Multi-Slice Low-Rank Tensor Decomposition Based Multi-Atlas Segmentation: Application to Automatic Pathological Liver CT Segmentation
Changfa Shi, Min Xian, Xiancheng Zhou, Haotian Wang, Heng-Da Cheng
MMulti-Slice Low-Rank Tensor Decomposition BasedMulti-Atlas Segmentation: Application to AutomaticPathological Liver CT Segmentation
Changfa Shi a,b , Min Xian c, ∗ , Xiancheng Zhou a , Haotian Wang c ,Heng-Da Cheng b a Mobile E-business Collaborative Innovation Center of Hunan Province, Hunan Universityof Technology and Business, Changsha 410205, China b Department of Computer Science, Utah State University, Logan, UT 84322, USA c Department of Computer Science, University of Idaho, Idaho Falls, ID 83402, USA
Abstract
Liver segmentation from abdominal CT images is an essential step for livercancer computer-aided diagnosis and surgical planning. However, both the ac-curacy and robustness of existing liver segmentation methods cannot meet therequirements of clinical applications. In particular, for the common clinicalcases where the liver tissue contains major pathology, current segmentationmethods show poor performance. In this paper, we propose a novel low-ranktensor decomposition (LRTD) based multi-atlas segmentation (MAS) frame-work that achieves accurate and robust pathological liver segmentation of CTimages. Firstly, we propose a multi-slice LRTD scheme to recover the underlyinglow-rank structure embedded in 3D medical images. It performs the LRTD onsmall image segments consisting of multiple consecutive image slices. Then, wepresent an LRTD-based atlas construction method to generate tumor-free liveratlases that mitigates the performance degradation of liver segmentation due tothe presence of tumors. Finally, we introduce an LRTD-based MAS algorithmto derive patient-specific liver atlases for each test image, and to achieve accu-rate pairwise image registration and label propagation. Extensive experimentson three public databases of pathological liver cases validate the effectivenessof the proposed method. Both qualitative and quantitative results demonstratethat, in the presence of major pathology, the proposed method is more accurateand robust than state-of-the-art methods.
Keywords:
Pathological liver segmentation, Low-rank tensor decomposition,Multi-atlas segmentation, Tensor robust PCA, (cid:63) M -product. ∗ Corresponding author. Tel: +1-208-757-5425.
Email address: [email protected] (Min Xian)
Preprint submitted to Elsevier February 25, 2021 a r X i v : . [ ee ss . I V ] F e b . Introduction According to GLOBOCAN 2018 estimates, liver cancer is the sixth mostcommon cancer and the fourth leading cause of cancer mortality around theworld (Bray et al., 2018). Liver segmentation (the extraction of the liver fromits surrounding tissue) of abdominal computed tomography (CT) images is a keystep and a prerequisite for liver cancer computer-aided diagnosis (CAD), surgi-cal planning and other interventional procedures. However, in current clinicalpractice, liver segmentation from abdominal CT images is still predominantlyperformed manually by expert radiologists in a slice-by-slice manner. Due tothe abundance of the CT images for each patient, the manual segmentation istime-consuming, and subject to observer error and personal bias. Therefore,it is highly desirable to develop fully automated liver segmentation approachesthat can efficiently and automatically extract the liver boundary without anyuser intervention.Numerous liver segmentation methods have been published in the last fewdecades. They can be generally classified into traditional image-based methods,model-based methods, and deep learning-based methods (Erdt et al., 2012).The traditional image-based methods are mainly relies on image intensity infor-mation to perform liver segmentation, such as intensity thresholding (Kobashiand Shapiro, 1995) and region growing (Rusk´o et al., 2009), and they tend tohave poor performance for clinical liver cases. The model-based methods, suchas active shape model (ASM) (Heimann and Meinzer, 2009) and multi-atlassegmentation (MAS) (Iglesias and Sabuncu, 2015), have yielded remarkable re-sults in liver CT segmentation (Heimann et al., 2009), where shape, appearanceand spatial location information of the liver tissue were incorporated into thesegmentation framework as prior knowledge. In recent years, deep learning-based methods achieve popularity in the field of medical image analysis dueto their tremendous success in the computer vision community, and they haveachieved state-of-the-art performance in medical image segmentation (Litjenset al., 2017). The main advantage of deep learning-based methods is that themost relevant features are automatically generated and selected for the givenproblem, rather than being manually engineered. However, when applied toliver CT image segmentation, the model-based methods proved to have com-parable performance to deep learning-based methods (Ahn et al., 2019). Themain reason is that the performance of deep learning-based methods highly de-pends on the availability of massive amounts of training data, which cannot befully met in the case of liver CT image segmentation. Furthermore, comparedto model-based methods, the interpretability of deep learning-based methods ispoor, which, however, is of paramount importance in clinical applications.Nevertheless, the drawback of aforementioned liver segmentation methods isthat they still cannot meet the performance requirements of clinical applications.In particular, for common clinical cases where the liver tissue contains majorpathology, current liver segmentation methods still show poor performance. It ismainly because the hypotheses of most current liver segmentation methods areonly applicable to segment liver tissue of healthy or minor pathology conditions,2 a) (b) (c)Figure 1: Examples demonstrating challenges in accurate and robust pathological liversegmentation in CT scans, consisting of liver tissue with (a) hypodense tumor, (b) hyperdensetumor, and (c) tumors located at liver boundary. rather than of major pathology.Specifically, the first challenge of pathological liver segmentation is the pres-ence of large tumors, which exhibit totally different intensity values from thatof the normal liver tissue. The large tumors can cause various undesired shapedeformation of the liver tissue. Moreover, the tumors show very large variabilityof size and image appearance, resulting in tumors with different contrast levels,such as hypodense tumors (Fig. 1a) and hyperdense tumors (Fig. 1b). Thesecond challenge of pathological liver segmentation is the complex spatial vari-ability of tumors at liver boundary or inside liver parenchyma. In particular,when the tumors are at liver boundary (Fig. 1c), there is a very high probabil-ity that the peripheral liver tissue will be excluded from the final segmentationresults, leading to under-segmentation. It is mainly due to the existence ofblurred boundaries between the peripheral tumors and the nearby tissue (e.g.,the muscle tissue) exhibiting similar image appearance.Therefore, in the routine clinical setting, pathological liver segmentationis still mostly performed manually by expert radiologists, which is very labor-intensive, and subject to high intra- and inter-operator variability. Furthermore,since the liver tissue with large tumors exhibits totally different shape and im-age appearance from that of the normal liver tissue, it is also very challengingeven for the expert radiologists to perform the segmentation task manually. Re-cently, a few pathological liver segmentation methods have been proposed inthe literature (refer to Section 2.1 for a comprehensive survey). Nevertheless,the performance of these segmentation approaches is still unsatisfactory in thepresence of major pathology.In this paper, to address the above-mentioned issues, we integrate the gen-eral (cid:63) M -product based low-rank tensor decomposition (LRTD) theory into thewidely used MAS framework, and propose a novel automatic method for ac-curate and robust pathological liver segmentation of abdominal CT images. Itis well-known that the segmentation accuracy of the MAS framework highlydepends on the quality of the constructed atlases and pairwise image registra-3ions (Iglesias and Sabuncu, 2015). When employed to perform pathologicalliver CT segmentation, the main challenge to traditional MAS framework is thepresence of large tumors, which exhibit totally different intensity values fromthat of the normal liver tissue, leading to liver atlases of low quality and largeerrors in pairwise image registrations. Inspired by the recently popular LRTDtheory (Kolda and Bader, 2011), also known as Tensor Robust Principal Com-ponent Analysis (TRPCA) (Lu et al., 2020), in the fields of signal processingand computer vision, we first propose a multi-slice LRTD scheme for low-rankstructure learning in 3D medical images. Specifically, we partition each liver CTimage into smaller segments consisting of multiple consecutive image slices, andperform the LRTD on each segment sequentially. Then we present an LRTD-based atlas construction method to obtain tumor-free liver atlases that mitigatethe performance degradation of liver segmentation caused by the presence oftumors. Specifically, given a data tensor D , the LRTD model decomposes itinto two parts: (1) a low-rank component L corresponding to the tumor-freeliver images via tensor rank minimization, and (2) a sparse component E cor-responding to the sparse tumors via (cid:96) -norm minimization. Thus, the LRTDmodel fits the goal of deriving tumor-free liver images very well. Furthermore,we introduce an LRTD-based MAS algorithm to derive patient-specific liver at-lases for each test image, and to yield accurate pairwise image registration andlabel propagation.In order to evaluate the performance of our proposed MAS-based segmen-tation framework, and to show its clinical applicability to pathological liversegmentation, we extensively tested it using three public clinical CT databases,and also compared it with state-of-the-art methods. The experimental resultsdemonstrate that the proposed method yields higher accuracy and robustnessthan that of state-of-the-art methods.The main contributions of the proposed LRTD-based MAS framework canbe summarized as follows:(1) A general multi-slice LRTD scheme is proposed to recover the underly-ing low-rank structure embedded in 3D medical images. In particular, thediscrete cosine transform (DCT) converts the calculation of tensor singu-lar value decomposition (t-SVD) to matrix SVD in the transform domain,which enhances the computational efficiency and maintains spatial relation-ship. Then the tensor singular value thresholding (t-SVT) algorithm is usedto recover the underlying low-rank structure. The general scheme is also ap-plicable to other medical imaging modalities and organs ( Qin et al. (2019);Xian et al. (2018); Khaleel et al. (2018)) (Section 4.1).(2) An LRTD-based atlas construction method is developed to produce tumor-free liver atlases. In the MAS framework, atlases with tumor regions willlead to inaccurate pairwise image registration and performance degradation(Section 4.1).(3) An LRTD-based MAS algorithm is proposed to achieve more accurate androbust liver segmentation. An atlas selection strategy is first implemented4o obtain patient-specific liver atlases for each test image. Then, basedon the selected tumor-free liver atlases, it generates a tumor-free test im-age that yields accurate pairwise image registration and label propagation(Section 4.2).(4) We conducted extensive experiments to compare the proposed method tostate-of-the-art methods using three public clinical CT databases. It isshown that our method is more accurate and robust than state-of-the-artmethods in the presence of major pathology (Section 6).
