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Fast Multi-class Dictionaries Learning with Geometrical Directions in MRI Reconstruction

Zhifang Zhan, Jian-Feng Cai, Di Guo, Yunsong Liu, Zhong Chen, Xiaobo Qu

arXiv:1503.02945v2cs.CVmath.OCphysics.med-ph

TL;DR

Undersampled MRI needs sparse representations that preserve image structure while keeping reconstruction computationally practical. The paper introduces fast orthogonal dictionaries trained separately on geometrically classified patches and embeds them in an efficient reconstruction model. Phantom and brain experiments report better artifact suppression and edge preservation than compared methods, with faster computation than typical K-SVD-based reconstruction.

  • Problem

    Pre-specified dictionaries may capture limited image features, while K-SVD-based adaptive reconstruction can be time consuming and miss some patches during training.

  • Method

    FDLCP trains fast orthogonal dictionaries separately on image patches classified by geometrical direction and uses them in a sparse reconstruction model solved with fast ADMM.

  • Results

    The method suppresses artifacts and preserves image edges better than compared state-of-the-art MR reconstruction methods, while computing much faster than typical K-SVD methods.

  • Takeaways & Limitations

    The approach supports undersampled MRI reconstruction with improved image quality and reduced computation time within the evaluated phantom and brain imaging experiments.

Abstract

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Objective: Improve the reconstructed image with fast and multi-class dictionaries learning when magnetic resonance imaging is accelerated by undersampling the k-space data. Methods: A fast orthogonal dictionary learning method is introduced into magnetic resonance image reconstruction to providing adaptive sparse representation of images. To enhance the sparsity, image is divided into classified patches according to the same geometrical direction and dictionary is trained within each class. A new sparse reconstruction model with the multi-class dictionaries is proposed and solved using a fast alternating direction method of multipliers. Results: Experiments on phantom and brain imaging data with acceleration factor up to 10 and various undersampling patterns are conducted. The proposed method is compared with state-of-the-art magnetic resonance image reconstruction methods. Conclusion: Artifacts are better suppressed and image edges are better preserved than the compared methods. Besides, the computation of the proposed approach is much faster than the typical K-SVD dictionary learning method in magnetic resonance image reconstruction. Significance: The proposed method can be exploited in undersapmled magnetic resonance imaging to reduce data acquisition time and reconstruct images with better image quality.

I. INTRODUCTION

CS-MRI reduces MRI acquisition time through compressed sensing, but fixed transforms can lose image structure under heavy undersampling. FDLCP addresses this with fast, geometrically classified multi-class dictionaries and reports improved reconstruction quality and computation time.

  • Compressed sensing reduces MRI data acquisition time by reconstructing sparse signals from relatively few measurements.
  • Pre-specified dictionaries capture limited image features, and combined transforms can still lose image structures under highly undersampled data.
  • K-SVD-based adaptive dictionary methods improve reconstruction quality but are time consuming during iterative MR reconstruction and may miss patches excluded from training.
  • FDLCP trains fast orthogonal dictionaries separately on image patches sharing geometrical directions, producing a sparser approximation for the target image.Training uses small-scale SVD and thresholding, while reconstruction uses a sparse multi-class dictionary model.
  • Experiments on phantom and brain MRI data report fewer artifacts, lower reconstruction error, and less computation time than DLMRI, WaTMRI, and PBDW.
  • The reconstruction model promotes sparse representations while enforcing consistency with the undersampled k-space measurements.

B. Fast Dictionary Learning (FDL)

Fast dictionary learning replaces slower K-SVD-style training with orthogonal dictionaries learned by thresholding and SVD. The paper extends this approach to geometrically classified patch classes for sparser MR image representation.

  • Adaptive dictionaries approximate image patches as sparse linear combinations of learned atoms.
  • K-SVD alternates sparse coding and dictionary-atom updates, but its original form has relatively low training speed in CS-MRI.
  • Orthogonal dictionaries reduce dictionary dimension and are expected to support adaptive sparse representation with faster computation.
  • Fast Dictionary Learning solves orthogonal dictionary training by alternating hard-thresholded sparse coding with SVD-based dictionary updates.
  • B. Fast Dictionary Learning (FDL): The proposed method trains dictionaries on geometrically classified patches and reconstructs images with a fast alternating direction method of multipliers.

