0833

SVD Compression for Nonlinear Encoding Imaging with Model-based Deep Learning Reconstruction

Zhehong Zhang¹, Kartiga Selvaganesan¹, Yonghyun Ha², Chenhao Sun², Anja Samardzija¹, Heng Sun¹, Gigi Galiana^1,2, and R. Todd Constable^1,2,3,4
¹Department of Biomedical Engineering, School of Engineering and Applied Science, Yale University, New Haven, CT, United States, ²Department of Radiology and Biomedical Imaging, School of Medicine, Yale University, New Haven, CT, United States, ³Interdepartmental Neuroscience Program, School of Medicine, Yale University, New Haven, CT, United States, ⁴Department of Neurosurgery, School of Medicine, Yale University, New Haven, CT, United States

Synopsis

Keywords: Image Reconstruction, Signal Representations, Nonlinear Encoding

Model-based deep learning reconstruction with a nonlinear encoding matrix poses unique challenges to GPU memory, due to the densely connected computational graph nodes in the physics model part. In this work, SVD compression is demonstrated as necessary for such networks, and it is applied to the highly nonlinear case of Bloch-Siegert encoding from a low-field MR scanner. The redundancy across all nonlinear encoding dimensions is exploited for compression. With the compressed encoding matrix, the model-based network is feasible to implement. It outperforms the traditional reconstruction at all levels of simulated Gaussian noise and has advantages over commonly used regularization terms.

Introduction

Nonlinear encoding, in contrast to the linear encoding associated with Fourier transformation, requires an alternative approach to image reconstruction, either to account for the field imperfection^1,2 or to decode the signal projected by nonlinear gradient coils^3–6 or RF arrays⁷. The general signal model is as follows:
$$b_{i,c} (t)= \int \rho(r) S_c (r) exp⁡(-j\phi_i (r,t))dr+n_{i,c} (t)=\int \rho(r) E_{i,c} (r,t)dr+n_{i,c} (t)$$
where the signal $$$b_{i,c}(t)$$$, mixed with noise $$$n_{i,c}(t)$$$, is from the underlying object $$$\rho(r)$$$ encoded by the $$$i$$$th time-varying position-dependent phase $$$\phi_i(r,t)$$$ and the $$$c$$$th receiving coil $$$S_c(r)$$$. Image reconstruction is formulated as
$$\hat{\rho}=\arg{\min}_{\rho}\|b-E\rho\|_2^2$$
Previous work has shown that model-based deep learning can improve image reconstruction of nonlinearly encoded data⁸. It reduces the need for rigorous field mapping of nonlinear gradient waveforms. In that work, implementation relied on playing the same nonlinear waveform during each readout, whereby the nonlinear encoding can be treated as a Point Spread Function (PSF) modulation after Fast Fourier Transform (FFT).

For a general case of nonlinear encoding, like that provided by RF encoding^7,9, the encoding matrix is not based on FFT, and there are various nonlinear terms occurring across readouts. A reasonable $$$E$$$ requires O(100GB), which is impractical for saving and loading. The approaches of spatial-temporal decomposition^2,9,10 avoid expressing $$$E$$$ explicitly, but do not alleviate the memory issue in backpropagation, as the computational graph still has nodes densely connected in the physics model part of the network, and thus the memory requirement often exceeds GPU capacity.

Here we present a compression method to reduce the data size and make model-based networks feasible for the reconstruction of nonlinear encoding imaging. We apply the method to the low-field nonuniform imaging MR device^7,9, shown in Fig. 1. It features gradient-free RF spatial encoding, with a nonplanar 1MHz imaging slice selected from the nonuniform electromagnet. Off-resonance RF pulses with 130kHz offset transmitted by a 3x3 coil array induce spatially varying Bloch-Siegert shifts¹¹ for in-plane nonlinear encoding. One can create various encoding patterns by switching on or off individual coil elements. Low-field MR scanners have lower SNR, and because of nonlinear encoding, certain regions may receive limited encoding, leading to noisy and blurry reconstruction. Both these issues can be improved with a model-based reconstruction. This study demonstrates the feasibility and advantage of a state-of-the-art model-based network MoDL¹² (Fig. 2) implemented with the proposed compression method.

Method

Unlike conventional linear encoding which has orthogonal basis vectors while redundancy only occurs at the coil encoding dimension, nonlinear encoding usually has redundancy in the frequency encoding, phase encoding, and coil sensitivity encoding dimensions, which has great potential for matrix size reduction. Similar to the concept reported previously¹³, Singular Value Decomposition (SVD)¹⁴ is used to decompose the encoding matrix $$$E$$$
$$E=U\Sigma V^*$$
where $$$U$$$ and $$$V$$$ both form a set of orthonormal vectors. The total energy of $$$E$$$ is defined as the sum of the squares of its singular values.

