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Sparse Bayesian Learning for Label Efficiency in Cardiac Real-Time MRI

Felix Terhag, Philipp Knechtges, Achim Basermann, Anja Bach, Darius Gerlach, Jens Tank, Raúl Tempone

TL;DR

This work addresses the labeling burden in real-time cardiac MRI, where outer-slice segmentations are unreliable and large numbers of frames would otherwise require manual annotation. It introduces Sparse Bayesian Learning to identify a small set of frequencies, dominated by $f_{ ext{heart}}$ and $f_{ ext{resp}}$, that describe ventricular volume over time and transfers information from reliably segmented inner slices to poorly segmented outer slices via a shared prior learned through a Type-II likelihood framework. Hyperparameters $\alpha_m$ and $\sigma^2$ are estimated by EM with MacKay updates, enabling automatic pruning of inactive frequencies and a low-labeling-budget strategy driven by minimizing the posterior covariance spread through a greedy algorithm with a guaranteed bound $(1-e^{-1/c_f})$. Empirical results on real patient data with univentricular hearts demonstrate that a small subset of labeled frames suffices for accurate volume prediction and that the trace-based greedy labeling consistently reduces predictive uncertainty compared to random labeling, providing a practical, uncertainty-aware approach to real-time MRI analysis.

Abstract

Cardiac real-time magnetic resonance imaging (MRI) is an emerging technology that images the heart at up to 50 frames per second, offering insight into the respiratory effects on the heartbeat. However, this method significantly increases the number of images that must be segmented to derive critical health indicators. Although neural networks perform well on inner slices, predictions on outer slices are often unreliable. This work proposes sparse Bayesian learning (SBL) to predict the ventricular volume on outer slices with minimal manual labeling to address this challenge. The ventricular volume over time is assumed to be dominated by sparse frequencies corresponding to the heart and respiratory rates. Moreover, SBL identifies these sparse frequencies on well-segmented inner slices by optimizing hyperparameters via type -II likelihood, automatically pruning irrelevant components. The identified sparse frequencies guide the selection of outer slice images for labeling, minimizing posterior variance. This work provides performance guarantees for the greedy algorithm. Testing on patient data demonstrates that only a few labeled images are necessary for accurate volume prediction. The labeling procedure effectively avoids selecting inefficient images. Furthermore, the Bayesian approach provides uncertainty estimates, highlighting unreliable predictions (e.g., when choosing suboptimal labels).

Sparse Bayesian Learning for Label Efficiency in Cardiac Real-Time MRI

TL;DR

This work addresses the labeling burden in real-time cardiac MRI, where outer-slice segmentations are unreliable and large numbers of frames would otherwise require manual annotation. It introduces Sparse Bayesian Learning to identify a small set of frequencies, dominated by and , that describe ventricular volume over time and transfers information from reliably segmented inner slices to poorly segmented outer slices via a shared prior learned through a Type-II likelihood framework. Hyperparameters and are estimated by EM with MacKay updates, enabling automatic pruning of inactive frequencies and a low-labeling-budget strategy driven by minimizing the posterior covariance spread through a greedy algorithm with a guaranteed bound . Empirical results on real patient data with univentricular hearts demonstrate that a small subset of labeled frames suffices for accurate volume prediction and that the trace-based greedy labeling consistently reduces predictive uncertainty compared to random labeling, providing a practical, uncertainty-aware approach to real-time MRI analysis.

Abstract

Cardiac real-time magnetic resonance imaging (MRI) is an emerging technology that images the heart at up to 50 frames per second, offering insight into the respiratory effects on the heartbeat. However, this method significantly increases the number of images that must be segmented to derive critical health indicators. Although neural networks perform well on inner slices, predictions on outer slices are often unreliable. This work proposes sparse Bayesian learning (SBL) to predict the ventricular volume on outer slices with minimal manual labeling to address this challenge. The ventricular volume over time is assumed to be dominated by sparse frequencies corresponding to the heart and respiratory rates. Moreover, SBL identifies these sparse frequencies on well-segmented inner slices by optimizing hyperparameters via type -II likelihood, automatically pruning irrelevant components. The identified sparse frequencies guide the selection of outer slice images for labeling, minimizing posterior variance. This work provides performance guarantees for the greedy algorithm. Testing on patient data demonstrates that only a few labeled images are necessary for accurate volume prediction. The labeling procedure effectively avoids selecting inefficient images. Furthermore, the Bayesian approach provides uncertainty estimates, highlighting unreliable predictions (e.g., when choosing suboptimal labels).

Paper Structure

This paper contains 13 sections, 2 theorems, 33 equations, 7 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Methods
  3. Modeling
  4. Finding sparse frequencies
  5. Sparse Bayesian learning and hyperparameter estimation
  6. Minimizing labeling work
  7. Optimality of the greedy algorithm
  8. Application
  9. Results
  10. Finding sparse frequencies
  11. Minimizing labeling work
  12. Conclusion
  13. Supplementary Figures

Key Result

Lemma 5

For $f_i$ defined in eq:obj_mono_nondec, it holds $f_i(J)>0$ for all $J\subset \Omega$ and $i\in \Omega\setminus J$.

Figures (7)

  • Figure 1: Scheme of the procedure. A neural network segments the raw images. An expert separates the segmentations into "good" and "bad" slices. The good slices are used to identify the dominating frequencies by empirically deriving a sparse prior in the frequency domain. The prior is used to determine the optimal order to label the images. An expert manually segments the "bad" slices frame by frame until the posterior provides sufficient certainty. The MRIs are short-axis views of univentricular hearts acquired by hypofon_study.
  • Figure 2: Values for $\alpha_m$, where $m>0$, after the expectation maximization procedure of the SBL algorithm. The largest peak is at the heart rate, with smaller peaks at double the heart rate. The respiratory rate is about $0.25$ Hz.
  • Figure 3: Posterior distribution and ground-truth data for two slices of two hearts. (a) Heart 1 is much less regular. The posterior distribution reflects this, with a broader posterior.
  • Figure 4: Histogram of 30,000 random draws of $c_{(\mathcal{X} ,\mathcal{Y} ,i)}$, as defined in \ref{['eq:estimated_wsc']}, to estimate the weak-submodularity constant. The highest obtained value is $\approx2.52$, and only $1.2\%$ of the samples are greater than 1.
  • Figure 5: Results of optimizing the posterior covariance. (a, b) Resulting covariance of the greedy approaches with 10,000 random draws. Shaded areas display the range of the random draws, where darker blue marks the inner 90%, and the blue curve represents the mean for (a) covariance trace, (b) determinant, and (c, d) resulting negative log-likelihood on Hearts 1 and 2. The wider band on Heart 1 reflects the higher irregularity in the heartbeat in this heart.
  • ...and 2 more figures

Theorems & Definitions (8)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Lemma 5
  • proof
  • Theorem 6
  • proof