2. Related Work
Li et al. (2020) proposed a pathological liver CT segmentation method bycombining level set, sparse shape composition, and graph cut methods. Theinitial liver shape was obtained by a level set method integrated with intensitybias and position constraint, followed by sparse shape composition and graphcut based segmentation refinements. Raju et al. (2020) proposed a user-guideddomain adaptation framework for pathological liver CT segmentation, whichused prediction-based adversarial domain adaptation to guide mask predictionsby the user interactions. Dakua et al. (2016) proposed a semi-automatic methodfor pathological liver CT segmentation, where a stochastic resonance algorithmwas utilized to enhance the contrast of the liver images, followed by cellularautomaton and level set segmentation methods. Umetsu et al. (2014) proposeda graph cut based method for segmenting liver CT cases with unusual shapesand pathological lesions, where a sparse representation based patient-specificprobabilistic atlas reinforced by the lesion bases was incorporated with the graphcut method. Nevertheless, in the presence of major pathology, the performanceof the above-mentioned liver CT segmentation methods is still not satisfactory,and further improvements are needed.
The LRTD theory has been widely studied and applied in the fields of signalprocessing and computer vision (Sidiropoulos et al., 2017; Cichocki et al., 2015;Sobral and Zahzah, 2017). Recently, it has gained wide applications in the fieldof medical image computing (Madathil et al., 2019), including medical imagereconstruction, super-resolution, denoising, and analysis. The images from mostmedical imaging modalities are currently acquired as 3D volumes, such as CT,Magnetic Resonance (MR). Since the LRTD models the 3D volumes in theirnative format (i.e., tensor) rather than vectorizing them into 1D vectors, it cansimultaneously exploits both the spatial and temporal correlations embedded inthe 3D volumes (Madathil et al., 2019).Liu et al. (2020) and Roohi et al. (2017) proposed multi-dimensional ap-proaches to the problem of dynamic MRI reconstruction of under-sampled k-space by formulating it as an LRTD problem, and the recovery performance5as superior to that of other reconstruction methods. Shi et al. (2015) pro-posed an MR image super-resolution method that integrated both local andglobal information for effective image recovery via total variation and low-ranktensor regularizations, respectively. A few LRTD-based image denoising meth-ods were proposed for MR (Khaleel et al., 2018; Fu and Dong, 2016) and CTimages (Sagheer and George, 2019) to reduce noise and artifacts introduced dur-ing image acquisition. Jiang et al. (2020) proposed a functional connectivitynetwork estimation approach by the assumption that the functional connectiv-ity networks have similar topology across subjects via the LRTD. Qin et al.(2019) proposed an LRTD-based method for accurately recovering vessel struc-tures and intensity information from the X-ray coronary angiography (XCA)sequences. Xu et al. (2016) proposed an LRTD-based one-step method foraxial alignment in 360-degree anterior chamber optical coherence tomography.To the best of our knowledge, the proposed method is the first MAS frameworkdirectly utilizing the LRTD for medical image segmentation.
The proposed LRTD-based MAS framework is partly inspired by two recentpapers ( Liu et al. (2015) and Shi et al. (2017)). They both proposed an atlas-based organ segmentation method by utilizing low-rank matrix decomposition(LRMD) theory (Zhou et al., 2014) to handle clinical cases with pathology. Inthis study, we significantly extend their methods in the following directions:(1) We propose a new MAS framework based on the LRTD, rather than theLRMD as in their original methods. To perform the LRMD, the 3D datatensor is first reformatted to a matrix by vectorizing voxel intensity valuesof each CT scan to form the column vectors. The local spatial informationis thus completely lost, and the multi-dimensional structure embedded inthe tensor data is disregarded, leading to considerable performance degra-dation. While our tensor-based method can fully exploit the intrinsic three-dimensional structural information of the CT scans, resulting in much moreaccurate data decomposition results.(2) The whole image volumes were directly employed to perform the LRMD intheir original methods, whereas we propose a new multi-slice LRTD scheme,where each liver CT image is first partitioned into smaller segments con-sisting of multiple consecutive image slices, then the LRTD is performed oneach segment sequentially. Since smaller image segment exhibits fewer over-all structural changes, it will lie on a low-rank subspace (Lee et al., 2018).It can thus yield more accurate data decomposition results than directlyperforming the LRMD on whole image volumes.
3. Mathematical Notations and Tensor Preliminaries
In this section, the notations and preliminaries of tensor used in the rest ofthe paper are briefly described. Throughout this paper, we follow the notation6nd terminology of Kolda and Bader (2011), Lu et al. (2019) and Kilmer et al.(2019).
