This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TMI.2018.2878226, IEEE Transactions on Medical Imaging Non-local Low-rank Cube-based Tensor Factorization for Spectral CT Reconstruction Weiwen Wu, Fenglin Liu, Yanbo Zhang, Senior Member, IEEE, Qian Wang and Hengyong Yu, Senior Member, IEEE Abstract—Spectral computed tomography (CT) the DECT usually employs conventional detectors (i.e., energy- reconstructs material-dependent attenuation images from integrating detectors), and its results are often corrupted by the projections of multiple narrow energy windows which is beam hardening and spectral blurring. Besides, there are only meaningful for material identification and decomposition. two different energy source/detector pairs. As a result, only two Unfortunately, the multi-energy projection datasets usually or three basis material maps can be accurately decomposed. The have lower signal-noise-ratios (SNR). Very recently, a photon-counting detectors (PCDs) illuminate the prospects of spatial-spectral cube matching frame (SSCMF) was multi-energy CT in practical applications because PCDs can proposed to explore the non-local spatial-spectral distinguish each incident photon energy by recording pulse similarities for spectral CT. This method constructs a group height [4]. Theoretically speaking, compared with the by clustering up a series of non-local spatial-spectral cubes. conventional energy integrating detector, a PCD can improve The small size of spatial patches for such a group makes the signal-to-noise ratio with reduced dose by accounting the SSCMF fail to fully encode the sparsity and low-rank number of received photons. However, the PCD has different properties. The hard-thresholding and collaboration responses to individual photon’s energy. This can lead to filtering in the SSCMF also cause difficulty in recovering spectral distortions, including charge sharing, K-escape, the image features and spatial edges. While all the steps are fluorescence x-ray emission and pulse pileups. These operated on 4-D group, the huge computational cost and distortions can further corrupt the spectral CT projection memory load might not be affordable in practice. To avoid datasets with complicated noises [5]. Therefore, it is difficult to the above limitations and further improve image quality, we obtain higher signal-noise-ratio (SNR) projections and first formulate a non-local cube-based tensor instead of reconstruct satisfactory spectral CT images. Alternatively, high group to encode the sparsity and low-rank properties. Then, quality spectral images can be achieved with higher-powered as a new regularizer, the Kronecker- Basis-Representation PCD or superior reconstruction methods [6]. In this work, we (KBR) tensor factorization is employed into a basic spectral mainly focus on improving image quality by developing a more CT reconstruction model to enhance the capability of image powerful reconstruction algorithm. feature extraction and spatial edge preservation, generating Many attempts have been made to reconstruct high quality a non-local low-rank cube-based tensor factorization spectral CT images. According to the employed prior (NLCTF) method. Finally, the split-Bregman method is knowledge, in our opinion, all of these efforts can be divided adopted to solve the NLCTF model. Both numerical into two categories: empirical-knowledge and prior-image- simulations and preclinical mouse studies are performed to knowledge based methods [7]. The empirical-knowledge based validate and evaluate the NLCTF algorithm. The results methods first convert the spectral images into a unified and show that the NLCTF method outperforms other state-of- image-independent transformation domain, and then formulate the-art competing algorithms. a sparsity/low-rank reconstruction model of the transform Index Terms—spectral CT, image reconstruction, Kronecker- coefficients in terms of an L0-norm, nuclear-norm or L1-norm. Basis-Representation, tensor factorization, non-local image Considering the diversity of targets, different empirical- similarity. knowledge methods were employed, such as total variation (TV) [8], tensor-based nuclear norm [9], PRISM (prior rank, I. INTRODUCTION intensity and sparsity model) [10, 11], tensor PRISM [12, 13], HE spectral computed tomography (CT) has obtained a superiorization-based PRISM [14], piecewise linear tight great achievement in terms of tissue characterization [1], frame transform [15], total nuclear variation (TVN) [16], patch- Tlesion detection and material