Magnetic resonance imaging (MRI) is non-invasive and offers excellent spatial resolution with various contrast mechanisms to reveal different properties of underlying anatomy. However, MRI is associated with an inherently slow acquisition process since the signals of an MR image are acquired sequentially in signal space (k-space). One possible approach to accelerate the acquisition process is to under-sample k-space signals. Parallel imaging and the application of compressed sensing (CS) are the main methods of accelerating data acquisition by signal under-sampling [[1], [2], [3], [4], [5]].

Applying CS to MR image (CS-MRI) acquisition, images can be reconstructed from a sub-Nyquist sampling, assuming that the images are compressible. Under such an assumption, images can be reconstructed through nonlinear optimization using iterative algorithms. However, this method has an extremely long reconstruction time compared to the traditional Fourier-transform-based method.

Recently, deep learning methods have been applied to the CS-MRI reconstruction problem. These proposed methods have the advantage of being able to quickly reconstruct images in a single pass using an appropriately trained network. Some studies have demonstrated that convolutional neural networks (CNNs) are equal to or sometimes outperform conventional iterative methods in terms of both reconstruction quality and computation time [6].

CS reconstruction methods using CNNs (hereafter, CNN-CS) can be approximately categorized as an unrolling-based approach or an end-to-end approach. The unrolling-based approaches use a cascade of layers or CNNs to represent iterative reconstruction for CS-MRI. Hammernik et al. proposed the Variational Network, which is composed of a variational model based on generalized compressed sensing reconstruction formulation and deep learning, and uses an unrolled gradient descent scheme to learn all the parameters in the formulation [7]. Aggarwal et al. proposed Model Based Deep Learning (MoDL) [8], which uses a variational framework containing data consistency items and a learned CNN to capture the redundant information of the image, and unrolls it into a deep network. In addition to those methods, ADMM-CSNet [9] and ISTA-Net [10] were proposed as model-based unrolling iterative reconstruction methods. Yang et al. used the alternating direction method of multipliers (ADMM) [11] optimization algorithm to automatically update the parameters through the unrolling process [9]. Inspired by ADMM-CSNet, ISTA-Net was proposed in which the Iterative Soft Thresholding Algorithm (ISTA) [12] was transformed into a deep network form.

The most popular deep learning examples categorized as the end-to-end type are DnCNN [13], U-Net [14], and the Generative Adversarial Network (GAN) [15]. The introduction of residual learning [16] that learns the residuals between the true image and an input image can boost the training process of those CNNs. The application of the GAN to CS-MRI was first demonstrated by Mardani et al., who proposed GANCS [17], which uses a mixture of pixel-wise L1 and L2 loss functions and Least Squared Generative Adversarial Networks (LSGANs) [18]. Yang et al. proposed the Deep De-Aliasing Generative Adversarial Network (DAGAN) [19], which contains perceptual loss [20] as a loss function to evaluate the visual error. While all of the above methods belong to the end-to-end approach, a hybrid architecture that works in both k-space and the image space has also been proposed. Souza et al. constructed a hybrid network that uses two U-Nets, one for the image domain and one for the k-space domain [21].

Eo et al. proposed KIKI-net [22], in which a k-space network and image space network were alternately applied such that two loss functions in k-space and the image domain were iteratively minimized. Apart from sequential modular reconstruction chains, Zhu et al. proposed the AUTOMAP [23] that learns a reconstruction mapping between the sensor-domain data and the image-domain output.

MR signals and images are complex in nature, and research on image reconstruction using complex CNNs has attracted attention. Deepcomplex MRI [24] proposed parallel imaging that adopts a deep residual convolutional neural network that considers the correlation between the real and imaginary parts of complex MR images. Deepcomplex MRI has achieved better results than real-value networks. In MR fingerprinting, the method using complex CNN was shown to be superior to real function CNN with 2-channel input for real and imaginary parts [25]. Dedmari et al. proposed the Complex Dense Fully Convolutional Network (CDFNet) [26], which uses densely connected fully convolutional blocks to support deep learning operations on complex-valued data. They demonstrated that CDFNet outperforms the standard state-of-the-art and real-valued CNN in terms of recovering fine structures and high frequency textures. Because MR images are obtained as complex-valued images, the phase of the image provides important information about tissue and magnetic fields, and image reconstruction that ignores this phase information is limited in reconstruction performance and use.

Various approaches have been made to reconstruct complex-valued images in CNN-CS. We divide existing complex-valued CNN-CS methods into two types in this paper.

Almost all of the end-to-end type CNN-CS methods split a complex-valued image into a real and an imaginary part, and the CNN has two-channel inputs and outputs corresponding to the real and imaginary parts of an image. This method is the simplest and straightforward way to compute complex-valued images, but it has been pointed out that this approach discards some of the complex algebraic structure of the data and does not necessarily maintain the phase information of the data as it moves throughout the network [27]. Lee et al. proposed another method that learns the magnitude and phase of images using two CNNs [28]. In this method, the absolute value image and the phase image need to be trained separately by each CNN.

This type of method treats complex-valued image data using a complex-valued CNN. Yang et al. improved the ADMM-CSNet to correspond to complex values [9]. Additionally, CNN-CS methods that use complex-valued image data in the convolution layers have been studied by Cole et al. [27] and Wang et al. [24]. In Reference [27], real-valued and complex-valued versions of an ISTA-net-based unrolled CNN and a U-Net-based end-to-end CNN were compared from various perspectives. In Reference [24], a complex CNN was used in a cascaded network for parallel imaging. In both studies, it was reported that complex-valued CNNs showed superior results compared to real-valued CNNs. However, the best way to handle complex-valued data in CNNs has still not been fully clarified.

Most of the methods use complex values in the transfer between layers; however, the calculations within a layer are conducted using real-valued data. Not only the computation in the convolution layer, but also the activation function needs to correspond to complex numbers. For instance, modReLU [29], ℂReLU [30], and zReLU [30] have been proposed as the activation function to be used in a complex-valued CNN.

In this paper, we propose a novel CNN-CS reconstruction method that allows the computation of complex-valued images while using real-valued CNNs by exploiting unique signal under-sampling patterns. A property of Fourier transforms is that Fourier transforms of the real and imaginary parts of a complex image become an even and an odd function, respectively. Even-functioning or odd-functioning an MR signal is accomplished by adding or subtracting the MR signal and its spatially inverted complex conjugate signal. If sampling points are chosen so that symmetrical points with respect to the origin of k-space are collected, then even-functioning or odd-functioning an MR signal is possible. Therefore, it is possible to calculate MR signals corresponding to the real and imaginary parts of a complex MR image independently. The proposed method is different from conventional methods that simply split the complex-valued image into real and imaginary parts for CNN calculation in that k-space signals corresponding to the real and imaginary parts of the complex-valued image are calculated before training the CNN. This property of the method mathematically ensures the split of the complex-valued image into real and imaginary parts. Computing with real-valued CNNs has many advantages. First, the structure of the CNNs becomes simpler to construct and easier to train because they have fewer training parameters compared with complex-valued CNNs. Second, approximations for complexification are not made in the CNN, allowing for more accurate computations. Third, a single CNN can reconstruct both the real and imaginary parts of complex-valued images, allowing for data augmentation, i.e., the number of images used for training is twice as many as when computing with complex-valued CNNs. In this study, we examined the reconstruction performance of the proposed method in comparison to other CNN-CS methods and examined the effectiveness of the proposed method.