A third-order tensor and its entries are denoted by capital boldface Eulerscript letters and small symbols, respectively, e.g., x ijk is the ( i, j, k )-th entryof the tensor X ∈ R n × n × n . Matrix is denoted by capital boldface letters,e.g., X ∈ R n × n . The two-dimensional horizontal, lateral, and frontal slicesof a third-order tensor X are denoted by X ( i, : , :), X (: , j, :), and X (: , : , k ), re-spectively. The frontal slices are often denoted more compactly as X ( k ) , i.e., X ( k ) = X (: , : , k ). The mode- k fibers of a third-order tensor X are vectors de-fined by fixing all indices but the k -th, e.g., the mode-3 fibers are denoted by X ( i, j, :).The inner product between two matrices is defined as (cid:104) X , Y (cid:105) = tr ( X ∗ Y ),where X ∗ and tr ( · ) denote the conjugate transpose of X and the matrix trace,respectively. The inner product between two tensors can then be defined as (cid:104) X , Y (cid:105) = (cid:80) n k =1 (cid:68) X ( k ) , Y ( k ) (cid:69) . Three tensor norms are used: (cid:107) X (cid:107) the (cid:96) -norm(i.e., the number of non-zero entries in X ), (cid:107) X (cid:107) = (cid:80) ijk | x ijk | the (cid:96) -norm,and (cid:107) X (cid:107) ∞ = max ijk | x ijk | the infinity norm. Mode-3 product . The mode-3 product ( × ) is an operator between atensor X ∈ R n × n × n and a matrix M ∈ R n × n . The result is another tensor¯ X ∈ R n × n × n defined as the following (Kolda and Bader, 2011):¯ X = X × M = M · X (3) , (1)where X (3) ∈ R n × n n denotes the mode-3 unfolding of X , defined as a ma-trix whose columns consist of the mode-3 fibers. The mode-3 product can beinterpreted geometrically as performing a linear transform on X along the thirddimension via transform matrix M . We can denote it more compactly as M ( X ). t-product . The t-product ( ∗ ) between two tensors X ∈ R n × n × n and Y ∈ R n × m × n is a multiplication operator that preserves the order of thetensor. The result tensor Z ∈ R n × m × n is defined as following (Kilmer andMartin, 2011): Z = X ∗ Y = fold (cid:0) bcirc ( X ) · unfold ( Y ) (cid:1) , (2)where bcirc ( X ) ∈ R n n × n n is a block circulant matrix, which can be re-garded as a new matricization of X , fold and unfold are a pair of operatorson tensors (see Eq. A.1 and Eq. A.2 in Appendix A).For tensor X ∈ R n × n × n , its discrete Fourier transform (DFT) along thethird dimension is denoted as ¯ X = X × F n = F n ( X ), where F n ∈ C n × n isthe DFT matrix. By using Matlab convention, we also have ¯ X = fft ( X , [] ,
3) .Conversely, X can be derived from ¯ X via inverse DFT, i.e., X = X × F − n =7 − n ( X ) = ifft ( ¯ X , [] , bcirc ( X ) can be block-diagonalized to a special block diagonal matrix ¯ X , whose main diagonal blocksare the frontal slices of ¯ X (see Eq. A.3 in Appendix A), via the DFT matrix F n ∈ C n × n (Kolda and Bader, 2011):( F n ⊗ I n ) · bcirc ( X ) · ( F − n ⊗ I n ) = ¯ X , (3)where ⊗ denotes the Kronecker product.Based on the block-diagonalized property of bcirc ( X ) in Eq. 3, the t-product can also be defined as the matrix-matrix product in the DFT do-main (Lu et al., 2020): Z = ( ¯ X (cid:52) ¯ Y ) × F − n = F − n (cid:0) F n ( X ) (cid:52) F n ( Y ) (cid:1) , (4)where (cid:52) denotes the face-wise product, defined as the matrix-matrix productbetween corresponding frontal slices of the two tensors. (cid:63) M -product . A new tensor-tensor product operator, called (cid:63) M -product, isproposed in Kilmer et al. (2019). It can convert the data into other transformdomains under any invertible matrix M , rather than the specific DFT domainas in the t-product. The superiority of (cid:63) M -product compared to the t-productis demonstrated in Kilmer et al. (2019).Let M ∈ R n × n be any invertible matrix satisfying: M ∗ M = MM ∗ = l I n , (5)where l > X ∈ R n × n × n and Y ∈ R n × m × n are two tensors.Then the (cid:63) M -product X (cid:63) M Y results in a tensor Z ∈ R n × m × n defined asbelow (Kilmer et al., 2019; Lu et al., 2019): Z = X (cid:63) M Y = ( ¯ X (cid:52) ¯ Y ) × M − = M − (cid:0) M ( X ) (cid:52) M ( Y ) (cid:1) , (6)where ¯ X = M ( X ) = X × M denotes the tensor in the transform domaininduced by the invertible transformation matrix M . Note that the t-productbecomes a special case of the (cid:63) M -product when M = F n .Based on the (cid:63) M -product, the following main concepts of tensor can bedefined: conjugate transpose, identity tensor, orthogonal tensor, and f-diagonaltensor (see Appendix A). Also, we can obtain the following theorem definingthe tensor SVD (t-SVD): Theorem 1 ( T-SVD
Kilmer et al. (2019)) . Let X ∈ R n × n × n , and M ∈ R n × n be any invertible matrix. Then X can be factorized as: X = U (cid:63) M S (cid:63) M V ∗ , (7) where U ∈ R n × n × n and V ∈ R n × n × n are orthogonal tensors, and S ∈ R n × n × n is an f-diagonal tensor. An illustration of the t-SVD factorization is shown in Fig. A.9 in AppendixA. Empirically, we can obtain the factorization results by computing matrixSVDs in the transform domain, refer Algorithm 3 in Appendix A.8 ensor nuclear norm . The minimization of the tensor rank is known asNP-hard due to the non-convexity nature (Hillar and Lim, 2013). Fortunately,as in the matrix case, the tensor nuclear norm can be employed as a convexrelaxation of the tensor rank for the LRTD (Lu et al., 2019).Let M ∈ R n × n be any invertible matrix satisfying Eq. 5, the tensor nuclearnorm of X ∈ R n × n × n is defined as (cid:107) X (cid:107) ∗ = l (cid:80) n i =1 (cid:107) ¯ X ( i ) (cid:107) ∗ (Lu et al., 2019).It has been proven that the tensor nuclear norm (cid:107) X (cid:107) ∗ is the convex envelopof the tensor average rank rank a ( X ) = l (cid:80) n i =1 rank( ¯ X ( i ) ) within the unit ballof the tensor spectral norm (cid:107) X (cid:107) = (cid:107) ¯ X (cid:107) (Lu et al., 2019). We will thus usethe tensor nuclear norm (cid:107) · (cid:107) ∗ to characterize the low-rank latent structure of atensor.
4. Methods
The main workflow of the proposed segmentation framework is depicted inFig. 2. In the training stage, two models used in the MAS framework areconstructed: population-specific probabilistic atlas (PA) and tumor-free liveratlases (Section 4.1). In the testing stage, for a given test abdominal CT image,we first implement an atlas selection strategy to derive patient-specific liveratlases, then generate a tumor-free test image based on the selected tumor-freeliver atlases, and finally perform the main steps of the MAS algorithm to extractthe liver tissue from the tumor-free test image (Section 4.2).
To derive patient-specific liver atlases, and to substantially reduce the com-putational cost of the MAS framework, we propose an LRTD-based liver atlasconstruction method. Specifically, we first group the training data into mul-tiple smaller clusters corresponding to different type cases, then construct apopulation-specific PA and tumor-free liver atlases for each cluster separately.
We first employ the same spectral clustering based algorithm (von Luxburg,2007) as in Shi et al. (2017). It involves two major steps: (1) training dataalignment: Since the similarity measure used in the clustering step is basedon both intensity images and liver shapes, the training data needs to be firstaligned; and (2) data clustering using the spectral clustering algorithm. Pleaserefer to Shi et al. (2017) for details.After the training data is partitioned into multiple clusters, we construct apopulation-specific PA and tumor-free liver atlases for each cluster separately.Nevertheless, most of the training images include major pathology. To miti-gate the performance degradation of liver segmentation due to the presence oftumors in the constructed atlases, we cast the procedure of generating tumor-free liver atlases as an LRTD problem (Kolda and Bader, 2011), also known asTRPCA (Lu et al., 2020). Specifically, we propose an LRTD-based liver atlasconstruction method, called LRTD-PA, based on the following two empirical9 esting Stage
Training DataTest Image
Constructing Population- Speci fi c Probabilistic Atlas Cluster 1 Cluster 2 Cluster 3 Deriving Intensity Images of the Tumor-Free Atlases Final SegmentationAtlas Selection Deriving Tumor-Free Test Image Multi-Atlas Segmentation Training Stage
Cluster 1 Cluster 2 Cluster 3Cluster 1 Cluster 2 Cluster 3 ...... ... ... ... ......
Cluster 3 ... ...
