Learned K-Space Acquisition Patterns

Updated 13 December 2025

Learned k-space acquisition patterns are data-driven strategies that jointly optimize sampling locations and reconstruction algorithms for accelerated MRI.
They leverage probabilistic relaxations and trajectory-based methods to create hardware-feasible, variable-density sampling schemes tailored to anatomical structures.
Empirical results demonstrate improvements of 1–4 dB in PSNR and enhanced SSIM, validating their superior performance over traditional hand-crafted schemes.

Learned k-space acquisition patterns refer to data-driven strategies that jointly optimize the locations in k-space to be measured and the corresponding reconstruction algorithm, typically formulated for accelerated magnetic resonance imaging (MRI) and related inverse problems. Unlike traditional hand-crafted sampling schemes (e.g., uniform, variable-density random, radial), learned patterns are parameterized through differentiable surrogates and co-adapted to anatomy, noise, and downstream networks via end-to-end training or bi-level optimization. This paradigm has produced quantifiable improvements in image quality, acceleration, and robustness across various anatomical targets and hardware constraints.

1. Mathematical Formulation of Learned k-Space Acquisition

The central problem is to acquire a subset of k-space coefficients that maximizes image reconstruction quality under a sample budget constraint. Formally, for an image $x\in\mathbb{C}^N$ and measurements $y = M \odot x$ where $M\in\{0,1\}^N$ is a binary mask ( $\|M\|_0 = M$ ), one aims to learn both $M$ and reconstruction parameters $\theta$ to solve

$\min_{M,\theta} \;\mathbb{E}_{x}\;\mathbb{E}_{m\sim p_\Phi}[ L_\mathrm{rec}(R_\theta(m\odot x), x) ] + \lambda \cdot \mathrm{Reg}(M, \theta)$

Here, $p_\Phi$ denotes the trainable probability distribution over masks, $L_\mathrm{rec}$ is typically MSE or (1–SSIM), and Reg includes entropy or hardware constraints (Huijben et al., 2020).

Several parameterizations are widely used:

Probabilistic/mask-based: A logit vector $\Phi\in\mathbb{R}^N$ is optimized and transformed via softmax to define per-location sampling probabilities. Differentiable relaxations (e.g., Gumbel-softmax, straight-through estimators), ensure gradients can flow through the (approximate) sampling process (Huijben et al., 2020, Zhang et al., 2020).
Trajectory-based: For non-Cartesian sampling, the trajectory is parameterized via control points, splines, or neural-ODEs (e.g., $k(t)$ solved by $dk/dt = f_\theta(k(t),t)$ ), subject to hardware constraints on gradient strength and slew rate (Alush-Aben et al., 2020, Peng et al., 2022).
Bi-level optimization: Some methods formulate an upper-level loss over masks and lower-level classical regularized reconstruction, differentiating through the inner variational solver (Sherry et al., 2019).

2. Joint Training Mechanisms and Differentiable Mask Sampling

End-to-end differentiability is enabled via sampling relaxations:

Gumbel-softmax / Concrete distribution: Draws Gumbel noise $g_n$ and forms

$s_n = \frac{\Phi_n + g_n}{\tau},\qquad \hat{m}_n = \frac{\exp(s_n)}{\sum_{i} \exp(s_i)}$

As $\tau \rightarrow 0$ , $\hat{m}$ approaches a discrete mask. For fixed sample count, a “top-M Gumbel-softmax trick” selects the largest $M$ relaxed entries (Huijben et al., 2020).

Bernoulli sampling with straight-through estimators: Binary masks $U_m = 1_{z_m < P_m}$ are sampled (with $z_m \sim \mathcal{U}(0,1)$ ), and backward gradients use the identity $\partial U_m / \partial w_m \approx \partial P_m / \partial w_m$ (Zhang et al., 2020, Zhang et al., 2023, Zhang et al., 2022).
Hard equality constraints and renormalization: The mask is probabilistically normalized in each minibatch to enforce a strict sampling ratio, ensuring hardware feasibility and fair comparison across methods (Zhang et al., 2023).
Trajectory optimization: Non-Cartesian learning requires continuous differentiability through the non-uniform FFT (NUFFT), enabled by backpropagation through spline control points and ODE solvers (Alush-Aben et al., 2020, Peng et al., 2022).

