Magnetic Resonance Imaging (MRI) is widely used in clinical practice, but suffers from prolonged acquisition time. Although deep learning methods have been proposed to accelerate acquisition and demonstrate promising performance, they rely on high-quality fully-sampled datasets for training in a supervised manner. However, such datasets are time-consuming and expensive-to-collect, which constrains their broader applications. On the other hand, self-supervised methods offer an alternative by enabling learning from under-sampled data alone, but most existing methods rely on further partitioned under-sampled k-space data as model's input for training, which causes an input distribution shift between the the training stage and the inference stage. Additionally, their models have not effectively incorporated comprehensive image priors, leading to degraded reconstruction performance. In this paper, we propose a novel re-visible dual-domain self-supervised deep unfolding network to address these issues when only under-sampled datasets are available. Specifically, by incorporating re-visible dual-domain loss, all under-sampled k-space data are utilized during training to mitigate the input distribution shift caused by further partitioning. This design enables the model to implicitly adapt to all under-sampled k-space data as input. Additionally, we design a Deep Unfolding Network based on Chambolle and Pock Proximal Point Algorithm (DUN-CP-PPA) to achieve end-to-end reconstruction. By employing a Spatial-Frequency Feature Extraction (SFFE) block to capture both global and local representations, the model effectively integrates imaging physics with comprehensive image priors to enhance reconstruction performance. Experiments on both single-coil and multi-coil datasets demonstrate that our method outperforms state-of-the-art approaches in terms of reconstruction performance and generalization capability.

In this study, we proposed a computer-aided diagnosis (CADx) framework under dual-energy spectral CT (DECT), which operates directly on the transmission data in the pre-log domain, called CADxDE, to explore the spectral information for lesion diagnosis. The CADxDE includes material identification and machine learning (ML) based CADx. Benefits from DECT's capability of performing virtual monoenergetic imaging with the identified materials, the responses of different tissue types (e.g., muscle, water, and fat) in lesions at each energy can be explored by ML for CADx. Without losing essential factors in the DECT scan, a pre-log domain model-based iterative reconstruction is adopted to obtain decomposed material images, which are then used to generate the virtual monoenergetic images (VMIs) at selected n energies. While these VMIs have the same anatomy, their contrast distribution patterns contain rich information along with the n energies for tissue characterization. Thus, a corresponding ML-based CADx is developed to exploit the energy-enhanced tissue features for differentiating malignant from benign lesions. Specifically, an original image-driven multi-channel three-dimensional convolutional neural network (CNN) and extracted lesion feature-based ML CADx methods are developed to show the feasibility of CADxDE. Results from three pathologically proven clinical datasets showed 4.01% to 14.25% higher AUC (area under the receiver operating characteristic curve) scores than the scores of both the conventional DECT data (high and low energy spectrum separately) and the conventional CT data. The mean gain >9.13% in AUC scores indicated that the energy spectral-enhanced tissue features from CADxDE have great potential to improve lesion diagnosis performance.

Incorporating deep learning (DL) into multiple-input multiple-output (MIMO) detection has been deemed as a promising technique for future wireless communications. However, most of the DL-based MIMO detection algorithms are lack of interpretation on internal mechanisms. In this paper, we analyze the performance of the DL-based MIMO detection to better understand its strengths and weaknesses. We investigate and compare two different models: data-driven DL detector with neural networks activated by rectifier linear unit (ReLU) function and model-driven DL detector based on traditional detection algorithms. We show that the data-driven DL detector asymptotically approaches to the maximum a posterior (MAP) detector in various scenarios but requires a large amount of training samples to converge in time-varying channels. On the other hand, the model-driven DL detector utilizes the expert knowledge to alleviate the impact of channels and achieves relatively high detection accuracy with a small set of training data. Simulation results confirm our analytical results and demonstrate the effectiveness of the DL-based MIMO detection for both linear and nonlinear signal systems.