Clustering Training Data
Figure 2: The main workflow of the proposed framework for pathological liver CT segmenta-tion, consisting of training and testing stages. observations: (1) the aligned training images are linearly correlated with eachother and form a low-rank third-order tensor (Sagheer and George, 2019); and(2) the portions that cannot be represented by the low-rank part are the grosserrors or outliers (e.g., tumors), which can also be considered sparse comparedto the whole image tensor.Let { I i | i = 1 , ..., N c } be the pre-aligned training images with their cor-responding label images { L i | i = 1 , ..., N c } of cluster c . We represent eachtraining image I i as a third-order image tensor D i ∈ R w × h × d by stacking allvoxel intensity values of the axial slices frontal-slice-wisely, where w , h and d denote the width, height and the number of axial CT slices, respectively. Thenwe construct a third-order image repository tensor X ∈ R w × h × ( dN c ) by con-catenating all training image tensors { D i | i = 1 , ..., N c } frontal-slice-wisely.Mathematically, the LRTD model decomposes the image repository tensor X into two components according to the following minimization:( ˆ L , ˆ E ) = arg min L , E rank( L ) + λ (cid:107) E (cid:107) s . t . X = L + E , (8)where L represents the low-rank component corresponding to the tumor-freetraining images via tensor rank minimization, E represents the sparse compo-nent corresponding to the sparse tumors via (cid:96) -norm minimization, and λ is atrade-off factor between the two components. Since E is employed to explicitlymodel the sparse gross errors via the (cid:96) -norm, the LRTD model fits the purpose10f generating tumor-free training images very well, and our proposed LRTD-PAis robust to handle major pathology.Nevertheless, the minimization problem in Eq. 8 is known to be computa-tionally intractable (NP-hard), due to the non-convex property of the tensorrank and (cid:96) -norm (Hillar and Lim, 2013; Natarajan, 1995). Fortunately, it hasbeen proven that solving the following relaxed dual convex minimization prob-lem, called Tensor Principal Component Pursuit (TPCP) (Zhang et al., 2020b;Lu et al., 2020), can achieve the same decomposition accuracy:( ˆ L , ˆ E ) = arg min L , E (cid:107) L (cid:107) ∗ + λ (cid:107) E (cid:107) s . t . X = L + E , (9)where the tensor nuclear norm (cid:107) · (cid:107) ∗ is defined as in Section 3.2. The tensornuclear norm and (cid:96) -norm are the convex surrogates of the tensor rank and (cid:96) -norm, respectively. Under certain incoherence conditions, it has been proventhat TPCP can exactly recover the underlying low-rank L and sparse E com-ponents with high probability (Lu and Zhou, 2019).However, empirically we find that when performing the LRTD directly onthe image repository tensor X that consists of all slices of the training images,the results are unsatisfactory. It is mainly because the differences between con-secutive image slices in terms of both background and liver tissue accumulatealong the axial direction, resulting in overall rapid changes over the whole imagevolume. The image repository tensor X thus does not possess strong low-rankproperty any more. We hypothesize that performing the LRTD on smaller seg-ments consisting of multiple consecutive image slices can lead to better results,since smaller segments exhibit fewer overall changes, they will lie on a low-ranksubspace (Lee et al., 2018). Furthermore, both the computational cost andmemory usage will be reduced substantially. To this end, we propose a multi-slice LRTD scheme to recover the underlying low-rank structure embedded in3D medical images. Specially, we first partition each training image tensor D i into smaller segments { D ij | j = 1 , ..., N s } consisting of multiple consecutiveimage slices of length K , then construct a third-order image repository tensor X j for each segment { D ij | i = 1 , ..., N c } frontal-slice-wisely, and finally per-form the LRTD on each segment tensor X j sequentially. Mathematically, themulti-slice LRTD scheme solves the following minimization for each segmenttensor X j :( ˆ L j , ˆ E j ) = arg min L j , E j (cid:107) L j (cid:107) ∗ + λ (cid:107) E j (cid:107) s . t . X j = L j + E j . (10)Eq. 10 is the optimization problem of proposed liver atlas constructionmethod LRTD-PA, through which the tumor-free training images lie in the low-rank component ˆ L j , while the tumors are extracted in ˆ E j . After obtaining { ( ˆ L j , ˆ E j ) | j = 1 , ..., N s } , we separately stack them frontal-slice-wisely to ob-tain ˆ L and ˆ E . The tumor-free training images in the low-rank component ˆ L areemployed to construct the population-specific PA ¯ I c , from which we can thenobtain intensity images of the tumor-free liver atlases { ˆ I i | i = 1 , ..., N c } . Fig.11 .. ...... D
11 1N N c Image Slice Partition X L +E ... ...... Image Slice Combination D c D D LRTD ... s D D D
21 2N ... s D D D ... s c D N c D X X N s ... L +E N s N s LRTD L +E LRTD ^ ^^ ^^ ^
Figure 3: An example of the multi-slice LRTD applied to training images of cluster c . Alltraining image tensors { D i | i = 1 , ..., N c } are first partitioned into smaller segments { D ij | j =1 , ..., N s } consisting of multiple consecutive image slices of length K = 5, then the LRTD isperformed on each segment tensor X j sequentially, and finally the decomposed low-rank andsparse components of all the segment tensors are separately combined to obtain tumor-freetraining images ˆ L and tumor images ˆ E . c . The iterative procedure for constructing population-specific PA andtumor-free liver atlases for each cluster using proposed LRTD-PA is summarizedin Algorithm 1. Note that since the LRTD is performed in a segment-by-segmentfashion, to maintain a smooth transition between neighboring segments in thetumor-free training images, we expand each segment to overlap with one neigh-boring slice at each end, and average the decomposition results in overlappingslices. Thus the smallest possible segment length K is 2. In the past few years, many optimization algorithms have been proposedto solve the LRTD problem (Sidiropoulos et al., 2017), and stable recovery ofthe low-rank L and sparse E components can be guaranteed (Lu and Zhou,2019; Zhang et al., 2020a). To achieve both efficiency and scalability, we usethe alternating direction method of multipliers (ADMM) algorithm (Boyd et al.,2010; Lu et al., 2020) to solve the TPCP problem in Eq. 10. The ADMM al-gorithm is a first-order optimization method where both the objective functionand the constraints exhibit separable structures. It has been widely utilized forsolving convex optimization problems with well-established convergence prop-erties (Boyd et al., 2010).Algorithm 4 (Appendix B) summarizes the derived ADMM algorithm for12 lgorithm 1 Population-Specific Probabilistic Atlas and Tumor-Free Liver At-lases Construction Procedure
Input:
Training images of class c : { I i | i = 1 , ..., N c } , and the maximumnumber of iterations: N max . Output:
The probability atlas ¯ I c , and intensity images of the tumor-free liveratlases { ˆ I i | i = 1 , ..., N c } for class c . Select the training image containing smallest tumor region as the initialtemplate ¯ I . Compute the non-rigid transformation T i that warps I i to the template¯ I : I i ← T i ( I i ) , i = 1 , ..., N c , and obtain the corresponding third-order imagetensors: D i ← I i , i = 1 , ..., N c . Obtain the population-specific probability atlas iteratively: for k = 1 to N max do3.1 Compute low-rank parts via multi-slice LRTD on X :[ I kLR , I kLR , ..., I kLR Nc ] ← MS-LRTD( X = [ D k , D k , ..., D kN c ] ) . Obtain the new template using low-rank parts: ¯ I k ← N c (cid:80) N c n =1 I kLR i . Compute the non-rigid transformation T ki that warps I kLR i to thetemplate ¯ I k , i = 1 , ..., N c . Align I ki to the space of the template ¯ I k using T ki : I ( k +1) i ← T ki ( I ki ) , i = 1 , ..., N c , and obtain the corresponding third-order image ten-sors: D ( k +1) i ← I ( k +1) i , i = 1 , ..., N c . end for ¯ I c ← ¯ I N max . Generate the tumor-free liver atlases of class c : Compute the non-rigid transformation T i that warps I i to the proba-bility atlas ¯ I c : I (cid:48) i ← T i ( I i ) , i = 1 , ..., N c , and obtain the corresponding third-order image tensors: D (cid:48) i ← I (cid:48) i , i = 1 , ..., N c . Compute low-rank parts via multi-slice LRTD on X (cid:48) :[ I (cid:48) LR , I (cid:48) LR , ..., I (cid:48) LR Nc ] ← MS-LRTD( X (cid:48) = [ D (cid:48) , D (cid:48) , ..., D (cid:48) N c ] ) . ˆ I i ← T − i ( I (cid:48) LR i ) , i = 1 , ..., N c . solving the TPCP problem (Eq. 10) in the proposed liver atlas constructionmethod LRTD-PA. The detailed mathematical derivation procedure is givenin Appendix B. Algorithm 5 (Appendix B) shows the optimization proce-dure