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Reducing Uncertainty in Undersampled MRI Reconstruction with Active Acquisition

Zizhao Zhang, Adriana Romero, Matthew J. Muckley, Pascal Vincent, Lin Yang, Michal Drozdzal

arXiv:1902.03051v1cs.CV

TL;DR

Undersampled MRI measurements can leave artifacts and uncertainty in reconstructed images, while MRI acquisition is slow. This paper jointly estimates reconstruction uncertainty and selects additional k-space measurements during inference; across a knee MRI dataset and ImageNet, it reports stronger active-acquisition performance, uncertainty correlated with error, and improved reconstruction architecture.

  • Problem

    MRI acquisition is slow, and undersampled measurements can produce blurred or aliased images that fall short of high-fidelity reconstruction.

    Source: §1 · p. 1

  • Method

    The system jointly trains a reconstruction network to predict an image and pixel-wise uncertainty and an evaluator to select the next k-space measurement during inference.

    Source: §1 · p. 2

  • Results

    Across a knee MRI dataset and ImageNet, the evaluator outperforms active-acquisition heuristics, uncertainty estimates correlate with reconstruction errors, and the reconstruction architecture surpasses prior architectures.

    Source: §1 · p. 2, §6 · p. 8

  • Takeaways & Limitations

    The authors conclude that the method may support more applicable MRI-acceleration solutions while maintaining reconstruction quality.

    Source: §6 · p. 8

    For DICOM data, the paper counts N/2 possible measurements because of Fourier-transform conjugate symmetry.

    Source: §5.1 · p. 6

Abstract

from arXiv · show

The goal of MRI reconstruction is to restore a high fidelity image from partially observed measurements. This partial view naturally induces reconstruction uncertainty that can only be reduced by acquiring additional measurements. In this paper, we present a novel method for MRI reconstruction that, at inference time, dynamically selects the measurements to take and iteratively refines the prediction in order to best reduce the reconstruction error and, thus, its uncertainty. We validate our method on a large scale knee MRI dataset, as well as on ImageNet. Results show that (1) our system successfully outperforms active acquisition baselines; (2) our uncertainty estimates correlate with error maps; and (3) our ResNet-based architecture surpasses standard pixel-to-pixel models in the task of MRI reconstruction. The proposed method not only shows high-quality reconstructions but also paves the road towards more applicable solutions for accelerating MRI.

1. Introduction

MRI acquisition is slow, while undersampled measurements can produce artifacts that compromise image fidelity. The paper proposes jointly estimating reconstruction uncertainty and selecting additional measurements adaptively, with the goal of improving reconstruction as acquisition proceeds.

1. Introduction

  • MRI examinations can take as long as an hour because k-space is acquired sequentially, motivating methods to accelerate acquisition.The slow acquisition can lead to uncomfortable examinations and high health care costs.Source: §1 · p. 1
  • Undersampled k-space can yield blurred or aliased images, so reconstruction systems aim to recover high-fidelity images from those measurements.Source: §1 · p. 1
  • With fixed trajectories, 10× acceleration has the highest reconstruction-error variance among the three tested factors, and variance decreases as more measurements are introduced.The passage describes a trade-off between acquisition speedup and reconstruction-error variance.Source: §1 · p. 2, Figure 2 · p. 2
  • Partial measurements can support multiple plausible reconstructions, motivating pixel-wise uncertainty estimates that may inform image assessment and guide active acquisition.The paper proposes using uncertainty reduction from additional measurements as an acquisition signal.Source: §1 · p. 2
  • The proposed system jointly produces a high-fidelity reconstruction and pixel-wise uncertainty, while an evaluator selects subsequent k-space measurements during inference.The prediction is iteratively refined toward reducing reconstruction error and final uncertainty.Source: §1 · p. 2

2. Related Work

Prior work addresses MRI reconstruction, uncertainty estimation, and acquisition planning, but the cited acquisition approaches generally use fixed trajectories. The paper contrasts its jointly trained adaptive approach with those fixed strategies and with an inference-time optimization method.