Deep neural networks (DNNs) are shown to be excellent solutions to staggering and sophisticated problems in machine learning. A key reason for their success is due to the strong expressive power of function representation. For piecewise linear neural networks (PLNNs), the number of linear regions is a natural measure of their expressive power since it characterizes the number of linear pieces available to model complex patterns. In this article, we theoretically analyze the expressive power of PLNNs by counting and bounding the number of linear regions. We first refine the existing upper and lower bounds on the number of linear regions of PLNNs with rectified linear units (ReLU PLNNs). Next, we extend the analysis to PLNNs with general piecewise linear (PWL) activation functions and derive the exact maximum number of linear regions of single-layer PLNNs. Moreover, the upper and lower bounds on the number of linear regions of multilayer PLNNs are obtained, both of which scale polynomially with the number of neurons at each layer and pieces of PWL activation function but exponentially with the number of layers. This key property enables deep PLNNs with complex activation functions to outperform their shallow counterparts when computing highly complex and structured functions, which, to some extent, explains the performance improvement of deep PLNNs in classification and function fitting.

Deep learning (DL) has emerged as an effective tool for channel estimation in wireless communication systems, especially under some imperfect environments. However, even with such unprecedented success, DL methods are often regarded as black boxes and are lack of explanations on their internal mechanisms, which severely limits their further improvement and extension. In this paper, we present preliminary theoretical analysis on DL based channel estimation for single-input multiple-output (SIMO) systems to understand and interpret its internal mechanisms. As deep neural network (DNN) with rectified linear unit (ReLU) activation function is mathematically equivalent to a piecewise linear function, the corresponding DL estimator can achieve universal approximation to a large family of functions by making efficient use of piecewise linearity. We demonstrate that DL based channel estimation does not restrict to any specific signal model and asymptotically approaches to the minimum mean-squared error (MMSE) estimation in various scenarios without requiring any prior knowledge of channel statistics. Therefore, DL based channel estimation outperforms or is at least comparable with traditional channel estimation, depending on the types of channels. Simulation results confirm the accuracy of the proposed interpretation and demonstrate the effectiveness of DL based channel estimation under both linear and nonlinear signal models.

Energy-resolved computed tomography (ErCT) with a photon counting detector concurrently produces multiple CT images corresponding to different photon energy ranges. It has the potential to generate energy-dependent images with improved contrast-to-noise ratio and sufficient material-specific information. Since the number of detected photons in one energy bin in ErCT is smaller than that in conventional energy-integrating CT (EiCT), ErCT images are inherently more noisy than EiCT images, which leads to increased noise and bias in the subsequent material estimation. In this work, we first deeply analyze the intrinsic tensor properties of two-dimensional (2D) ErCT images acquired in different energy bins and then present a F ull- S pectrum-knowledge-aware Tensor analysis and processing (FSTensor) method for ErCT reconstruction to suppress noise-induced artifacts to obtain high-quality ErCT images and high-accuracy material images. The presented method is based on three considerations: (1) 2D ErCT images obtained in different energy bins can be treated as a 3-order tensor with three modes, i.e., width, height and energy bin, and a rich global correlation exists among the three modes, which can be characterized by tensor decomposition. (2) There is a locally piecewise smooth property in the 3-order ErCT images, and it can be captured by a tensor total variation regularization. (3) The images from the full spectrum are much better than the ErCT images with respect to noise variance and structural details and serve as external information to improve the reconstruction performance. We then develop an alternating direction method of multipliers algorithm to numerically solve the presented FSTensor method. We further utilize a genetic algorithm to tackle the parameter selection in ErCT reconstruction, instead of manually determining parameters. Simulation, preclinical and synthesized clinical ErCT results demonstrate that the presented FSTensor method leads to significant improvements over the filtered back-projection, robust principal component analysis, tensor-based dictionary learning and low-rank tensor decomposition with spatial-temporal total variation methods.