A. Fast Dictionary Learning on Classified Patches

The method learns orthogonal dictionaries separately for image patches grouped by geometrical direction, using fast SVD-based updates and thresholding to improve sparsity efficiently.

  • Motivation: Adaptive dictionaries must enforce patch sparsity while remaining computationally efficient, but a single dictionary may miss distinct features in some patches.The method addresses both sparse representation and training efficiency.
  • Direction estimation: Geometrical directions are selected by rearranging patch pixels into 1D vectors and preserving the largest 25% of Haar-wavelet coefficients.The selected direction provides the sparsest representation among candidate directions.
  • Patch classification: Patches are classified by estimated geometrical direction, and an orthogonal dictionary is trained separately for each class.Each class contains patches sharing the same direction.
  • Sparsity benefit: The classified-patch method achieves the sparsest representation and fastest approximation-error decay in the illustrated comparison.It outperforms non-adaptive Haar wavelets, DCT, and unclassified adaptive dictionary learning in the figure’s sparsity comparison.
  • Dictionary training: For each class, sparse coding uses hard thresholding, while dictionary updates use SVD in alternating iterations.The training procedure alternates coefficient computation and dictionary updating.

3. For each geometric direction { } 1, , Q ω ω ω ∈ 

The trained multi-class dictionaries are used for sparse coding and evaluated by relative l2 norm error to quantify approximation quality.

  • Training procedure: For each geometric-direction class, the algorithm alternates sparse coding, SVD, and dictionary updates until convergence, then outputs the class dictionary.The resulting collection contains one learned dictionary for each direction class.
  • Evaluation: Relative l2 norm error measures sparse-approximation quality when a specified percentage of the largest coefficients is retained.Lower error indicates closer agreement with the ground-truth image and stronger sparsifying ability.

B. Sparse Reconstruction Model with Multi-class Dictionary

The reconstruction model combines undersampled k-space fidelity with sparse patch representations under geometrically organized, trained multi-class dictionaries.

  • Objective terms: The l2 term enforces fidelity to undersampled k-space data, while the l1 term promotes sparse representations of image patches.The l1 penalty replaces the training-stage l0 penalty to obtain a convex reconstruction problem.
  • Patch structure: Overlapping patches are extracted with a one-pixel shift under a periodic-boundary assumption.For 8×8 patches, the overlap factor is set to 64.
  • Transform construction: The combined transform is an adaptive tight frame constructed from geometrical operators, patch extraction, and the class-specific dictionaries.Its rows form a tight frame in image space.
  • Model: The target MR image is reconstructed by minimizing data inconsistency while enforcing sparsity through trained patch dictionaries.The dictionaries are embedded with geometrical-direction information.

C. Numerical Algorithm

The numerical solver splits the sparse and fidelity terms with an auxiliary variable and applies ADMM-style updates to recover the image efficiently.

  • Variable splitting: An auxiliary variable α=Φx separates the l1 sparsity term from the l2 data-fidelity term.The constrained formulation is solved with alternating direction methods.
  • Stopping rule: The solver uses the early stopping criterion ||y−F_Ux||_2≤ε, with ε=10^-4 reported to perform well across the imaging data.The stopping condition is checked after the first inner iteration.
  • Iterations: The algorithm alternates sparse-coefficient updates, image updates from a closed-form normal equation, and multiplier updates.The reconstructed image is returned after the stopping condition is met.
  • Image update: The image update combines the Fourier encoding operator, measured k-space data, transform coefficients, and auxiliary variables.Equation (20) provides the closed-form update for x.

D. The Complete Procedure of The Proposed Method

The proposed procedure forms a reference image, classifies patches by geometrical direction, learns dictionaries, and reconstructs the sparse MR image. The reference can be updated once to improve classification, dictionary learning, and reconstruction.