3. Reconstruction Networks Coupled with Acquisition Learning

Reconstruction networks are tightly integrated with acquisition optimization:

Model-based unrolled networks: Many approaches employ unrolled proximal-gradient or ADMM forms with learnable step sizes and proximal (regularization) blocks (often small CNNs), ensuring explicit data consistency with the current sampling mask or trajectory (Huijben et al., 2020, Zhang et al., 2023, Zhang et al., 2022, Alkan et al., 2023).
Feature fusion components: For multi-echo or multi-contrast MRI, recurrent or cross-echo fusion blocks are inserted to leverage redundancy and improve reconstructions conditioned on the sampling pattern (Zhang et al., 2023, Zhang et al., 2022).
Multi-resolution or hybrid learning: Some methods alternate between trajectory-only updates (using parameter-free density-compensated adjoint) and full joint learning of trajectory plus deep network, to stabilize optimization and ensure dense center-k-space coverage (R et al., 2021).

4. Empirical Characteristics of Learned Patterns

Learned acquisition masks and trajectories consistently exhibit the following empirically validated behavior:

Variable-density adaptation: Central low-frequency regions (high energy/contrast) are sampled with near-unity density, while high-frequency support is covered more sparsely but non-uniformly to maximize recoverable detail (Huijben et al., 2020, Sherry et al., 2019, Xue et al., 2020, Zhang et al., 2020). Mask densities decay anisotropically according to anatomical smoothness (Sherry et al., 2019).
Task adaptivity: Sampling is optimized for the downstream task (image reconstruction, segmentation, or quantitative mapping), and changing the loss $L_\mathrm{rec}$ to e.g. cross-entropy will drive significant differences in optimal sampling (Huijben et al., 2020, Zhang et al., 2022).
Hardware-aware trajectories: For non-Cartesian and dynamic cases, trajectories are smooth, center-dense, physically feasible, and differ by anatomy—brain imaging yields more centrally saturated trajectories than knee or cardiac imaging, which require broader high-frequency coverage (Peng et al., 2022, Shor et al., 2023, Aggarwal et al., 8 Aug 2024).
Empirical gains: Learned patterns outperform uniform, low-pass, and variable-density baselines by 1–4 dB in PSNR and 0.01–0.09 in SSIM, with performance consistently validated across datasets (fastMRI, 3D FSE, multi-contrast) and prospective settings (Huijben et al., 2020, Zhang et al., 2020, Alkan et al., 2023, Zhang et al., 2023).

Study	Mask/Trajectory Type	Net Gain vs Baseline
(Huijben et al., 2020)	DPS mask, PGD network	+1.5–2 dB PSNR, +0.04 SSIM
(Zhang et al., 2020)	LOUPE binary, MoDL	+1–2 dB PSNR, +0.01 SSIM
(Alkan et al., 2023)	3D continuous, PGD	+4.4 dB (R=5×), +2.0 dB (R=10×)
(Shor et al., 2023)	Multi-frame dynamic	+1.2 dB PSNR, +0.04 SSIM
(Zhang et al., 2022)	Multi-echo, ADMM, RNN	–14% QSM RMSE

5. Extensions: Domain Generalization, Adaptive and Sequential Acquisition

Recent advances treat k-space acquisition as an adaptive process or as a robust optimization for domain generalization:

Domain robustness: Introducing stochastic or adversarial perturbations to mask or trajectory parameters during training simulates scanner or domain shifts (gradient errors, anatomical variance), leading to improved generalization under cross-domain settings and reduced structured artifacts (Wattad et al., 6 Dec 2025). Acquiring with learned patterns plus perturbation can substantially mitigate degradation under distribution shift.
Reinforcement learning of sequential policies: Framing acquisition as a Markov decision process enables learning sequential selection policies via DQN/DDQN (for Cartesian) or PPO (for non-Cartesian/radial), conditioned on interim reconstructions or anatomical priors (Pineda et al., 2020, Xu et al., 5 Aug 2025). Anatomy-aware rewards and cross-attention network architectures maximize information gain in cardiac or other structured scenarios.
Multi-task and dynamic applications: The learned frameworks extend naturally to optimize for multi-contrast (T1, T2*, QSM) (Zhang et al., 2023, Zhang et al., 2022), dynamic MRI via multiple coordinated frame-wise trajectories (Shor et al., 2023), and even active CT or non-medical inverse problems (Huijben et al., 2020).

