Calibrationless Reconstruction of Uniformly -Undersampled Multi -Channel MR Data with Deep Learning Estimated ESPIRiT Maps

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Calibrationless Reconstruction of Uniformly-Undersampled

Multi-Channel MR Data with Deep Learning Estimated

ESPIRiT Maps

Junhao Zhang1,2, Zheyuan Yi1,2,3, Yujiao Zhao1,2, Linfang Xiao1,2, Jiahao

Hu1,2,3, Christopher Man1,2, Vick Lau1,2, Shi Su1,2, Fei Chen3, Alex

T.L.Leong1,2, and Ed X. Wu1,2*

1Laboratory of Biomedical Imaging and Signal Processing

The University of Hong Kong, Hong Kong SAR, People’s Republic of China

2Department of Electrical and Electronic Engineering

The University of Hong Kong, Hong Kong SAR, People’s Republic of China

3Department of Electrical and Electronic Engineering

Southern University of Science and Technology, Shenzhen, People’s Republic of

China

*Correspondence to:

Ed X. Wu, Ph.D.

Department of Electrical and Electronic Engineering

The University of Hong Kong, Hong Kong SAR, China

Tel: (852) 3917-7096

Email: ewu@eee.hku.hk

SUBMITTED TO MAGNETIC RESONANCE IN MEDICINE

Short Running Title: Deep learning of coil maps for ESPIRiT reconstruction

Keywords: Calibrationless reconstruction, ESPIRiT, deep learning

Deep learning of coil maps for ESPIRiT reconstruction Zhang & Wu et al. Arxiv 2022

Page 2 of 29

ABSTRACT

Purpose: To develop a truly calibrationless reconstruction method that derives

ESPIRiT maps from uniformly-undersampled multi-channel MR data by deep

learning.

Methods: ESPIRiT, one commonly used parallel imaging reconstruction technique,

forms the images from undersampled MR k-space data using ESPIRiT maps that

effectively represents coil sensitivity information. Accurate ESPIRiT map estimation

requires quality coil sensitivity calibration or autocalibration data. We present a U-Net

based deep learning model to estimate the multi-channel ESPIRiT maps directly from

uniformly-undersampled multi-channel multi-slice MR data. The model is trained

using fully-sampled multi-slice axial brain datasets from the same MR receiving coil

system. To utilize subject-coil geometric parameters available for each dataset, the

training imposes a hybrid loss on ESPIRiT maps at the original locations as well as

their corresponding locations within the standard reference multi-slice axial stack.

The performance of the approach was evaluated using publicly available T1-weighed

brain and cardiac data.

Results: The proposed model robustly predicted multi-channel ESPIRiT maps from

uniformly-undersampled k-space data. They were highly comparable to the reference

ESPIRiT maps directly computed from 24 consecutive central k-space lines. Further,

they led to excellent ESPIRiT reconstruction performance even at high acceleration,

exhibiting a similar level of errors and artifacts to that by using reference ESPIRiT

maps.

Conclusion: A new deep learning approach is developed to estimate ESPIRiT maps

directly from uniformly-undersampled MR data. It presents a general strategy for

calibrationless parallel imaging reconstruction through learning from coil and

protocol specific data.

Deep learning of coil maps for ESPIRiT reconstruction Zhang & Wu et al. Arxiv 2022

Page 3 of 29

INTRODUCTION

Magnetic Resonance Imaging (MRI) is a critical technology in modern medicine.

MRI can generate anatomical and functional images with high resolution and flexible

contrasts1,2. However, MRI comes at the expense of long scan time when compared to

other medical imaging modalities. To speed up MRI data acquisition, parallel imaging

has been developed and used routinely nowadays in the clinic3. It takes advantage of

the spatial encoding effect of the MR multi-channel receiving coil system. Two

seminal parallel imaging reconstruction methods are sensitivity encoding (SENSE) in

image space4 and generalized partially parallel acquisitions (GRAPPA) in k-space5.

SENSE uses prior knowledge of coil sensitivity profiles to separate the folded pixels

in the image space that arise from the uniform undersampling in k-space. GRAPPA

synthesizes the missing k-space data using the GRAPPA weight kernels across all

channels. Both approaches need calibration data either from additional pre-scan or

autocalibration signals. Often there exists inconsistency between calibration data and

undersampled data, e.g., due to subject motion, causing artifacts in the reconstructed

images6,7. ESPIRiT8, following SENSE and GRAPPA, is an effective hybrid-space

reconstruction method. It utilizes k-space kernel operations to derive a set of

eigenvector maps, i.e., ESPIRiT maps, to effectively represent coil sensitivity

information. They are then incorporated in a generalized SENSE reconstruction.