of the proposed multi-slice LRTD scheme. The convergence propertiesof the ADMM algorithm with two blocks, as in this study, have been well es-tablished (Boyd et al., 2010). Suppose that the size of the data tensor X is n × n × n with n ≥ n , the main computational cost of the ADMM algo-13ithm lies in computing the t-SVT operator D τ in the L subproblem (Eq. B.5 inAppendix B). For any general invertible matrix M ∈ R n × n , the per-iterationcomputational complexity of the ADMM algorithm is O ( n n n + n n n ) (Luand Zhou, 2019). After the tumor-free liver atlases are constructed, we propose an LRTD-based MAS algorithm to perform liver CT segmentation. Specifically, given atest image I t , we first implement an atlas selection strategy to obtain patient-specific liver atlases, and to substantially improve the computational efficiency.Then based on intensity images of the selected tumor-free liver atlases, we gener-ate a tumor-free test image I LR t . Finally, the main steps of the MAS algorithmare performed to extract the liver tissue from I LR t .Considering that the liver tissue exhibits high anatomical variability, weconstruct patient-specific liver atlases for each test image via an atlas selec-tion strategy (Shi et al., 2017) to achieve more accurate liver segmentation.Specifically, we first choose the best training data cluster based on the simi-larity between the test image I t and the population-specific PAs. The cluster c is selected when the normalized cross correlation (NCC) between I t and thepopulation-specific PA ¯ I (cid:48) c after its warping into the space of I t , i.e., N CC ( ¯ I (cid:48) c , I t ),is the largest. The NCC between two images I i and I j is defined as: N CC ( I i , I j ) = (cid:80) x ∈ Ω (cid:0) I i ( x ) − ¯ I i (cid:1) (cid:0) I j ( x ) − ¯ I j (cid:1)(cid:113)(cid:80) x ∈ Ω (cid:0) I i ( x ) − ¯ I i (cid:1) (cid:113)(cid:80) x ∈ Ω (cid:0) I j ( x ) − ¯ I j (cid:1) , (11)where ¯ I i and ¯ I j denote the mean values of the images within the overlappingregion Ω. Then the tumor-free intensity images and their corresponding labelimages within the chosen cluster c are regarded as the patient-specific liveratlases for the test image I t .Generally, the test image I t is directly employed to perform image registra-tions in the MAS algorithm. However, we empirically find that most of the testimages contain tumors. In order to achieve accurate pairwise image registrationand label propagation, we use the proposed LRTD-PA to generate a tumor-free test image I LR t . Specifically, given the test image I t , we first constructa third-order data tensor X t ∈ R w × h × ( d ( N c +1) ) by concatenating the warpedimage repository tensor of the chosen cluster X (cid:48) ∈ R w × h × ( dN c ) with the testimage tensor D t ∈ R w × h × d frontal-slice-wisely. After performing the multi-sliceLRTD on the data tensor X t , we can obtain the tumor-free test image I LR t , asshown in Algorithm 2. Fig. 4 shows an example of generating the tumor-freeimage I LR t for a test image I t using the proposed LRTD-PA.After obtaining the tumor-free test image I LR t , it is the input to the MASalgorithm. The main steps of the MAS algorithm (i.e., image registration, labelpropagation, and label fusion) are then performed to obtain the liver segmen-tation result. 14 lgorithm 2 Tumor-Free Test Image Derivation Procedure
Input:
Intensity images of the tumor-free atlases within the chosen cluster c : { ˆ I i | i = 1 , ..., N c } , the test image: I t , and the maximum number of iterations: N max . Output:
The tumor-free test image I LR t . Compute the non-rigid transformation T i that warps ˆ I i to the target image I t : ˆ I i ← T i ( ˆ I i ) , i = 1 , ..., N c , and obtain the corresponding third-order imagetensors: ˆ D i ← ˆ I i , i = 1 , ..., N c . Obtain the test image tensor: D t ← I t . Obtain the tumor-free test image iteratively: for k = 1 to N max do3.1 Compute the low-rank test image via multi-slice LRTD: I kLR t ← MS-LRTD( X t = [ ˆ D k , ˆ D k , ..., ˆ D N c k , D t ] ) . Compute the non-rigid transformation T ki that warps ˆ I i to I kLR t :ˆ I i ( k +1) ← T ki ( ˆ I i ) , i = 1 , ..., N c , and obtain the corresponding third-orderimage tensors: ˆ D ( k +1) i ← ˆ I i ( k +1) , i = 1 , ..., N c . end for I LR t ← I N max LR t . ... ( ( ... ( ( ... ( ( I LR + = t t I Figure 4: An example of generating the tumor-free image I LR t for a test liver image I t (indicated by the pink dashed-line rectangle in X t ) using the proposed low-rank tensor de-composition based probabilistic atlas (LRTD-PA). The data tensor X t is decomposed into alow-rank component L corresponding to the tumor-free liver images, and a sparse component E corresponding to the sparse tumors (indicated by green rectangles). For the purpose ofillustration, it only shows one specific 2D slice of the CT volume. Firstly, all intensity images of the tumor-free liver atlases within the chosencluster c { ˆ I i | i = 1 , ..., N c } are non-rigidly warped to the space of I LR t , resultingin N c non-rigid transformations { T i | i = 1 , ..., N c } . These non-rigid transfor-15 able 1: Specifications of the CT scans from the three databases. Database Numberof scans In-planematrixsize In-planeresolution[mm] Numberof slices Slicethickness[mm]SLIVER07-Train 20 512 ×
512 0.58-0.81 64-394 0.7-5.03Dircadb1 20 512 ×
512 0.56-0.86 74-260 1.0-4.0CT-ORG-Test 21 512 ×
512 0.65-1.37 75-841 0.8-5.0mations then propagate the corresponding label images { L i | i = 1 , ..., N c } intothe space of I LR t , resulting in the propagated atlas labels { L (cid:48) i | i = 1 , ..., N c } .Finally, the N c propagated atlas labels are combined via label fusion toobtain the liver segmentation result. Specifically, we adopt the joint label fusion(JLF) algorithm (Wang et al., 2013) considering its high performance. The JLFis a statistical local weighted label fusion algorithm, where the optimal weightsfor label fusion are obtained by minimizing the total expectation of labelingerrors. Moreover, the pairwise correlation between atlases is explicitly modeledas the joint probability of two atlases creating a segmentation error at a voxel.Please refer to Wang et al. (2013) for more details on the JLF algorithm. Toobtain the final segmentation result from the posterior probability map of liverlikelihood produced by the JLF algorithm, it performs a series of simple post-processing operations on the probability map, consisting of thresholding usingOtsu’s method (Otsu, 1979), morphological opening operator to remove smallunconnected components and the noise, and morphological closing operator tofill small cavities.
5. Experiments
In order to evaluate the performance of the proposed MAS framework, and toshow its clinical applicability, we tested it on a clinical dataset of 61 abdominalCT scans of pathological liver cases from three publicly available databases.Table 1 outlines the specifications of the CT scans from the three databases.Both SLIVER07-Train and 3Dircadb1 include 20 abdominal CT scanswith corresponding ground truths, provided by the organizers of the Segmen-tation of the Liver Competition 2007 (SLIVER07) (Heimann et al., 2009), andIRCAD (Soler et al., 2009), the French Research Institute against DigestiveCancer, respectively. The CT-ORG-Test consists of 21 abdominal CT scanswith corresponding ground truths (Rister et al., 2020). Most of the cases in https://wiki.cancerimagingarchive.net/display/Public/CT-ORG%3A+CT+volumes+with+multiple+organ+segmentations In order to perform quantitative evaluations of segmentation accuracy, thefollowing two volume and surface based metrics are employed:(1) Jaccard index ( JI ) (Jaccard, 1901) measures the volumetric overlap betweentwo segmentation results A and B , defined as: JI ( A, B ) = | V ( A ) ∩ V ( B ) || V ( A ) ∪ V ( B ) | × , (12)where V ( X ) denotes the binary volume of segmentation result X .(2) Average symmetric surface distance ( ASD ) is the average distance betweenthe surfaces of two segmentation results A and B (Heimann et al., 2009): ASD ( A, B ) = (cid:80) s A ∈ S ( A ) d (cid:0) s A , S ( B ) (cid:1) + (cid:80) s B ∈ S ( B ) d (cid:0) s B , S ( A ) (cid:1) | S ( A ) | + | S ( B ) | , (13)where d (cid:0) v, S ( X ) (cid:1) is the shortest Euclidean distance from a voxel v to thesurface voxels of segmentation result X .The units of JI and ASD are percent and millimeters, respectively. For JI ( ASD ), the larger (smaller) the value is, the more accurate the segmentationresult will be.In order to determine whether the differences in segmentation accuracy be-tween our method and other compared methods were statistically significant inthe experiments, the paired t -test was carried out with a significance level of p < .
05. The null hypothesis is that the mean values of the same evaluationmetric are exactly the same for the compared methods.