6. Broader Implications and Limitations

Learned k-space acquisition enables adaptive, information-centric MRI protocols:

Task-aware acquisition: The sampling adapts to both the data statistics (anatomy, noise) and the inductive bias of the reconstruction network, frequently yielding more informative or robust measurements than hand-crafted variable-density schemes (Huijben et al., 2020, Wattad et al., 6 Dec 2025).
Differentiable mask/trajectory optimization: Use of Gumbel-softmax and straight-through estimators provides full backpropagation through discrete mask sampling, outperforming REINFORCE-style stochastic estimators in terms of variance and stability (Huijben et al., 2020, Zhang et al., 2020).
Hardware realization: Physically feasible, gradient-/slew-limited trajectories are producible via spline/ODE-based parameterizations with projection or penalty methods, ensuring that learned patterns can directly inform pulse-sequence design (Alush-Aben et al., 2020, R et al., 2021, Peng et al., 2022).
Robustness and domain transfer: Learning acquisition patterns in simulation with modeled acquisition uncertainty improves real-world and cross-scanner reliability, opening routes for actively adaptive MRI (Wattad et al., 6 Dec 2025).

Limitations cited include reliance on retrospective data, possible over-regularization that discourages slice- or subject-specific policies (adaptive acquisition often collapses to non-adaptive masks in multi-coil networks), and the need for further work on hardware-constrained, multi-coil, prospective or on-scanner implementations (Bakker et al., 2022, R et al., 2021). Extension to multi-modal, eddy-current corrected, and patient-adaptive protocols is an active area of research.

7. Representative Algorithms and Comparative Analysis

A selection of influential frameworks and their core mechanisms:

Framework	Mask Param.	Recon Network	Sampling Domain	Notable Methods
DPS (Huijben et al., 2020)	Gumbel-softmax	Unrolled PGD (CNN)	Cartesian	Top-M, annealing
LOUPE (Zhang et al., 2020)	Bernoulli-ST	Unrolled MoDL	Multi-coil Cat.	Binary mask, ST grad
FLAT (Alush-Aben et al., 2020)	B-spline	3D U-Net	3D Non-Cart.	Spline+hard const.
AutoSamp (Alkan et al., 2023)	Free points	Unrolled PGD	Non-Cart. 3D	Infomax, NUFFT
mcLARO/LARO (Zhang et al., 2023, Zhang et al., 2022)	Sigmoid+renorm	Unrolled ADMM	Multi-echo Cart.	Feature fusion, mask per echo
RL Radial (Xu et al., 5 Aug 2025)	Action policy	Cross-attn actor-critic	Radial	Golden angle+PPO
Hybrid-MR (R et al., 2021)	Points+proj.	U-Net, Primal-Dual	Non-Cart. 2D	Multi-res, hybrid

Each approach is distinguished by its parameterization, relaxation mechanism, and class of feasible acquisition strategies (masks or continuous trajectories). Joint optimization remains universal. Empirical evidence demonstrates superiority over fixed variable-density or classical patterns for both reconstruction and--with appropriate augmentations--domain transfer performance.

References: (Huijben et al., 2020, Sherry et al., 2019, Zhang et al., 2023, Pineda et al., 2020, Alush-Aben et al., 2020, Peng et al., 2022, Zhang et al., 2020, R et al., 2021, Zhang et al., 2022, Bakker et al., 2022, Wattad et al., 6 Dec 2025, Shor et al., 2023, Aggarwal et al., 8 Aug 2024, Xue et al., 2020, Alkan et al., 2023, Xu et al., 5 Aug 2025)