4) Implementation Details:

The method is optimized per subject and diffusion shell using an adaptively densified Gaussian representation, and evaluations report improvements in DWI, tensor metrics, and orientation recovery.

  • Each subject and diffusion shell is optimized separately using approximately 1.4 × 10^5 initialized and 2.4 × 10^5 adaptively densified Gaussian primitives.Each primitive contains a learnable center, diagonal spatial covariance, positive-semidefinite diffusion tensor, and regularized second-order even angular residual.Source: 4) Implementation Details · p. 6
  • The proposed method consistently achieves the best missing-direction DWI reconstruction across HCP shells and tested joint acceleration settings.Its advantage becomes more evident under stronger acceleration with fewer observed diffusion directions.Source: 1) DWI Reconstruction and… · p. 6
  • Compared with baselines, the method better preserves anatomical edges, suppresses undersampling artifacts, and maintains direction-dependent diffusion contrast across diffusion weightings.Source: 1) DWI Reconstruction and… · p. 6
  • Downstream evaluation fits diffusion tensors identically across methods and compares FA, MD, PEVs, and tensor glyphs with reference results.Source: 2) Downstream Diffusion Metric and… · p. 6
  • The method achieves higher FA and MD reconstruction accuracy across tested acceleration conditions, preserving gains beyond image-level fidelity.Source: 2) Downstream Diffusion Metric and… · p. 6
  • In white-matter regions, it maintains lower PEV angular error and more continuous local directional structure than baseline methods.Source: 2) Downstream Diffusion Metric and… · p. 6

Optimization steps

Primitive budget

  • Increasing the primitive budget improves DWI, FA, and MD PSNR, but incremental gains progressively diminish at larger capacities.The improvements from 300k to 350k and from 350k to 400k are smaller than earlier gains.Source: 3) Analysis of Gaussian Optimization… · p. 6

C. Ablation Study

The ablations test primitive response components, optimization choices, sampling, and auxiliary losses, showing that the complete design performs best and that geometry refinement is especially consequential.

  • The ablation study evaluates tensor and SH response components, positive-semidefinite constraints, observed-response calibration, densification, geometry refinement, direction selection, and zero-filled anchoring.Source: C. Ablation Study · pp. 7, 9

Primitive response design

Primitive response design

Optimization and sampling

V. DISCUSSION

The framework couples shared spatial Gaussian support with direction-conditioned diffusion responses, distinguishing anatomical support from diffusion attenuation. Its current scope is limited by simplified acquisition and response assumptions, motivating extensions to raw multi-coil, multi-shell, and richer microstructural settings.

Method distinction

  • Unlike dynamic 4D Gaussian Splatting, the method keeps shared spatial 3D Gaussians and conditions each response on diffusion direction.For a fixed shell, direction is modeled on S2 through a continuous tensor–residual response rather than a temporal coordinate.Source: §V · p. 9, 2) Ablation on Optimization and… · p. 10

Physical interpretation

  • Spatial covariance defines anatomical support, while the diffusion tensor controls signal attenuation under diffusion gradients.The SH residual models partial-volume effects and departures from ideal single-tensor behavior.Source: 2) Ablation on Optimization and… · p. 10
  • The regularized even-SH residual adds angular flexibility in the softplus latent domain without replacing the tensor branch with an unconstrained interpolator.This design complements the tensor attenuation anchor rather than using an unconstrained angular model alone.Source: 2) Ablation on Optimization and… · p. 10

Limitations

  • Experiments use processed HCP DWI with retrospective single-coil-equivalent undersampling rather than raw multi-coil data.A raw-data implementation would require coil sensitivity estimation, scanner-specific trajectories, and noise modeling.Source: 2) Ablation on Optimization and… · p. 10
  • The implementation uses one tensor per Gaussian, diagonal spatial covariance, and scan-specific self-supervised optimization.Future extensions include richer microstructural responses, full anisotropic covariance, raw multi-coil acquisitions, and amortized initialization.Source: 2) Ablation on Optimization and… · p. 10, §VI · p. 10

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