To reduce image noise without deteriorating the important edge information,it first preprocessed all training and test images via the 3D anisotropic diffusionfilter (Perona and Malik, 1990). To perform the pairwise non-rigid image regis-trations, it utilized the publicly available elastix toolbox (Klein et al., 2010), http://elastix.isi.uu.nl able 2: Parameter settings for the proposed segmentation framework. Parameter Value Description k K λ λ Parameter in Eq. 10 for LRTD-PA. λ =1 / (cid:112) max( n , n ) n is the default value for λ assuggested in Lu et al. (2020), where ( n , n , n )is the size of the data tensor. N max (Wang and Yushke-vich, 2013) was employed to perform label fusion. In Algorithm 1, the maskedtraining images via liver binary masks were used to obtain all the non-rigidtransformations, which were then propagated to the original training images.While only the original training and test images were used in Algorithm 2.The parameter settings for the proposed MAS framework were optimized vialeave-one-out cross-validation (LOOCV) using the SLIVER07-Train database.Table 2 lists the parameter settings for the proposed segmentation framework.In the experiments, it compared the proposed LRTD-based MAS frameworkwith two other closely related MAS frameworks: conventional MAS and theLRMD-based MAS. The latter one can be considered as a special case of theproposed framework, by substituting LRTD with LRMD to generate tumor-freeliver images. Refer to Shi et al. (2017) for more details. To make the compar-isons between different frameworks fair, the same training data clustering, non-rigid registration, and the JLF steps were utilized. Furthermore, we comparedthe proposed segmentation framework with other state-of-the-art methods.All the MAS frameworks were implemented in Python on Ubuntu 18.04.The SimpleITK library (Yaniv et al., 2018) was utilized to perform the imageprocessing. All the tests in this study were run on a PC equipped with an IntelCore i7 processor and 32 GB RAM. The average time to segment one test imagewas about 95 min, most of which was spent performing the pairwise non-rigidimage registrations and the JLF, with an average time of about 3 min (per imageregistration) and 15 min, respectively. able 3: Quantitative comparative results of the masked tumor-free liver images with differentchoices of the transform matrix M using the SLIVER07-Train database. Metrics Initial FFT DWT DCT σ ± ± ± ± H [bits] 1.38 ± ± ± ± Time [ms] 172.25 ± ± ± For each metric, the mean and standard deviation of the overall datasets are given.Bold values are the best result in that column.
6. Results M An important implementation in the proposed liver atlas construction methodLRTD-PA is to choose the best transform matrix M of (cid:63) M -product in Section3.2, which converts the data tensor X into other transform domain as ¯ X . Tothis end, we tested the effect of three commonly used transforms (Kernfeld et al.,2015), i.e., FFT, DCT, and Daubechies-4 discrete wavelet transform (DWT), onthe image intensity standard deviation σ and entropy H of the masked tumor-free liver images via liver binary masks. Since the smaller the σ and H are,the more homogeneous and smooth the appearance of the generated tumor-freeliver image will be (Liu et al., 2015).Table 3 shows the σ and H of the masked tumor-free liver images with dif-ferent choices of the transform matrix M using the SLIVER07-Train database.Although all the three transform matrices largely reduce the values of σ and H compared to their initial values of 32 .
44 and 1 .
38, respectively, the DCT achievesthe smallest mean σ of 15 .
38, which is less than half of its initial value. Also,the DCT yields the smallest mean H of 1 .
13. The time needed to perform themulti-slice LRTD on each image segment is also given in Table 3. The DCT isthe most efficient and only takes 63 .
68 ms, while FFT costs nearly three times ofthat for the same task. Therefore, we choose the DCT as the transform matrix M for the proposed LRTD-PA. The main aim of the proposed multi-slice LRTD scheme is to generate ac-curate tumor-free liver images, while coping with the rapid changes of bothbackground and liver tissue over the whole image volume. An important hyper-parameter for the multi-slice LRTD scheme is the number of consecutive imageslices K , which determines how many neighboring slices are used to performthe LRTD. To select the best K for this study, it tested the effect of differentchoices of K (from 2 to 11) on the image intensity standard deviation σ of themasked tumor-free liver images via liver binary masks.19 Number of Consecutive Image Slices K I m ag e I n t e n s i t y S t a nd a r d D e v i a t i o n Figure 5: Image intensity standard deviation σ of the masked tumor-free liver images withdifferent choices of the number of consecutive image slices K for the multi-slice LRTD schemeusing the SLIVER07-Train database. Fig. 5 shows the σ of the masked tumor-free liver images with differentchoices of K using the SLIVER07-Train database. It can be seen that the valueof σ first decreases as the number of K increases. However, the value of σ beginsto increase when K >
5, and the multi-slice LRTD scheme achieves the smallestvalue of σ when K = 5. Therefore, we choose K = 5 for the multi-slice LRTDscheme in this study. Note here that we only give the results of σ with regardsto different K values, since the use of image intensity entropy H yields the samebest choice K = 5. In this section, to show the effectiveness of the proposed LRTD-based MASframework, we applied it to the challenging task of pathological liver segmenta-tion of CT images in the 3Dircadb1 database. It also compared the proposedmethod with two other closely related MAS frameworks, i.e., conventional MASand the LRMD-based MAS, to verify its superiority. To make the compar-isons between different MAS frameworks fair, the same training data clustering,non-rigid registration, and the JLF steps were utilized.Fig. 6 shows the visually comparative results of the three population-specificliver PAs generated by the conventional PA, the LRMD-PA, and the proposedLRTD-PA. We can easily see that large areas of the PAs constructed by the con-ventional method appear totally different from that of the normal liver tissue(indicated by black arrows). Thus, the conventional method is strongly influ-enced by the presence of major pathology in training data. In comparison, the20 a) (b) (c)Figure 6: Comparative results of the three population-specific probabilistic atlases (PAs) gen-erated by the conventional PA (1st column), the LRMD-PA (2nd column), and the proposedLRTD-PA (3rd column). Each row shows the PA of one training data cluster. The areasindicated by black arrows in the PAs of the conventional PA appear totally different fromthat of the normal liver tissue. liver tissue of the PAs constructed by the LRMD-PA and our method appearsmore homogeneous, due to the use of tumor-free training images, which alsoleads to much more accurate pairwise image registrations. Nevertheless, com-pared with the LRMD-PA, our results are even more homogeneous and smooth,especially in areas containing major pathology and vessels. It is mainly be-cause our tensor-based method can fully exploit the high correlations betweenneighboring slices of the 3D CT scans. While the LRMD-PA reformats thedata tensor to a matrix by vectorizing voxel intensity values of each CT scan toform the column vectors, where the local spatial information is totally lost, andthe multi-dimensional structure embedded in the tensor data is ignored, caus-ing severe performance degradation. Therefore, our method can handle majorpathology more effectively, and can mitigate the performance degradation ofliver segmentation caused by the presence of tumors in the constructed atlases.Fig. 7 gives the tumor-free liver images for three challenging pathologicalcases generated by the proposed LRTD-PA method, consisting of liver tissuewith hypodense tumor, hyperdense tumor, and hypo- and hyperdense tumors21 a) (b) (c)Figure 7: Results of the tumor-free liver images for three challenging pathological cases gener-ated by the proposed LRTD-PA method, consisting of liver tissue with hypodense tumor (1strow), hyperdense tumor (2nd row), and hypo- and hyperdense tumors (3rd row) indicated byblack arrows. In each row, (a) the original images, (b) the corresponding tumor-free images(low-rank components), and (c) sparse tumors (sparse components) are displayed sequentially. (indicated by black arrows). For all the three challenging cases, our methodsuccessfully eliminates the tumors with different contrast levels from the originalimages as the sparse components. And the liver tissue in the decomposed low-rank components appears much more homogeneous and smooth than that inthe original images. Thus, by eliminating the influence of major pathology,the proposed LRTD-based MAS algorithm can achieve accurate pairwise imageregistration and label propagation. Furthermore, it shows that the proposedmulti-slice LRTD scheme is able to successfully recover the underlying low-rankstructure embedded in 3D medical images.Fig. 8 shows the visual results of liver segmentation generated by the con-ventional MAS, the LRMD-based MAS, and our proposed LRTD-based MASframeworks on three challenging pathological cases, consisting of liver tissuewith hypodense tumor, hyperdense tumor, and hypo- and hyperdense tumors(indicated by white arrows). Each row shows one case. The first case is a verychallenging one, where the peripheral hypodense tumor (indicated by the upperleft white arrow) has very similar intensity values to that of the nearby mus-cle tissue. We can see that the segmentation results of both the conventional22 a) (b) (c) (d) (e)Figure 8: Results of liver segmentation by (a) the conventional MAS, (b) the LRMD-basedMAS, and (c) the proposed LRTD-based MAS frameworks on three challenging pathologicalcases, consisting of liver tissue with hypodense tumor (1st row), hyperdense tumor (2nd row)and hypo- and hyperdense tumors (3rd row), indicated by white arrows. The ground truthsare delineated by green contours, while the segmentation results are shown as pink regions. (d)The 3D visualization of average symmetric surface distance (ASD) errors of our method. Thered and blue regions indicate over- and under-segmentation, respectively. (e) The distance tocolor bar.