Markov random field (MRF) has been widely used to incorporate a priori knowledge as penalty or regularizer to preserve edge sharpness while smoothing the region enclosed by the edge for pieces-wise smooth image reconstruction. In our earlier study, we proposed a type of MRF reconstruction method for low-dose CT (LdCT) scans using tissue-specific textures extracted from the same patient's previous full-dose CT (FdCT) scans as prior knowledge. It showed advantages in clinical applications. This paper aims to remove the constraint of using previous data of the same patient. We investigated the feasibility of extracting the tissue-specific MRF textures from an FdCT database to reconstruct a LdCT image of another patient. This feasibility study was carried out by experiments designed as follows. We constructed a tissue-specific MRF-texture database from 3990 FdCT scan slices of 133 patients who were scheduled for lung nodule biopsy. Each patient had one FdCT scan (120 kVp/100 mAs) and one LdCT scan (120 kVp/20 mAs) prior to biopsy procedure. When reconstructing the LdCT image of one patient among the 133 patients, we ranked the closeness of the MRF-textures from the other 132 patients saved in the database and used them as the a prior knowledge. Then, we evaluated the reconstructed image quality using Haralick texture measures. For any patient within our database, we found more than eighteen patients' FdCT MRF texures can be used without noticeably changing the Haralick texture measures on the lung nodules (to be biopsied). These experimental outcomes indicate it is promising that a sizable FdCT texture database could be used to enhance Bayesian reconstructions of any incoming LdCT scans.

Reducing the exposure to X-ray radiation while maintaining a clinically acceptable image quality is desirable in various CT applications. To realize low-dose CT (LdCT) imaging, model-based iterative reconstruction (MBIR) algorithms are widely adopted, but they require proper prior knowledge assumptions in the sinogram and/or image domains and involve tedious manual optimization of multiple parameters. In this paper, we propose a deep learning (DL)-based strategy for MBIR to simultaneously address prior knowledge design and MBIR parameter selection in one optimization framework. Specifically, a parameterized plug-and-play alternating direction method of multipliers (3pADMM) is proposed for the general penalized weighted least-squares model, and then, by adopting the basic idea of DL, the parameterized plug-and-play (3p) prior and the related parameters are optimized simultaneously in a single framework using a large number of training data. The main contribution of this paper is that the 3p prior and the related parameters in the proposed 3pADMM framework can be supervised and optimized simultaneously to achieve robust LdCT reconstruction performance. Experimental results obtained on clinical patient datasets demonstrate that the proposed method can achieve promising gains over existing algorithms for LdCT image reconstruction in terms of noise-induced artifact suppression and edge detail preservation.

Cerebrovascular diseases, i.e., acute stroke, are a common cause of serious long-term disability. Cerebral perfusion computed tomography (CPCT) can provide rapid, high-resolution, quantitative hemodynamic maps to assess and stratify perfusion in patients with acute stroke symptoms. However, CPCT imaging typically involves a substantial radiation dose due to its repeated scanning protocol. Therefore, in this paper, we present a low-dose CPCT image reconstruction method to yield high-quality CPCT images and high-precision hemodynamic maps by utilizing the great similarity information among the repeated scanned CPCT images. Specifically, a newly developed low-rank tensor decomposition with spatial-temporal total variation (LRTD-STTV) regularization is incorporated into the reconstruction model. In the LRTD-STTV regularization, the tensor Tucker decomposition is used to describe global spatial-temporal correlations hidden in the sequential CPCT images, and it is superior to the matricization model (i.e., low-rank model) that fails to fully investigate the prior knowledge of the intrinsic structures of the CPCT images after vectorizing the CPCT images. Moreover, the spatial-temporal TV regularization is used to characterize the local piecewise smooth structure in the spatial domain and the pixels' similarity with the adjacent frames in the temporal domain, because the intensity at each pixel in CPCT images is similar to its neighbors. Therefore, the presented LRTD-STTV model can efficiently deliver faithful underlying information of the CPCT images and preserve the spatial structures. An efficient alternating direction method of multipliers algorithm is also developed to solve the presented LRTD-STTV model. Extensive experimental results on numerical phantom and patient data are clearly demonstrated that the presented model can significantly improve the quality of CPCT images and provide accurate diagnostic features in hemodynamic maps for low-dose cases compared with the existing popular algorithms.