2. Related Work

  • MRI reconstruction research includes compressed-sensing and deep-learning methods, while adversarial losses have also been explored to enhance perceptual reconstruction quality.The cited compressed-sensing approaches are described as potentially over-smoothed and time-consuming; prior deep-learning methods improve reconstruction error, but their perceptual results are described as not compelling.Source: §2 · p. 2
  • Medical-imaging uncertainty has been studied mainly for segmentation, while this paper focuses on data uncertainty caused by partially observed k-space.The passage distinguishes epistemic uncertainty from aleatoric uncertainty and notes that reconstruction errors may be harder for experts to detect than segmentation errors.Source: §2 · pp. 2–3
  • Earlier k-space trajectory approaches use fixed inference-time trajectories, whereas the paper's jointly trained adaptive acquisition uses image physics, the object, and reconstruction information to select measurements.A cited on-the-fly eigenvalue approach adapts to object-specific encoding physics but requires solving an optimization problem at inference time.Source: §2 · p. 3

3. Background and notation

The background defines undersampled k-space and its zero-filled image reconstruction through Fourier transforms and a binary sampling mask. It also explains how the paper simulates measurements from DICOM magnitude images and relates image- and frequency-domain distances.

3. Background and notation

  • The paper represents fully sampled k-space as y and defines undersampled k-space as S ⊙ y, with the zero-filled image obtained by applying the inverse Fourier transform.The binary mask S defines the Cartesian acquisition trajectory; applying the FFT maps a reconstructed image back to k-space.Source: §3 · p. 3
  • Because hospital DICOM images store magnitude images rather than raw k-space, the paper simulates measurements by applying the FFT to the magnitude image.Source: §3 · p. 3
  • Parseval’s theorem equates the squared l2 distance between two images with the squared l2 distance between their Fourier-domain representations.Source: §3 · p. 3

4. Method

The framework pairs a reconstruction network that outputs an image and uncertainty estimates with an evaluator that scores k-space rows. At inference, evaluator scores guide iterative measurement selection until acquisition is complete or a stopping criterion is met.

4.1. Reconstruction network

  • The cascaded c-ResNet uses FC-ResNets and data-consistency layers to estimate missing information while preserving observed measurements.Three identical encoder-decoder modules are interleaved with data-consistency layers and linked by shortcuts between residual blocks.Source: §4.1 · pp. 3–4

4.2. Uncertainty estimates

  • The network predicts pixel-wise Gaussian uncertainty alongside the reconstruction, training by conditional log-likelihood and using uncertainty to identify likely errors and trigger stopping.Uncertainty is modeled as a Gaussian centered at the reconstruction, with predicted variance; it highlights regions likely to contain large reconstruction errors.Source: §4.2 · p. 4

4.3. Evaluator network

  • The evaluator scores k-space rows by processing spectral maps and an acquisition-trajectory embedding, with similarity-based targets avoiding polarized scores that hinder ranking.Each spectral map corresponds to one k-space row; the target score approaches 1 when reconstructed and ground-truth maps are similar and 0 otherwise.Source: §4.3 · p. 4

4.4. Joint adversarial training

  • Adversarial training alternates updates to the reconstruction and evaluator networks, encouraging reconstructions to receive high evaluator scores.The overall reconstruction objective combines cascaded outputs, with β controlling the evaluator loss's influence and K denoting the number of cascaded FC-ResNets.Source: §4.4 · p. 5

4.5. Active acquisition

  • At inference, evaluator scores select the next unobserved measurement; the process updates the input and repeats until acquisition is complete or a stopping criterion is met.A low global uncertainty score is given as an example stopping criterion.Source: §4.5 · p. 5

5. Experiments

Experiments on knee MRI and ImageNet evaluate reconstruction architectures, uncertainty estimates, and active acquisition. The results favor evaluator-guided sampling over tested heuristics and show that uncertainty tracks reconstruction quality.

5. Experiments

  • Experiments use a large-scale Knee DICOM dataset and ImageNet, with knee results reported on 5,048 test images and classification evaluated on grayscale 128×128 ImageNet images.Knee images were drawn from three axial slices at close-to-central volume positions; the ImageNet classification model was a ResNet50 trained on the preprocessed images.Source: §5 · pp. 5–6

5.1. Comparison of reconstruction architectures

  • At kMA = 21% (∼5x speedup), the vanilla c-ResNet outperforms the compared pixel-wise baselines on MSE and SSIM.The full method has a slight performance drop because it also optimizes uncertainty estimates and the evaluator, rather than directly optimizing MSE alone.Source: §5.1 · p. 6
  • ResNet-based architectures perform better than the tested alternatives across kMA values, and the models scale gracefully to unseen kMAs from 38% to 50%.A single model was trained with random acquisition trajectories from 10% to 37% kMA.Source: §5.1 · p. 6

