Step-by-Step Reconstruction Using Learned Dictionaries
Jon Tamir1
1Electrical and Computer Engineering, The University of Texas at Austin, Austin, TX, United States

Synopsis

Compressed sensing introduced sparse representation of signals under a sparsifying transform, enabling significant reductions in number of measurements for signal reconstruction. In the context of MRI, this has led to dramatic reductions in scan time. Dictionary learning builds off of compressed sensing and tailors the sparsifying transform to the data itself. By learning both the dictionary and the sparse representation, further scan time reductions are possible. This talk will introduce dictionary learning for MRI reconstruction through hands-on software examples.

Target Audience

MRI scientists interested in learning and/or implementing image reconstruction algorithms.

Outcomes and Objectives

As a result of attending this session, participants should be able to:
  • Understand the main concepts of dictionary learning
  • Learn about applications of dictionary learning to MRI reconstruction
  • Implement dictionary learning algorithms for MRI reconstruction

Purpose

As signal modeling and representations have evolved in the theoretical and applied communities, so too have reconstruction methods for MRI. In the last ten years, MRI reconstruction has shifted from “Nyquist-era” modeling, to heavily data-driven modeling. With these representations, more and more prior knowledge is assumed about the image itself and baked in to the reconstruction algorithm. In turn, this enables dramatic reductions in scan times, as fewer measurements are needed to faithfully recover the underlying image representation.

Compressed sensing [1-3] was the first mainstream push into this paradigm, where the image was assumed to be sparse in some fixed transform domain such as wavelets or finite differences. Moving beyond sparsity in a fixed, known basis, it was soon recognized that the sparsifying transform could be designed from the data itself. Dictionary learning captures this concept, where each signal is described by a linear combination of a small number of units, or “atoms”, from a dictionary [4-6]. The signal can be image patches [7,8], temporal dynamics [9], or other multi-dimensional combinations. Given a pre-trained dictionary, image reconstruction can follow a similar procedure as in the case of compressed sensing: find the coefficients of the image under the sparsifying transform. However, the dictionary and the sparse representation can also be solved jointly from the under-sampled data itself [8].

The purpose of this tutorial is to review a general formulation for dictionary learning, discuss its benefits and drawbacks as related to classical methods, and explore applications of dictionary learning to MRI reconstruction. The guiding principles will be demonstrated through hands-on coding exercises. Connections to deep learning will be discussed [10].

Methods

Dictionary learning will be introduced in both analysis and synthesis form. The talk will review common approaches to dictionary design based on the synthesis form, focusing on methods such as simple alternating minimization [11] and K-SVD [7]. Richer representations including convolutional sparse coding [12-14] will be introduced. Tradeoffs between classical sparse representation and dictionary learning will be discussed.

Dictionary learning for MRI reconstruction will be demonstrated through hands-on software demos in Python. First, a dictionary for temporal relaxation curves will be pre-trained using simulated CPMG data [9]. The dictionary will be used to reconstruct under-sampled data from a variable flip-angle multi-echo spin-echo experiment and compared to a subspace-based (linear) reconstruction [15]. Next, a dictionary will be designed for image patches and used for MRI reconstruction [8]. Finally, convolutional sparse coding will be used to jointly learn the dictionary and the reconstruction directly from under-sampled data [13,14].

Discussion

The software demos will provide participants hands-on experience with dictionary learning and reference implementations for future use. By attending the tutorial, users will understand the benefits and drawbacks of dictionary learning for MRI reconstruction. Participants will gain an appreciation for the similarities and differences between compressed sensing and dictionary learning, as well as the similarities and differences between dictionary learning and deep learning.

Conclusion

Dictionary learning is a generalization of compressed sensing to learned sparsifying transforms, and allows the signal representation to be tailored to the data. Dictionary learning can be used in MRI reconstruction to reduce the number of measurements needed to faithfully reproduce the underlying signal.

Acknowledgements

No acknowledgement found.

References

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Proc. Intl. Soc. Mag. Reson. Med. 28 (2020)