SPICER Self-Supervised Learning for MRI with Automatic Coil Sensitivity Estimation and Reconstruction Yuyang Hu1 Weijie Gan2 Chunwei Ying3 Tongyao Wang4 Cihat Eldeniz3 Jiaming

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SPICER: Self-Supervised Learning for MRI with Automatic Coil

Sensitivity Estimation and Reconstruction

Yuyang Hu1,*, Weijie Gan2,*, Chunwei Ying3, Tongyao Wang4, Cihat Eldeniz3, Jiaming

Liu1, Yasheng Chen5, Hongyu An1,3,4,5, and Ulugbek S. Kamilov1,2

1Department of Electrical and Systems Engineering, Washington University in St. Louis, St. Louis, MO 63130, USA

2Department of Computer Science and Engineering, Washington University in St. Louis, St. Louis, MO 63130, USA

3Mallinckrodt Institute of Radiology, Washington University in St. Louis, St. Louis, MO 63110, USA

4Department of Biomedical Engineering, Washington University in St. Louis, St. Louis, MO 63130, USA

5Department of Neurology, Washington University in St. Louis, St. Louis, MO 63130, USA

*These authors have contributed equally to the work

June 7, 2024

Running head: Self-Supervised Learning for MRI with Automatic Coil Sensitivity Estimation and Recon-

struction

Address correspondence to:

Ulugbek S. Kamilov, PhD

One Brookings Drive

MSC 1045-213-1010J

St. Louis, MO 63130, USA.

Email: kamilov@wustl.com

This work was supported in part by the NSF CAREER award under CCF-2043134, NIH R01 EB032713,

RF1NS116565, R21NS127425 and NIH R01HL129241.

Approximate word count: 250 (Abstract) 3,700 (body)

Submitted to Magnetic Resonance in Medicine as a Research Article.

arXiv:2210.02584v2 [eess.IV] 6 Jun 2024

Abstract

Purpose: To introduce a novel deep model-based architecture (DMBA), SPICER, that uses pairs of noisy

and undersampled k-space measurements of the same object to jointly train a model for MRI reconstruction

and automatic coil sensitivity estimation.

Methods: SPICER consists of two modules to simultaneously reconstructs accurate MR images and es-

timates high-quality coil sensitivity maps (CSMs). The ﬁrst module, CSM estimation module, uses a

convolutional neural network (CNN) to estimate CSMs from the raw measurements. The second module,

DMBA-based MRI reconstruction module, forms reconstructed images from the input measurements

and the estimated CSMs using both the physical measurement model and learned CNN prior. With the

beneﬁt of our self-supervised learning strategy, SPICER can be eﬃciently trained without any fully-sampled

reference data.

Results: We validate SPICER on both open-access datasets and experimentally collected data, showing

that it can achieve state-of-the-art performance in highly accelerated data acquisition settings (up to 10×).

Our results also highlight the importance of diﬀerent modules of SPICER—including the DMBA, the CSM

estimation, and the SPICER training loss—on the ﬁnal performance of the method. Moreover, SPICER can

estimate better CSMs than pre-estimation methods especially when the ACS data is limited.

Conclusion: Despite being trained on noisy undersampled data, SPICER can reconstruct high-quality

images and CSMs in highly undersampled settings, which outperforms other self-supervised learning methods

and matches the performance of the well-known E2E-VarNet trained on fully-sampled groundtruth data.

Keywords: Parallel MRI, Image Reconstruction, Inverse Problems, Deep Learning, Coil Sensitivity Esti-

mation.

1 Introduction

Magnetic resonance imaging (MRI) is a medical imaging technology known to suﬀer from slow data acquisi-

tion. Parallel MRI (PMRI) is a widely-used acceleration strategy that relies on the spatial encoding provided

by multiple receiver coils to reduce the amount of data to acquire (1–4). The multi-coil under-sampled data

can be reconstructed by ﬁtting the missing k-space lines (1, 2) or in the image space using coil sensitivity

maps (CSMs) (3,4). Compressed sensing (CS) is a complementary technique used to further accelerate data

collection by using prior knowledge on the unknown image (sparsity, low-rankness) (5, 6).

Deep learning (DL) has recently emerged as a promising paradigm for image reconstruction in CS-

PMRI (7–9). Traditional DL methods train convolutional neural networks (CNNs) to map acquired measure-

ments to the desired images (10, 11). Recent work has shown that deep model-based architectures (DMBAs)

can perform better than generic CNNs by accounting for the measurement model of the parallel imaging

system (12–17). Most of these methods require pre-calibrated CSMs as an important element in their model.

However, CSM pre-estimation strategies rely on suﬃcient auto-calibration signal (ACS) lines, which limits

the acceleration rates for data acquisition. To address this limitation, recent work has proposed to jointly

estimate high-quality images and CSMs in an end-to-end manner (18–20). However, these methods still

require fully-sampled groundtruth images as training targets, which limits their applicability to settings

where groundtruth is diﬃcult to obtain or unavailable. On the other hand, there has also been a broad

interest in developing self-supervised DL methods that rely exclusively on the information available in the

undersampled measurements (17, 21–26).

Despite the rich literature on DMBAs and self-supervised DL, the existing work in the area has not

investigated joint image reconstruction and coil sensitivity estimation directly from noisy and undersampled

data. We bridge this gap by presenting Self-Supervised Learning for MRI with Automatic Coil Sensitivity

Estimation (SPICER) as a new self-supervised learning framework for parallel MRI that is equipped with

an automatic CSM estimator. SPICER is a synergistic combination of a powerful model-based architecture

and a ﬂexible self-supervised training scheme. The SPICER architecture consists of two branches: (a) a

CNN for estimating CSMs from possibly limited ACS data, while ensuring physically realistic predictions;

(b) a DMBA that uses the estimated CSMs for high-quality image reconstruction. The SPICER training

is performed using undersampled and noisy measurements without any fully-sampled groundtruth. For

training, SPICER necessitates at least one pair of undersampled and noisy measurements from each slice.