ESPIRiT reconstruction with L1 regularization can lead to improved performance

when compared to the traditional SENSE and GRAPPA8. However, the ESPIRiT

reconstruction still requires quality coil calibration or autocalibration data, hindering

its robust applications in various imaging scenario. For instance, in abdominal9 and

cardiac parallel imaging10,11, the geometric mismatch could easily occur due to motion

between pre-scan coil sensitivity calibration data and subsequent imaging data,

degrading reconstruction quality. Additionally, fast-spin-echo (FSE) and echo-planar-

imaging (EPI) are among the most common template sequences in modern MR

scanners due to their high acquisition efficiency. Integration of autocalibration in FSE

sequence can complicate the phase encoding ordering and requires careful

optimization12,13. For EPI based parallel imaging, it is also difficult to integrate

autocalibration in the sequence because uniform undersampling in phase encoding

direction is often a pre-requisite for effective reduction of geometric distortions due to

field inhomogeneity14,15.

Several calibrationless MRI reconstruction methods have been proposed for parallel

imaging16-20. For example, low-rank reconstruction methods have emerged as the

alternatives, enabling simultaneous autocalibration and k-space approximation. They

include simultaneous auto-calibrating and k-space estimation (SAKE), parallel-

imaging low-rank matrix modeling of local k-space neighbors (P-LORAKS) and

annihilating filter-based low-rank Hankel matrix approach (ALOHA)21-23. These

methods construct the entire undersampled k-space data into a structured low-rank

matrix to recover missing k-space data through an iterative procedure using k-space

data consistency and rank truncation. They exploit the inherent nature of multi-

channel data and finite image spatial support. Although powerful, many such low-

rank completion methods are computationally demanding, hampering their

applications for high-resolution volume imaging or direct 3D reconstruction24,25. Note

that the low-rank based ENLIVE method26 combines ESPIRiT and NLINV27. It is

Deep learning of coil maps for ESPIRiT reconstruction Zhang & Wu et al. Arxiv 2022

Page 4 of 29

computationally efficient, yet still requires variable density sampling and cannot

readily accomondate uniform undersampling.

Deep learning has demonstrated tremendous success in various fields and shown great

potential to significantly advance MRI28,29. For example, GRAPPA-Net is a full end-

to-end convolutional neural network (CNN) model that first fills the missing k-space

lines using non-linear CNN-based interpolation functions and then maps the filled k-

space to the corresponding image space30. Variational network are proposed to learn

the non-linear mapping from aliased image to alias-free image, where the deep

learning model is embedded in the iterative image reconstruction with generalized

compressed sensing concept31. RAKI, another end-to-end reconstruction model,

reconstructs the fully-sampled MR k-space from undersampled MR k-space data32. In

general, these parallel imaging reconstruction methods explicitly require the

additional consecutive central k-space lines or certain level of coil sensitivity

information. More recently, DeepSENSE33 is proposed to reconstruct undersampled

k-space data through deep learning of coil sensitivity maps, representing a new deep

learning strategy in coil sensitivity map space. Yet it is not entirely calibrationless

since it aims to estimate high-resolution coil sensitivity maps from the low-resolution

ones that are derived from a small number of consecutive central space lines. Given

that simple uniform undersampling in k-space can be and has been widely integrated

into various clinical imaging protocols for acceleration or artifact correction, it is

imperative to develop deep learning methods for predicting coil sensitivity

information directly from uniformly-undersampled k-space data without any

calibration data.

In this study, we propose a calibrationless reconstruction framework that directly

estimates ESPIRiT maps from multi-channel uniformly-undersampled MR data via

deep learning and applies such maps for subsequent ESPIRiT reconstruction. For

demonstration, a U-Net based model for axial reconstruction is trained using fully-

sampled multi-slice axial brain datasets from the same MR receiving coil system. To

utilize subject-coil geometric parameters available for each dataset, the training

imposes a hybrid loss on ESPIRiT maps at the original locations and their

corresponding or transformed locations within the standard reference multi-slice axial

stack (i.e., with a fixed geometric relation to the coil system). The results show that

the deep learning model could reliably estimate the ESPIRiT maps from uniformly-

undersampled axial multi-slice MR data, led to excellent ESPIRiT image

reconstruction performance.