Prior-image-based reconstruction (PIBR) methods have demonstrated great potential for radiation dose reduction in computed tomography applications. PIBR methods take advantage of shared anatomical information between sequential scans by incorporating a patient-specific prior image into the reconstruction objective function, often as a form of regularization. However, one major challenge with PIBR methods is how to optimally determine the prior image regularization strength which balances anatomical information from the prior image with data fitting to the current measurements. Too little prior information yields limited improvements over traditional model-based iterative reconstruction, while too much prior information can force anatomical features from the prior image not supported by the measurement data, concealing true anatomical changes. In this paper, we develop quantitative measures of the bias associated with PIBR. This bias exhibits as a fractional reconstructed contrast of the difference between the prior image and current anatomy, which is quite different from traditional reconstruction biases that are typically quantified in terms of spatial resolution or artifacts. We have derived an analytical relationship between the PIBR bias and prior image regularization strength and illustrated how this relationship can be used as a predictive tool to prospectively determine prior image regularization strength to admit specific kinds of anatomical change in the reconstruction. Because bias is dependent on local statistics, we further generalized shift-variant prior image penalties that permit uniform (shift invariant) admission of anatomical changes across the imaging field of view. We validated the mathematical framework in phantom studies and compared bias predictions with estimates based on brute force exhaustive evaluation using numerous iterative reconstructions across regularization values. The experimental results demonstrate that the proposed analytical approach can predict the bias-regularization relationship accurately, allowing for prospective determination of the prior image regularization strength in PIBR. Thus, the proposed approach provides an important tool for controlling image quality of PIBR methods in a reliable, robust, and efficient fashion.

Dynamic cerebral perfusion computed tomography (DCPCT) has the ability to evaluate the hemodynamic information throughout the brain. However, due to multiple 3-D image volume acquisitions protocol, DCPCT scanning imposes high radiation dose on the patients with growing concerns. To address this issue, in this paper, based on the robust principal component analysis (RPCA, or equivalently the low-rank and sparsity decomposition) model and the DCPCT imaging procedure, we propose a new DCPCT image reconstruction algorithm to improve low-dose DCPCT and perfusion maps quality via using a powerful measure, called Kronecker-basis-representation tensor sparsity regularization, for measuring low-rankness extent of a tensor. For simplicity, the first proposed model is termed tensor-based RPCA (T-RPCA). Specifically, the T-RPCA model views the DCPCT sequential images as a mixture of low-rank, sparse, and noise components to describe the maximum temporal coherence of spatial structure among phases in a tensor framework intrinsically. Moreover, the low-rank component corresponds to the "background" part with spatial-temporal correlations, e.g., static anatomical contribution, which is stationary over time about structure, and the sparse component represents the time-varying component with spatial-temporal continuity, e.g., dynamic perfusion enhanced information, which is approximately sparse over time. Furthermore, an improved nonlocal patch-based T-RPCA (NL-T-RPCA) model which describes the 3-D block groups of the "background" in a tensor is also proposed. The NL-T-RPCA model utilizes the intrinsic characteristics underlying the DCPCT images, i.e., nonlocal self-similarity and global correlation. Two efficient algorithms using alternating direction method of multipliers are developed to solve the proposed T-RPCA and NL-T-RPCA models, respectively. Extensive experiments with a digital brain perfusion phantom, preclinical monkey data, and clinical patient data clearly demonstrate that the two proposed models can achieve more gains than the existing popular algorithms in terms of both quantitative and visual quality evaluations from low-dose acquisitions, especially as low as 20 mAs.