We extensively validated SPICER on in-vivo MRI data for several acceleration factors. Our results show

that SPICER can achieve state-of-the-art performance on PMRI at high acceleration rates (up to 10×).

Moreover, SPICER can estimate better CSMs than pre-estimation methods especially when the ACS data

is limited.

This paper extends the preliminary work presented in the workshop paper (27). Compared to the method

in (27), SPICER uses a diﬀerent forward model, model-based deep learning architecture, and training loss

function. This paper also provides an expanded discussion on related work, additional technical details, as

well as completely new numerical results using real in-vivo MRI data acquired using a 32-channel coil.

2 Theory

2.1 Problem Formulation

Consider the following CS-PMRI measurement model

y=Ax +e,[1]

where x∈Cnis an unknown image, y= (y1,· · · ,ync) are the multi-coil measurements from nc≥1 coils,

e= (e1,· · · ,enc) is the noise vector, and A= (A1,· · · ,Anc) is the measurement operator (or forward

operator). The measurement model for each coil can be represented as

yk=P F Sk

| {z }

x+ek, k = 1,2, . . . , nc,[2]

where Sk∈Cn×nis the CSM of the kth coil, F∈Cn×nis the Fourier transform operator, P∈Cn×nis

the k-space sampling operator, and e∈Cnis the noise vector. Note that S= (S1,· · · ,Snc) varies for each

scan, since it depends on the relative location of the coils with the object being imaged. When Sare known,

image reconstruction can be formulated as regularized optimization

x= arg min

f(x) with f(x) = g(x) + h(x),[3]

where gis the data ﬁdelity term that quantiﬁes consistency with the observed data yand his a regularizer

that infuses prior knowledge on x. Examples of gand hused in CS-PMRI are the least-squares and total

variation (TV) functions (28)

g(x) = 1

2∥Ax −y∥2

2and h(x) = τ∥Dx∥1,[4]

where Ddenotes the image gradient and τ > 0 is a regularization parameter.

DL has recently gained popularity in MRI image reconstruction due to its excellent empirical perfor-

mance (7, 8, 29). Traditional DL methods are based on training CNNs (such as U-Net (30)) to map the

corrupted images (31,32) or the under-sampled measurements (33,34) to their desired fully sampled ground-

truth versions. There is also growing interest in DMBAs that can combine physical measurement models and

learned CNN priors. Well known examples of DMBAs are plug-and-play priors (PnP) (35,36), regularized by

denoising (RED) (37), and deep unfolding (DU) (38–40). In particular, DU has gained considerable recogni-

tion due to its ability to achieve the state-of-the-art performance, while providing robustness to changes in

data acquisition. DU architectures are typically obtained by unfolding iterations of an image reconstruction

algorithm as layers, representing the regularizer within image reconstruction as a CNN, and training the

resulting network end-to-end. Diﬀerent DU architectures can be obtained by using various optimization/re-

construction algorithms. In this paper, we will rely on a DU variant of the RED model as the basis of our

image reconstruction method (41).

2.2 Reconstruction using Pre-Calibrated CSMs

There are two widely-used image formation approaches in CS-PMRI (see recent review (8)): (a) reconstruc-

tion in the k-space domain and (b) reconstruction in the image domain. GRAPPA (2) is a well-known

example of (a) that ﬁlls in unacquired k-space values by linearly interpolating acquired neighboring k-space

samples. Recent work (42) extends GRAPPA by using a CNN to learn a non-linear interpolator in k-space.

SENSE (3) and ESPIRiT (4) are two well-known examples of (b) that ﬁrst pre-calibrate CSMs and then use

it to solve the inverse problem [1]. Our work in this paper adopts strategy (b), which will be the focus of

the subsequent discussion.

Pre-estimated CSMs can either be obtained by doing a separate calibration scan (43) or estimated directly

from the ACS region of the undersampled measurements. The drawback of the former approach is that it

extends the total scan time. ESPIRiT (4) is based on the latter approach. There are several issues and

challenges with the pre-estimated approaches (43, 44). One issue is that the inconsistencies between the

calibration scan and the accelerated scan can result in imaging artifacts. Another issue is that estimating

CSMs from a small number of ACS lines may not be suﬃciently accurate. DeepSENSE (45), a recent

supervised DL method, uses a CNN to learn a mapping from the ACS data to CSM references, obtained

by dividing the fully-sampled individual coil images by the sum-of-squares (SoS) reconstruction from the

fully-sampled measurements. While DeepSENSE improves over methods based on pre-estimating CSMs,

especially when the ACS data is limited, DeepSENSE still requires fully-sampled data to generate training

CSMs.

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SPICER:Self-SupervisedLearningforMRIwithAutomaticCoilSensitivityEstimationandReconstructionYuyangHu1,*,WeijieGan2,*,ChunweiYing3,TongyaoWang4,CihatEldeniz3,JiamingLiu1,YashengChen5,HongyuAn1,3,4,5,andUlugbekS.Kamilov1,21DepartmentofElectricalandSystemsEngineering,WashingtonUniversityinSt.Louis,St.Lo...

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SPICER Self-Supervised Learning for MRI with Automatic Coil Sensitivity Estimation and Reconstruction Yuyang Hu1 Weijie Gan2 Chunwei Ying3 Tongyao Wang4 Cihat Eldeniz3 Jiaming

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