METHODS

Proposed deep learning estimated ESPIRiT maps for ESPIRiT reconstruction

ESPIRiT takes advantage of the autocalibration feature of GRAPPA to derive

SENSE-like relative maps related to coil sensitivity, i.e., ESPIRiT maps. These maps

are mathematically derived from the null space of the calibration matrix from the coil

calibration data by singular value decomposition (SVD)8,34.

Deep learning of coil maps for ESPIRiT reconstruction Zhang & Wu et al. Arxiv 2022

Page 5 of 29

The framework of our proposed ESPIRiT reconstruction is shown in Figure 1A. It

mainly consists of two steps: estimating first set of multi-channel ESPIRiT maps from

uniformly-undersampled MR data and applying the estimated maps for ESPIRiT

reconstruction. Specifically, a deep learning model is developed for mapping aliased

images to the corresponding ESPIRiT maps (Figure 1B). The input of the model is

the multi-channel aliased MR images from uniformly-undersampled multi-channel

data, while the output is the corresponding multi-channel ESPIRiT maps.

An attention U-Net model35-37 is adopted (Figure 1C). It constitutes a key element of

our proposed architecture. Note that the complex-valued input and target/output are

treated as separate two real-valued channels for real and imaginary parts,

respectively31,38,39. The number of filter channels in U-Net model is 12, 64, 128, 256,

512 and 1024 for the convolutional layers from undersampled data to latent feature

spaces while 1024, 512, 256, 128, 64 and 12 for layers from latent features to multi-

channel ESPIRiT maps. Additionally, channel-wise attention block is included in the

model to help effectively process the input information across multiple channels40.

The coil sensitivity information in any MRI system is largely coil-specific. When

scanning a particular subject in clinical MRI setting, the exact coil sensitivity profiles

or ESPIRiT maps within any imaging slice also depend on the orientation/position of

the slice with respect to MR receiving coil system. For example, typical multi-slice

axial head scan often involves slightly different orientation/position with respect to

the standard reference multi-slice axial stack geometry (i.e., the MR coil system, too)

due to variation in head position and slice localization by scanner operator (Figure

1D). Such subject-coil geometry information is available during the scan and recorded

in the standard DICOM header. During the training of our proposed model, we

incorporate these subject-coil parameters for each dataset by imposing a hybrid loss

function. Specifically, the model is trained by minimizing a hybrid L1 loss on two sets

of multi-slice multi-channel ESPIRiT maps as described in Equation (1).

(1)

ZHere θ are all the parameter sets in the model and λ is a learnable parameter to

control the loss contributions. Specifically, Eijoriginal and Eijtransformed represent two

ESPIRiT maps for ith channel at jth slice within their original multi-slice locations and

their transformed locations within the standard reference multi-slice axial stack,

corresponding to blue and red stacks in Figure 1D, respectively. Note that this

reference stack has a fixed orientation and position relative to magnet and gradient

coil center. Its orientation and position typically have a fixed geometric relation to

the coil system. Thus Eijtransformed should be mostly coil specific and dataset

independent. Meanwhile Eijoriginal will be dataset dependent since each multi-slice

axial head scan can be prescribed with slightly different geometry. In practice,

Eijtransformed and Eijoriginal differ from each other in position and orientation but will be

very similar to certain extent due to their geometric proximity and the spatial

smoothness nature of ESPIRiT maps. Note that the second term in Equation (1) here

could be considered as an indirect way to initialize the model training. Therefore,

incorporating Eijtransformed as part of the loss function for training will facilitate the

learning process through improving stabilization and convergency indirectly. With

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CalibrationlessReconstructionofUniformly-UndersampledMulti-ChannelMRDatawithDeepLearningEstimatedESPIRiTMapsJunhaoZhang1,2,ZheyuanYi1,2,3,YujiaoZhao1,2,LinfangXiao1,2,JiahaoHu1,2,3,ChristopherMan1,2,VickLau1,2,ShiSu1,2,FeiChen3,AlexT.L.Leong1,2,andEdX.Wu1,2*1LaboratoryofBiomedicalImagingandSignalPro...

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Calibrationless Reconstruction of Uniformly -Undersampled Multi -Channel MR Data with Deep Learning Estimated ESPIRiT Maps

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