Image textures in computed tomography colonography (CTC) have great potential for differentiating non-neoplastic from neoplastic polyps and thus can advance the current CTC detection-only paradigm to a new level with diagnostic capability. However, image textures are frequently compromised, particularly in low-dose CT imaging. Furthermore, texture feature extraction may vary, depending on the polyp spatial orientation variation, resulting in variable results. To address these issues, this study proposes an adaptive approach to extract and analyze the texture features for polyp differentiation. Firstly, derivative (e.g. gradient and curvature) operations are performed on the CT intensity image to amplify the textures with adequate noise control. Then Haralick co-occurrence matrix (CM) is used to calculate texture measures along each of the 13 directions (defined by the first and second order image voxel neighbors) through the polyp volume in the intensity, gradient and curvature images. Instead of taking the mean and range of each CM measure over the 13 directions as the so-called Haralick texture features, Karhunen-Loeve transform is performed to map the 13 directions into an orthogonal coordinate system so that the resulted texture features are less dependent on the polyp orientation variation. These simple ideas for amplifying textures and stabilizing spatial variation demonstrated a significant impact for the differentiating task by experiments using 384 polyp datasets, of which 52 are non-neoplastic polyps and the rest are neoplastic polyps. By the merit of area under the curve of receiver operating characteristic, the innovative ideas achieved differentiation capability of 0.8016, indicating the CTC diagnostic feasibility.

Markov random field (MRF) model has been widely employed in edge-preserving regional noise smoothing penalty to reconstruct piece-wise smooth images in the presence of noise, such as in low-dose computed tomography (LdCT). While it preserves edge sharpness, its regional smoothing may sacrifice tissue image textures, which have been recognized as useful imaging biomarkers, and thus it may compromise clinical tasks such as differentiating malignant vs. benign lesions, e.g., lung nodules or colon polyps. This study aims to shift the edge-preserving regional noise smoothing paradigm to texture-preserving framework for LdCT image reconstruction while retaining the advantage of MRF's neighborhood system on edge preservation. Specifically, we adapted the MRF model to incorporate the image textures of muscle, fat, bone, lung, etc. from previous full-dose CT (FdCT) scan as a priori knowledge for texture-preserving Bayesian reconstruction of current LdCT images. To show the feasibility of the proposed reconstruction framework, experiments using clinical patient scans were conducted. The experimental outcomes showed a dramatic gain by the a priori knowledge for LdCT image reconstruction using the commonly-used Haralick texture measures. Thus, it is conjectured that the texture-preserving LdCT reconstruction has advantages over the edge-preserving regional smoothing paradigm for texture-specific clinical applications.

In the above paper (IEEE Trans. on Medical Imaging, vol. 35, no. 6, Mar. 2016, pp. 860-870), the first footnote should have noted the following information. Hao Zhang, Hao Han contributed equally to this work. Zhengrong Liang is the corresponding author.

Previous studies have shown that by minimizing the total variation (TV) of the to-be-estimated image with some data and/or other constraints, a piecewise-smooth X-ray computed tomography image can be reconstructed from sparse-view projection data. However, due to the piecewise constant assumption for the TV model, the reconstructed images are frequently reported to suffer from the blocky or patchy artifacts. To eliminate this drawback, we present a total variation-stokes-projection onto convex sets (TVS-POCS) reconstruction method in this paper. The TVS model is derived by introducing isophote directions for the purpose of recovering possible missing information in the sparse-view data situation. Thus the desired consistencies along both the normal and the tangent directions are preserved in the resulting images. Compared to the previous TV-based image reconstruction algorithms, the preserved consistencies by the TVS-POCS method are expected to generate noticeable gains in terms of eliminating the patchy artifacts and preserving subtle structures. To evaluate the presented TVS-POCS method, both qualitative and quantitative studies were performed using digital phantom, physical phantom and clinical data experiments. The results reveal that the presented method can yield images with several noticeable gains, measured by the universal quality index and the full-width-at-half-maximum merit, as compared to its corresponding TV-based algorithms. In addition, the results further indicate that the TVS-POCS method approaches to the gold standard result of the filtered back-projection reconstruction in the full-view data case as theoretically expected, while most previous iterative methods may fail in the full-view case because of their artificial textures in the results.

In the above-named article [ibid., vol. 33, no. 3, pp. 749-763, Mar. 2014], Y. Liu was incorrectly listed as the corresponding author. Z. Liang should have been indicated as the corresponding author.