MAS and the LRMD-based MAS exclude part of the peripheral pathologicalareas, thus resulting in under-segmentation. It is mainly due to the blurredboundaries between the peripheral hypodense tumor and the nearby muscle tis-sue, leading to inaccurate pairwise image registration and label propagation inthe peripheral pathological areas. While our method accurately delineates theperipheral pathological areas, and yields more accurate segmentation results.This is mainly because by generating more accurate tumor-free liver images,our method can more effectively mitigate the performance degradation of liversegmentation due to the presence of tumors, compared to both the conventionalMAS and the LRMD-based MAS.In the second and the third cases, the liver tumors are also located nearthe boundary of the liver tissue. As expected, due to the inaccurate pairwiseimage registration in the peripheral pathological areas, the segmentation re-sult of the conventional MAS excludes a large portion of the peripheral liver23 able 4: Comparative results of liver segmentation with three different MAS frameworksusing the 3Dircadb1 database.
Method JI [%] ASD [mm]Conventional MAS 84.90 ± ∗ ± ∗ LRMD-based MAS 90.53 ± ∗ ± ∗ Our LRTD-based MAS 92.11 ± ± For each metric, the mean and standard deviation of the overall datasets are given.Bold values are the best result in that column. ∗ indicates a statistically significantdifference between the marked result and the corresponding one of our method at asignificance level of 0 . tissue, and thus under-segmenting the live tissue. On the other hand, boththe LRMD-based MAS and our method successfully delineate the boundariesnear the peripheral pathological areas, and the segmentation results are muchmore accurate. This is because in both methods, the negative influence of thesetumors on liver segmentation results can be largely mitigated by generatingtumor-free liver images. However, our method recovers more finer edge de-tails and yields more accurate result than the LRMD-based MAS, due to theuse of more homogeneous and smooth tumor-free liver images, which lead tomore accurate pairwise image registration and label propagation. These re-sults demonstrate the strong robustness of our method to both hypodense andhyperdense tumor contrast levels.Table 4 lists the quantitative comparative results of liver segmentation withthree different MAS frameworks, i.e., the conventional MAS, the LRMD-basedMAS, and the proposed LRTD-based MAS, using the 3Dircadb1 database. Foreach metric, the mean and standard deviation of the overall datasets are given.It can be observed that our LRTD-based MAS yields the best segmentation ac-curacy with the lowest variances in terms of both JI and ASD , demonstratingthe robustness of our method on a variety of test data. Furthermore, our methodoutperforms the other two compared methods by a large margin, and statisti-cally significant improvements ( p < .
05) over both the conventional MAS andthe LRMD-based MAS were observed according to JI as well as ASD . In par-ticular, the mean JI and mean ASD of our method are 92 .
11 % and 1 .
37 mm,respectively. For the conventional MAS, the obtained mean JI ( ASD ) is quitelow (high), with a value below 85 % (above 2 .
80 mm). Compared to the LRMD-based MAS, our method achieves an average improvement of 1 .
58 % and 18 %according to JI and ASD , respectively. The above results therefore clearlyindicate that our proposed method is significantly more accurate and robustthan both the conventional MAS and the LRMD-based MAS in the presence ofmajor pathology. 24 .3. Comparison with State-of-the-art Methods
In this section, we compared our proposed MAS-based liver segmentationframework with state-of-the-art automatic methods using public databases toevaluate its performance relative to the wider context of existing works.We first compare our proposed liver segmentation framework with state-of-the-art automatic methods using the 3Dircadb1 database. Table 5 outlines thequantitative results of liver segmentation of all the compared methods, includingboth deep learning based methods and model-based methods. As can be seen,among all the 11 compared methods, our method achieves the best performance,i.e., the largest (smallest) value of JI ( ASD ). Also, our method yields very smallvariances in terms of both JI and ASD , suggesting its robustness on a varietyof datasets. Among deep learning based methods, the cascaded U-Net methodproposed by Christ et al. (2017) obtains the best accuracy, with the mean JI and ASD of 89 .
30 % and 1 .
50 mm, respectively. However, the performanceobtained by all other deep learning based methods is inferior to that of allthe model-based methods (including MAS, ASM and graph cut) in terms of
ASD . Although deep learning based methods are currently considered verygood for medical image segmentation, their performance is strongly dependenton the availability of massive amounts of annotated training data. Moreover,the lack of interpretability remains a major constraint to the adoption of deeplearning based methods in clinical applications, where interpretability of theobtained results is of paramount importance. Therefore, in clinical applicationswhere the amount of annotated training data is low and a high level of trust isrequired (such as this study), model-based methods are still preferable to deeplearning based methods.Furthermore, we compared the proposed liver segmentation framework withstate-of-the-art deep learning based method using the CT-ORG-Test database.The quantitative results of the compared methods are summarized in Table 6.We can see that as on the 3Dircadb1 database, our method is superior to the3D U-Net (with data augmentation) method proposed by Rister et al. (2020)in terms of JI , with an average improvement of 0 .
76 %. It achieved a relativelysmall value of
ASD compared to our method, but with a much larger variance,suggesting a low robustness. Also, their method required substantially moretraining data (119) than ours (20). Specifically, the mean JI and ASD of ourmethod are 91 .
60 % and 1 .
44 mm, respectively. Furthermore, the performanceobtained by Rister et al. (2020) degraded a lot when data augmentation wasnot applied, with the mean value of JI dropped from 90 .
84 % to 85 .
19 %, andthe mean value of
ASD increased from 1 .
09 mm to 1 .
21 mm.All the above experimental results indicate that the performance of ourmethod is more accurate and robust than that of state-of-the-art methods,including both deep learning based methods and model-based methods. Fur-thermore, the results demonstrate the strong robustness of our method againstdifferent tumor contrast levels. Therefore, the proposed MAS-based liver seg-mentation framework can be utilized for accurate and robust liver segmentationin the presence of major pathology. 25 able 5: Comparison of the proposed method and state-of-the-art automatic methods usingthe 3Dircadb1 database.
Method SegmentationFramework JI [%] ASD [mm]
Deep learning based methods
Jiang et al. (2019) AHCNet 89.57 N/ALi et al. (2018) H-DenseUNet 89.98 ± ± ± ± Model-based methods
Lu et al. (2018) GC 90.79 ± ± ± ± ± ± ± ± ± ± Our method
MAS ± ± Bold values are the best result in that column. ASM, GC and DM stand for activeshape model, graph cut and deep learning model, respectively. N/A stands for NotAvailable information.Table 6: Comparison of the proposed method and state-of-the-art deep learning based methodusing the CT-ORG-Test database.
Method Segmentation Framework JI [%] ASD [mm]Rister et al. (2020) 3D U-Net (Without DA) 85.19 1.21 ± ± MAS ± ± Bold values are the best result in that column. DA stands for Data Augmentation.