  • D. The Complete Procedure of The Proposed Method: The procedure comprises reference image forming, patch classification, dictionary learning, and sparse MR image reconstruction.The initial reference is reconstructed from undersampled k-space data using SIDWT and ADMM.
  • D. The Complete Procedure of The Proposed Method: Because artifacts in the initial reference may impair patch classification, the reference can be updated once before further learning and reconstruction.The paper states that this update can improve the reconstruction.
  • D. The Complete Procedure of The Proposed Method: Figure 3 compares phantom reconstructions and corresponding error magnitudes for four methods under Cartesian sampling with 33% data.Panels show a fully sampled image, four reconstructions, the undersampling pattern, and four error maps.

IV. RESULTS

The evaluation uses phantom and in vivo MR data with several undersampling patterns and compares FDLCP with three state-of-the-art CS-MRI methods. Reconstruction quality is assessed using RLNE and SSIM, with lower RLNE and higher SSIM indicating better agreement and structural similarity.

  • IV. RESULTS: Experiments evaluate phantom and in vivo MR reconstructions using Cartesian, 2D random, and pseudo radial sampling patterns.FDLCP is compared with WaTMRI, DLMRI, and PBDW.
  • IV. RESULTS: The comparison methods use specified implementations and parameters, including Square DLMRI with a 64-atom dictionary learned from 19,200 patches over 36 iterations.DLMRI uses 8×8 patches, sparsity level 13, and linearly varying error thresholds.
  • IV. RESULTS: Reconstruction performance is quantified by relative l2 norm error and structural similarity index.RLNE measures difference from the fully sampled image, while SSIM measures structural similarity and detail preservation.
  • IV. RESULTS: Lower RLNE indicates greater consistency with the fully sampled image, and RLNE below 0.1 is considered acceptable in the authors' experience.SSIM values closer to one indicate stronger structural similarity and detail preservation.

A. Experiments on Phantom Data

Experiments evaluate FDLCP on phantom and brain MRI under Cartesian, random, and pseudo-radial undersampling, comparing reconstruction quality and computation with established methods. FDLCP generally reduces artifacts and reconstruction error while preserving image features, with performance depending on patch and reference-update settings.

  • Phantom reconstruction: FDLCP reconstructs the phantom with minimal feature loss, achieving the lowest RLNE and highest SSIM among the compared methods.WaTMRI produces obvious artifacts, DLMRI ringing, and PBDW smooth-region artifacts or edge loss.
  • Brain imaging reconstruction: Under Cartesian sampling, FDLCP consistently reduces RLNE and improves SSIM across sampling rates while outperforming the other state-of-the-art methods.The experiments use the T2-weighted brain image shown in Fig. 4(a).
  • Brain imaging reconstruction: Under pseudo-radial sampling, FDLCP preserves image features better and achieves superior RLNE and SSIM metrics compared with the other methods.Additional brain-image tests using multiple sampling patterns also report better FDLCP performance.
  • Computation time: The proposed method runs much faster than DLMRI and PBDW but slower than WaTMRI, with reference formation, patch classification, and dictionary learning comprising 15% of total time.The reported runtime comparison depends partly on whether stages use Matlab or optimized MEX/C implementations.
  • Parameter settings: An 8×8 patch size is optimal because increasing patch size improves RLNE and SSIM initially, whereas oversized patches can contain multiple edge directions.With 8×8 patches, Q=71 is used; performance changes only slightly for Q∈[8,71].
  • Parameter settings: One reference-image update is optimal: RLNE and SSIM improve substantially from T=0 to T=1, while further updates yield marginal gains at higher computational cost.The typical configuration uses an SIDWT reference image, n×n=8×8, Q=71, and T=1.

E. Effect of Initial Reference Image

The experiments compare initial reference images and updating strategies while also positioning FDLCP against newer reconstruction methods. An SIDWT-based start reaches higher quality efficiently, and FDLCP remains competitive across broader method comparisons.