Projecting $$$E$$$ onto the subspace spanned by the first k singular vectors $$$\{v_1,\cdots,v_k\}$$$ yields a representation in the lower dimensional space $$$C^k, 1<k<rank(E)$$$. This is done by multiplying
$$E_k=EV_k$$
where $$$V_k=\{v_1,\cdots,v_k\}$$$. The energy ratio is the fraction of energy of $$$E$$$ retained after truncation. The acquired signal should also be projected to the new space.
$$b_k=bV_k$$
Therefore, the physics model part of MoDL is formulated in the lower dimensional space
$$\rho_{n+1}=\arg{\min}_{\rho}\sum\limits_{i=1}^{N_i}\sum\limits_{c=1}^{N_c}\|b_{k_{i,c}}-E_{k_{i,c}}\rho\|_2^2+\lambda\|\rho-z_n\|_2^2$$
where $$$z_n$$$ is the ’denoised’ version of $$$\rho_n$$$ at the $$$n$$$th unrolled iteration.

In this study, 500 images from fastMRI dataset¹⁵ and a 3T liver image serve as the ground truth, and signals were simulated with 256 readout points per echo, 128 phase encoding steps, and 20cmx20cm FOV received by the 3x3 coil array. A range of Gaussian noise levels was simulated and tested. MoDL was compared to traditional CG reconstructions with and without regularizations.

Results and Discussion

Fig. 4A shows the potential for compression across three encoding dimensions by plotting singular values and the corresponding energy ratios, which demonstrate redundancy in the nonlinear encoding scheme. Combining these dimensions lead to a >90% reduction of the encoding matrix and signals. Fig. 4B shows the benefit of compression through memory consumption change. Attempting MoDL with the uncompressed input exceeds the memory capacity of most GPUs, but is made feasible with the compressed input. Note that the spatial-temporally decomposed matrix brings additional graph node connections, compared to an explicitly formulated matrix.

Fig. 5 shows the performance of various reconstruction methods using the full and compressed $$$E$$$ matrix. The compressed $$$E$$$ has 1.9% of the original size but only causes slight degradation in PSNR, RMSE, and SSIM across all noise levels (-0.2dB, +1%, and -0.01, respectively). Applying MoDL, which is only feasible with the compressed $$$E$$$ matrix, improves PSNR, RMSE, and SSIM by +2~6dB, -6%~-11%, and 0.12~0.15, respectively. Compared to traditional regularization terms with optimized weightings, MoDL yields the best PSNR, RMSE, and SSIM, approximately by a margin of 2dB, -4%, and 0.06.

Conclusion

Nonlinear encoding imaging with model-based deep learning reconstruction presents unique challenges for memory requirements. SVD compression was proven feasible and necessary for the implementation of the model-based network. With the compressed data, MoDL was implemented and demonstrated to outperform traditional reconstructions at various noise levels.

Acknowledgements

No acknowledgement found.

References

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Figures

Figure 1. An overview of the low-field nonuniform imaging device developed at Yale University. (A) The prototype. (B) Adjustable B0 of the main magnet serves as a natural way for slice selection. (C) Off-resonance RF pulses from the 3x3 coil array induce spatially varying Bloch-Siegert shift for in-plane encoding. (D) Field cycling is used to provide a high field (300mT) for spin polarization to enhance the signal, and a low field (24mT) for readout to suppress dephasing. (E) Simultaneous transmitting and receiving are expected for fast acquisition.

Figure 2. The architecture of MoDL. It is an unrolled network, equivalent to an alternating recursive algorithm. It is combined with residual-learning-based denoisers that capture important features of the image, and data consistency terms that encourage the fidelity to the acquired signal. The encoding matrix is incorporated into the network, and the forward model is explicitly accounted for, posing unique challenges for memory, especially with nonlinear encoding imaging.

Figure 3. The necessity of compression. (A) The explicit expression of the encoding matrix is impractical to handle. (B) The spatial-temporal decomposition does not reduce the dense connection of the nodes in the computational graph. (C) Compression of the encoding matrix is necessary to make a model-based network feasible. It is done by multiplying the encoding matrix by its first few singular vectors, and thus it is projected to the lower-dimensional space. The feasibility of compression lies in the redundancy of nonlinear encoding.

Figure 4. The feasibility and the result of compression. (A) All the three encoding dimensions have redundancy, as is shown by the decreasing singular value curves and the saturating energy ratio curves. Combining these three dimensions lead to great potential for compression. (B) The compressed encoding matrix, whose size is 1.9% of the original one, reduces the memory consumption of MoDL significantly, feasible in almost all GPUs. Note that the spatial-temporally decomposed input brings additional graph node connection, compared to explicitly formulated E.

Figure 5. (A) Across all simulated Gaussian noise levels (defined as the ratio of the standard deviation of noise compared to the maximum value of the signal), the compression degrades the image only in a slight manner, and MoDL with compressed input outperforms the traditionally used conjugate-gradient method. (B) MoDL outperforms CG with several state-of-the-art traditional regularization terms. Low-field MR has intrinsically severe noise, and this nonlinear encoding scheme has certain regions poorly encoded. MoDL shows promise to address these issues.

Proc. Intl. Soc. Mag. Reson. Med. 31 (2023)

0833

DOI: https://doi.org/10.58530/2023/0833