5.2. Uncertainty analysis

  • Mean uncertainty correlates with reconstruction MSE, though the relationship weakens as both values increase.The study assessed 512 test images using random trajectories ranging from 10% to 95% kMA.Source: §5.2 · p. 6, Figure 7 · p. 6
  • During active acquisition, reconstruction quality improves and error and uncertainty decrease, reaching very low values around kMA = 30%.Higher uncertainty regions tend to have higher reconstruction error and are concentrated in complex, often high-frequency regions.Source: §5.3 · p. 7, Figure 8 · p. 7
  • On the Knee dataset, evaluator-guided acquisition is more measurement-efficient than random and low-to-high-frequency baselines.The comparison reports that order+C performs on par with random+C+R, while the proposed method exhibits higher measurement efficiency.Source: §5.3 · p. 7, Figure 9 · p. 7
  • On ImageNet, random+C+R matches the proposed method in classification accuracy, despite order+C+R performing better on MSE.The authors suggest semantic information may occur in arbitrary high-frequency image regions.Source: §5.3 · p. 8, Figure 10 · p. 8
  • Among the tested evaluator designs, only the proposed evaluator produces increasing scores and decreasing score variance as measurements accumulate.Scores were computed for spectral maps corresponding to unobserved measurements.Source: §5.3 · p. 8, Figure 11 · p. 8

6. Conclusions

The conclusion presents an active acquisition pipeline for undersampled MRI and reports validation on a large-scale knee dataset and ImageNet. It highlights its evaluator, uncertainty estimates, and reconstruction architecture.

6. Conclusions

  • The pipeline iteratively suggests k-space trajectories to reduce uncertainty during undersampled MRI reconstruction.The authors describe validation on a large-scale knee dataset and ImageNet.Source: §6 · p. 8
  • The evaluator consistently outperforms alternative active acquisition heuristics, uncertainty estimates correlate with reconstruction error, and the reconstruction architecture surpasses previously introduced architectures.The authors state that uncertainty estimates can be used to trigger the halt signal of active acquisition at inference time.Source: §6 · p. 8
  • The authors argue the method paves the way toward MRI acceleration that maintains high-fidelity reconstructions with low uncertainty.Source: §6 · p. 8

Supplementary Material

The supplementary material outlines network details, experimental checks, inference-time requirements, and additional active acquisition results.

Supplementary Material

1. Network Architectures

The section directs readers to diagrams and component descriptions for both networks in the proposed framework.

1. Network Architectures

  • Figure 12 diagrams the reconstruction network and evaluator.Source: §1 · p. 11
  • Table 2 describes the building blocks of the reconstruction network.Source: §1 · p. 11
  • Table 3 defines the components needed to replicate the evaluator network design.Source: §1 · p. 11

2. Conjugate Symmetry in DICOM data

DICOM MRI images store only magnitude, so the paper simulates k-space data from that magnitude. Conjugate symmetry means the top half of the simulated data contains all frequency information needed for reconstruction, so paired rows are selected together.

2. Conjugate Symmetry in DICOM data

  • The top half of simulated k-space contains all frequency information needed to recover the image magnitude.The data are simulated as y = F(abs(x)) from DICOM images, which store only abs(x).Source: §2 · p. 11
  • Selecting a k-space row automatically selects its conjugate-symmetric partner, limiting full observation of 128 × 128 k-space to at most 64 iterations.The authors state that this selection strategy changes only the cardinality of the selection process, not the generalizability of their approach or the baselines.Source: §2 · p. 11

3. Inference time

For the example MRI protocol, single-line selection requires inference in roughly 7 ms, while batch selection requires about 2.7 s. The section also describes the reconstruction and evaluator network architectures.

3. Inference time

  • 7 ms per line and 2.7 s per batch are the approximate inference-speed requirements for the example TSE protocol.The protocol acquires four k-space lines per 2D slice; the initial non-optimized pipeline takes about 15 ms on a single GPU.Source: §3 · p. 11
  • Figure 12 illustrates the reconstruction network on the left and the evaluator network on the right.Source: Figure 12 · p. 12
  • The reconstruction network repeats an FC-ResNet module three times and links residual blocks across adjacent modules with shortcut connections.Each module contains an encoder, three residual blocks, a decoder, and a data consistency layer.Source: Table 2 · p. 12
  • The evaluator creates spectral maps, embeds a single sampling-trajectory column into a 6-d vector, and replicates it across spatial locations.The evaluator network uses global average pooling.Source: Table 3 · p. 13

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