  • Initial reference image: An SIDWT-based initial reference produces higher-quality FDLCP reconstruction than a zero-filling reference at the corresponding early update stage.The paper uses the SIDWT-based reference for startup because it provides a good starting point.
  • Initial reference image: After two reference-image updates, reconstructions from SIDWT-based and zero-filling initial references are comparable, with nearly identical evaluation metrics.The SIDWT-based route requires one SIDWT and FDLCP reconstruction, whereas zero-filling requires two FDLCP reconstructions.
  • Comparison with other state-of-the-art methods: With l1-norm FDLCP, reconstruction error is lower than PANO, while l0-norm FDLCP reconstructs better than PBDWS, BPFA, and NLS.BPFA performs best among some T2-weighted cases, whereas NLS performs better for the T1-weighted brain image.
  • Comparison with other state-of-the-art methods: FDLCP is relatively faster than PANO and PBDWS, although NLS is fastest and BPFA slowest in the reported comparison.Runtime comparisons depend on implementation language and optimized MEX/C components.
  • Conclusion: Overall, FDLCP suppresses artifacts and preserves image edges while outperforming the compared state-of-the-art methods and running much faster than typical K-SVD-based MRI reconstruction.The conclusion identifies multi-feature patch classification and a projected fast iterative soft-thresholding algorithm as future or possible developments.

Appendix

The appendix describes geometric pixel reordering and compares sparsifying transforms and l0/l1 reconstruction models. It reports that FDLCP achieves stronger reconstruction results with l0 penalization and provides an implementation release.

  • Geometric reordering: The ω_G operator reorders patch pixels along predefined candidate geometrical directions, converting them into a one-dimensional vector.Directions are uniformly partitioned over [0, 2π].
  • Geometric reordering: For 8×8 patches, FDLCP uses 71 predefined directions to maximally explore geometric directions and fairly compare with PBDW.The setting follows PBDW’s default parameter choice.
  • Reconstruction models: The baseline reconstruction model uses a Curvelets or Contourlets transform and ADMM to solve the constrained optimization problem.The transform Ψ is selected according to the compared sparsifying transform.
  • Sparsifying transforms: FDLCP outperforms Curvelets and Contourlets in the illustrated reconstruction, achieving RLNE 0.0935 and SSIM 0.9626.Curvelets and Contourlets obtain RLNE values of 0.1634 and 0.1589, and SSIM values of 0.8407 and 0.8572, respectively.
  • Reconstruction models: Using an l0 norm penalty improves FDLCP image quality over the l1 penalty, reducing reconstruction error by 22%.The appendix reports RLNE and objective-function values that decrease and gradually stabilize empirically during iterations, with initial oscillation.
  • Implementation: The FDLCP implementation supports both l1 and l0 norm minimization, with code to be released on the authors’ website.The l0 formulation uses ADMM and hard thresholding for sparse coefficients.

APPENDIX D

Appendix D compares FDLCP with DLMRI at matched dictionary size and examines how DLMRI’s atom count affects reconstruction quality and computation. FDLCP preserves edges and achieves better reported metrics, while larger DLMRI dictionaries increase computation substantially.

  • Matched dictionary comparison: At 3712 total atoms, FDLCP preserves image edges better than DLMRI and achieves lower RLNE and higher SSIM.The matched count is based on 58 observed geometric directions multiplied by 64 atoms per direction.
  • Dictionary-size analysis: Increasing DLMRI dictionary atoms from 1024 to 4096 yields marginal RLNE and SSIM improvements but increases computation time by about 3.5 times.The reported trade-off motivates setting DLMRI’s atom count to 64 when computation time is considered.
  • Penalty comparison: The FDLCP l0 reconstruction experiment reports RLNE 0.0741 with l0 versus 0.0935 with l1, and SSIM 0.9707 versus 0.9626.The comparison uses a Cartesian undersampling pattern retaining 32% of the data.
  • Convergence analysis: Empirical convergence plots track RLNE and the objective function against iteration time for the l0 norm minimization problem.The plots provide an empirical, rather than theoretical, convergence assessment.
  • Dictionary-size analysis: DLMRI commonly sets its atom count equal to the number of pixels in a patch, producing promising reconstruction results in reported experiments.This convention corresponds to 64 atoms for an 8×8 patch.
  • Matched dictionary comparison: With 3712 atoms, DLMRI and FDLCP obtain RLNE values of 0.1202 and 0.0935, respectively, and SSIM values of 0.9013 and 0.9626.The values correspond to the reconstructed brain images shown in Fig. 5.
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