7. Conclusion
In this paper, we propose a novel automatic method for accurate and ro-bust pathological liver segmentation of CT images, by integrating the general (cid:63) M -product based LRTD theory into the widely used MAS framework. Ourmethod significantly enhances the traditional MAS framework in three direc-tions, by proposing a multi-slice LRTD scheme, an LRTD-based atlas construc-tion method, and an LRTD-based MAS algorithm.26o demonstrate the effectiveness of our proposed segmentation framework,we conducted extensive experiments using a total of 61 clinical CT scans ofpathological liver cases from three publicly available databases, and our methodyielded high performance. All the experimental results indicate that: • The proposed multi-slice LRTD scheme is able to successfully recover theunderlying low-rank structure embedded in 3D medical images. • The proposed liver atlas construction method LRTD-PA yields much morehomogeneous and smooth tumor-free liver atlases than both the conven-tional PA and the LRMD-PA methods. • The proposed LRTD-based MAS algorithm derives very patient-specificliver atlases for each test image, and achieves accurate pairwise imageregistration and label propagation. • The performance of our proposed segmentation framework is superior tothat of state-of-the-art methods, including deep learning based methodsand model-based methods.In future studies, we plan to further improve its performance in two aspects:(1) In the proposed multi-slice LRTD scheme, both the (cid:63) M -product and thetensor nuclear norm depend on the transformation matrix M utilized. We mayfurther increase the accuracy of tensor decomposition by learning the optimaltransformation matrix (Lu and Zhou, 2019). (2) Currently, the computationalcost of our method is still a little high. It is mainly due to the computationallyintensive nature of the pairwise non-rigid image registrations. We will considerusing deep learning-based registration methods, e.g., the VoxelMorph (Balakr-ishnan et al., 2019), to perform the image registrations to decrease the compu-tational time of our segmentation framework. Acknowledgment
This work was supported in part by the National Natural Science Foundationof China under Grant No. 61701178, and the Natural Science Foundation ofHunan Province of China (No. 2018JJ3256), and the China Scholarship Council(No. 201908430083).
Appendix A. Tensor Preliminaries bcirc ( X ) = X (1) X ( n ) . . . X (2) X (2) X (1) . . . X (3) ... ... . . . ... X ( n ) X ( n − . . . X (1) . (A.1)27 nfold ( X ) = X (1) X (2) ... X ( n ) , fold (cid:0) unfold ( X ) (cid:1) = X . (A.2)¯ X = bdiag ( ¯ X ) = ¯ X (1) ¯ X (2) . . . ¯ X ( n ) . (A.3) n n n n n n n n n n n n Figure A.9: An illustration of the t-SVD factorization of an n × n × n tensor X . Algorithm 3
The T-SVD Algorithm under Transformation Matrix
MInput: tensor X ∈ R n × n × n and invertible transformation matrix M ∈ R n × n . Output: U ∈ R n × n × n , S ∈ R n × n × n , V ∈ R n × n × n . Convert data into transform domain: ¯ X ← M ( X ) . Compute matrix SVD for each frontal slice:[ ¯ U ( k ) , ¯ S ( k ) , ¯ V ( k ) ] ← svd( ¯ X ( k ) ) , k = 1 , ..., n . Convert data back to spatial domain: U ← M − ( ¯ U ), S ← M − ( ¯ S ), V ← M − ( ¯ V ) Definition 1 (Conjugate transpose Kilmer et al. (2019)).
Let M ∈ R n × n be any invertible matrix. The conjugate transpose of a tensor X ∈ R n × n × n is the tensor X ∗ ∈ R n × n × n satisfying (cid:0) M ( X ∗ ) (cid:1) ( i ) = (cid:16) M ( X ) ( i ) (cid:17) ∗ for i =1 , , · · · , n . 28 efinition 2 (Identity tensor Kilmer et al. (2019)). Let M ∈ R n × n beany invertible matrix. The tensor I ∈ R n × n × n is an identity tensor if eachfrontal slice of ¯ I = M ( I ) is an n × n identity matrix I n . Definition 3 (Orthogonal tensor Kilmer et al. (2019)).
Let M ∈ R n × n be any invertible matrix. A tensor Q ∈ R n × n × n is orthogonal if it satisfies: Q ∗ (cid:63) M Q = Q (cid:63) M Q ∗ = I . (A.4) Definition 4 (F-diagonal tensor Kilmer et al. (2019)).
A tensor is f-diagonalif each of its frontal slice is diagonal.
Appendix B. The Closed-Form Solutions to the TPCP Problem ofLRTD-PA in Eq. 10 via the ADMM Algorithm
The augmented Lagrangian function to be minimized for the TPCP problemin Eq. 10 is given as follows (from now on the subscript j is omitted to simplifynotation): L µ ( L , E , Y ) = (cid:107) L (cid:107) ∗ + λ (cid:107) E (cid:107) + (cid:104) Y , X − L − E (cid:105) + µ (cid:107) X − L − E (cid:107) F , (B.1)where Y and µ > L k +1 = arg min L L µ k ( L , E k , Y k ) , E k +1 = arg min E L µ k ( L k +1 , E , Y k ) . (B.2)Then L and E are updated alternately by minimizing the augmented Lagrangianfunction with the other fixed. Finally, the Lagrange multiplier Y is updatedaccording to the following rule: Y k +1 = Y k + µ k ( X − L k +1 − E k +1 ) . (B.3)Furthermore, both minimization subproblems in Eq. B.2 have closed-form so-lutions.(i) L minimization subproblem : Theorem 2 ( Lu et al. (2020)) . Given a tensor W ∈ R n × n × n and τ > ,the optimal solution to the following minimization problem is given by: D τ ( W ) = arg min X τ (cid:107) X (cid:107) ∗ + 12 (cid:107) X − W (cid:107) F , (B.4) where D τ is the tensor singular value thresholding (t-SVT) operator (Lu et al.,2020) defined as: D τ ( W ) = U (cid:63) M S τ (cid:63) M V ∗ , where U (cid:63) M S (cid:63) M V ∗ = W is thet-SVD of W , S τ = M − (cid:0) S τ ( ¯ S ) (cid:1) , and S τ (¯ s ijk ) = max( | ¯ s ijk | − τ, · sgn(¯ s ijk ) is the shrinkage operator applied on ¯ S element-wisely, where sgn( · ) is the signfunction. L subproblem inEq. B.2 can be obtained as follows according to Theorem 2: L k +1 = arg min L L µ k ( L , E k , Y k )= arg min L (cid:107) L (cid:107) ∗ + µ k (cid:107) X − L − E k + Y k µ k (cid:107) F = D µk ( X − E k + Y k µ k ) . (B.5)(ii) E minimization subproblem : Theorem 3 ( Hale et al. (2008)) . Given a tensor W ∈ R n × n × n and τ > , Algorithm 4
The ADMM Algorithm for Solving the TPCP Problem in Eq. 10
Input:
Data tensor X , weighting parameter λ . Output: L = L k +1 , E = E k +1 . Initialization: L = E = Y = 0, µ = 10 − , µ max = 10 , ρ = 1 . , ε = 10 − , and k = 0. Solving the TPCP problem iteratively: while
Not Converged do2.1
Update L : L k +1 ← D µk ( X − E k + Y k µ k ). Update E : E k +1 ← S λµk ( X − L k +1 + Y k µ k ). Check the convergence conditions: if (cid:107) L k +1 − L k (cid:107) ∞ < ε and (cid:107) E k +1 − E k (cid:107) ∞ < ε and (cid:107) X − L k +1 − E k +1 (cid:107) ∞ < ε thenbreak . end if2.4 Update Y : Y k +1 ← Y k + µ k ( X − L k +1 − E k +1 ). Update µ : µ k +1 ← min( ρµ k , µ max ) . k ← k + 1. end while lgorithm 5 Optimization Procedure of the Multi-Slice LRTD Scheme
Input:
Aligned training image tensors of cluster c : { D i | i = 1 , ..., N c } ,weighting parameter λ , and the segment length K . Output: ˆ L , ˆ E . Partition D i into N s image segments consisting of multiple consecutiveimage slices of length K : [ D i , D i , · · · , D iN s ] ← D i , i = 1 , ..., N c . Perform the LRTD on each image segment tensor: for j = 1 to N s do2.1 X j ← Construct an image repository tensor for segment j by usingthe corresponding training segments { D ij | i = 1 , ..., N c } . Use the ADMM Algorithm (Algorithm 4) to perform LRTD:( ˆ L j , ˆ E j ) ← ADMM( X j , λ ) . end for3. ( ˆ L , ˆ E ) ← Stack { ˆ L j | j = 1 , ..., N s } , { ˆ E j | j = 1 , ..., N s } frontal-slice-wisely. the optimal solution to the following minimization problem is given by: S τ ( W ) = arg min X τ (cid:107) X (cid:107) + 12 (cid:107) X − W (cid:107) F , (B.6) where S τ is the shrinkage operator. Similarly, the closed-form solution for E subproblem in Eq. B.2 can bewritten as follows according to Theorem 3: E k +1 = arg min E L µ k ( L k +1 , E , Y k )= arg min E λ (cid:107) E (cid:107) + µ k (cid:107) X − L k +1 − E + Y k µ k (cid:107) F = S λµk ( X − L k +1 + Y k µ k ) . (B.7) References
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