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Learning Spatially Adaptive $\ell_1$-Norms Weights for Convolutional Synthesis Regularization

Andreas Kofler, Luca Calatroni, Christoph Kolbitsch, Kostas Papafitsoros

TL;DR

The paper tackles reconstructing images from undersampled, noisy data in low-field MRI by learning spatially adaptive per-filter regularization weights for a convolutional synthesis model. It combines a fixed pre-trained convolutional dictionary with a CNN-generated per-filter $\boldsymbol{\Lambda}$-map and solves a weighted $\ell_1$-regularized sparse coding problem via an unrolled FISTA algorithm, augmented by a high-pass pre-processing step. The approach demonstrates competitive performance with model-based deep learning (MoDL) and adaptive TV methods while offering improved interpretability through the learned parameter maps that reflect each filter's contribution. This method enables at-inference-time self-adaptation of the reconstruction to filter relevance, and provides a principled, convergent, and largely model-based alternative to fully end-to-end learned reconstructions. The results suggest practical utility for selecting and weighting filters in convolutional representations and point to future self-supervised extensions to estimate regularization maps without reference data.

Abstract

We propose an unrolled algorithm approach for learning spatially adaptive parameter maps in the framework of convolutional synthesis-based $\ell_1$ regularization. More precisely, we consider a family of pre-trained convolutional filters and estimate deeply parametrized spatially varying parameters applied to the sparse feature maps by means of unrolling a FISTA algorithm to solve the underlying sparse estimation problem. The proposed approach is evaluated for image reconstruction of low-field MRI and compared to spatially adaptive and non-adaptive analysis-type procedures relying on Total Variation regularization and to a well-established model-based deep learning approach. We show that the proposed approach produces visually and quantitatively comparable results with the latter approaches and at the same time remains highly interpretable. In particular, the inferred parameter maps quantify the local contribution of each filter in the reconstruction, which provides valuable insight into the algorithm mechanism and could potentially be used to discard unsuited filters.

Learning Spatially Adaptive $\ell_1$-Norms Weights for Convolutional Synthesis Regularization

TL;DR

The paper tackles reconstructing images from undersampled, noisy data in low-field MRI by learning spatially adaptive per-filter regularization weights for a convolutional synthesis model. It combines a fixed pre-trained convolutional dictionary with a CNN-generated per-filter -map and solves a weighted -regularized sparse coding problem via an unrolled FISTA algorithm, augmented by a high-pass pre-processing step. The approach demonstrates competitive performance with model-based deep learning (MoDL) and adaptive TV methods while offering improved interpretability through the learned parameter maps that reflect each filter's contribution. This method enables at-inference-time self-adaptation of the reconstruction to filter relevance, and provides a principled, convergent, and largely model-based alternative to fully end-to-end learned reconstructions. The results suggest practical utility for selecting and weighting filters in convolutional representations and point to future self-supervised extensions to estimate regularization maps without reference data.

Abstract

We propose an unrolled algorithm approach for learning spatially adaptive parameter maps in the framework of convolutional synthesis-based regularization. More precisely, we consider a family of pre-trained convolutional filters and estimate deeply parametrized spatially varying parameters applied to the sparse feature maps by means of unrolling a FISTA algorithm to solve the underlying sparse estimation problem. The proposed approach is evaluated for image reconstruction of low-field MRI and compared to spatially adaptive and non-adaptive analysis-type procedures relying on Total Variation regularization and to a well-established model-based deep learning approach. We show that the proposed approach produces visually and quantitatively comparable results with the latter approaches and at the same time remains highly interpretable. In particular, the inferred parameter maps quantify the local contribution of each filter in the reconstruction, which provides valuable insight into the algorithm mechanism and could potentially be used to discard unsuited filters.

Paper Structure

This paper contains 12 sections, 9 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Methods
  3. Problem Formulation
  4. Algorithm Unrolling
  5. High-Pass Filtering
  6. Network Architecture
  7. Experiments
  8. Dataset
  9. Comparisons
  10. Training and Implementation Details
  11. Results and Discussion
  12. Conclusion

Figures (4)

  • Figure 1: The proposed pipeline consists of three blocks: the first one is a high-pass pre-processing to construct the raw data $\mathbf{y}^\prime$. The second defines a re-parametrization of adaptive $\{\boldsymbol{\Lambda}_k\}_k$-maps in terms of a CNN $\text{NET}_\Theta$ given the input image $\mathbf{x}_0:=\mathbf A^{ \boldsymbol{\mathsf{H}}} \mathbf{y}$. The third consists of unrolling $T$ iterations of the FISTA algorithm for solving \ref{['eq:cdl_explicit_lambda_map_high_passed']}. Note that the pre-processing block includes a parameter $\beta>0$, which is learned together with the network weights $\Theta$.
  • Figure 2: An example of 12 out of $K=64$$\{\boldsymbol{\Lambda}_k\}_k$-maps for dictionary filters with filter size $k_f \times k_f = 11\times 11$ along with the corresponding sparse feature maps. The first two rows show the feature maps of the real and imaginary part of the solution of \ref{['eq:cdl_explicit_lambda_map_high_passed']}, while the third row shows the corresponding $\{\boldsymbol{\Lambda}_k\}_k$-maps. Results are ordered in a decreasing order w.r.t. the variance of the $\{\boldsymbol{\Lambda}_k\}_k$-maps. The results indicate that the network $\mathrm{NET}_{\Theta}$ can provide structural information associated with each filter and, moreover, can identify filters that are less relevant for the reconstruction (high values of the map).
  • Figure 3: An example of images reconstructed with the different methods for a noise level of $\sigma=0.3$. The SSIM and the PSNR, which are computed by masking out values outside of the defined mask, where no signal is present, are listed in the point-wise error images. To better highlight residual errors that can be attributed to the model limitations, the point-wise error images were scaled by a factor of 3.
  • Figure 4: A summary of SSIM and PSNR obtained on the test set by the different approaches. The proposed CDL-$\boldsymbol{\Lambda}$ achieves competitive results compared to the TV-$\boldsymbol{\Lambda}$kofler2023learning as well as to the deep learning-based method MoDL aggarwal2018modl.