decomposition [2], etc. As a based low-rank [17], tensor nuclear norm (TNN) with TV [18], special case, the dual-energy CT (DECT) uses two different structure tensor TV [19], nonlocal low-rank and sparse matrix energy settings to discriminate material components in terms of decomposition [20], multi-energy non-local means (MENLM) their energy-related attenuation characteristics [3]. However, [21], spatial spectral nonlocal means [22], etc. However, image similarities in non-local spatial space are usually ignored This work was supported in part by the National Natural Science among these methods. Very recently, considering the non-local Foundation of China (No. 61471070), National Instrumentation Program of similarity within spatial-spectral space, we proposed a spatial- China (No. 2013YQ030629), NIH/NIBIB U01 grant (EB017140) and China Scholarship Council (No. 201706050070). spectral cube matching frame (SSCMF) algorithm by stacking Wu and Liu* ([email protected]) are with the Key Lab of Optoelectronic up a series of similar small cubes (4 × 4 × 4) to form a 4-D Technology and Systems, Ministry of Education, Chongqing University, group and then operating hard-thresholding and collaboration Chongqing 400044, China. Asterisk indicates the corresponding author.. filtering on the group [7] . The length of the patches in a group Zhang, Wang and Yu*(E-mail: [email protected]) are with the Department of Electrical and Computer Engineering, University of is usually too small to accurately characterize the sparsity and Massachusetts Lowell, Lowell, MA 01854, USA. Asterisk indicates the low-rank property. The hard-thresholding and collaboration corresponding author. filtering are rough in image feature recovery and spatial edge 0278-0062 (c) 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TMI.2018.2878226, IEEE Transactions on Medical Imaging preservation. Besides, both the hard-thresholding filtering and spectral CT reconstruction model. Third, because of the collaboration filtering are applied on the formulated 4D group, advantages of the split-Bregman frame [37, 38] for our and it might not be affordable for such huge computational cost application in spectral CT, it was employed to solve the NLCTF and memory load in practice. model rather than the ADMM strategy. The prior-image-knowledge based methods explore both The rest of this paper is organized as follows. In section II, image sparsity and similarity by adopting high quality prior the mathematic model is constructed and the reconstruction images, such as constructing a redundant dictionary [23]. A method is developed. In section III, numerical simulations and dual-dictionary learning (DDL) method was applied to sparse- preclinical experiments are designed and performed to validate view spectral CT reconstruction [24]. A tensor dictionary and evaluate the proposed algorithm. In section IV, some learning (TDL) was introduced to explore the image similarity related issues are discussed and conclusions are made. among different energy bins [25]. Considering the similarity between the image gradient of different energy bins, the image II. METHOD gradient L -norm was incorporated into the TDL (L TDL) 0 0 2.A. KBR-based Tensor Factorization framework for sparse-view spectral CT reconstruction [26]. The spectral prior image constrained compressed sensing A 푁푡ℎ order tensor can be denoted as 퓧 ∈ 퓡퐼1×퐼2×퐼3×…×퐼푁 . algorithm (spectral PICCS)[27], TV-TV and total variation The KBR measure for a tensor 퓧 can be expressed as: spectral mean (TV-SM) methods [28] can also be considered 푁 푚(퓧) = ‖퓒‖0 + 훼 ∏ 푟푎푛푘(푿(푛)) , (1) as prior-image-knowledge based methods, where a high quality 푛=1 퐼1×퐼2×퐼3×…×퐼푁 image is treated as prior to constrain the final solution [29]. where ‖. ‖0 represents the L0 norm, 퓒 ∈ 퓡 is Very recently, an average-image-incorporated BM3D the core tensor of 퓧 with higher order singular value technology was developed to enhance the correlations among decomposition (HOSVD), 푿(푛) represents the unfolding energy bin images [30]. However, the high quality prior images matrix with the mode-n, and 훼 > 0 is a tradeoff parameter to may not be available in practice. In addition, they do not fully balance the roles of two terms. The first term in (1) constrains utilize the similarities within a single channel. the number of Kronecker bases for representing the target To handle the aforementioned issues, in this paper, a non- tensor, complying with intrinsic mechanism of the CP local low-rank cube-based tensor (NLCT) will be constructed