Automatic Lung Segmentation of Volumetric Low-Dose CT Scans Using Graph Cuts

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1 Automatic Lung Segmentation of Volumetric Low-Dose CT Scans Using Graph Cuts Asem M. Ali and Aly A. Farag Computer Vision and Image Processing Laboratory (CVIP Lab) University of Louisville, Louisville, KY Abstract. We propose a new technique for unsupervised segmentation of the lung region from low dose computed tomography (LDCT) images. We follow the most conventional approaches such that initial images and desired maps of regions are described by a joint Markov-Gibbs random field (MGRF) model of independent image signals and interdependent region labels. But our focus is on more accurate model identification for the MGRF model and the gray level distribution model. To better specify region borders between lung and chest, each empirical distribution of volume signals is precisely approximated by a linear combination of Gaussians (LCG) with positive and negative components. LCG models parameters are estimated by the modified EM algorithm. Initial segmentation (labeled volume) based on the LCG models is then iteratively refined by using the MGRF with analytically estimated potentials. In this framework the graph cuts is used as a global optimization algorithm to find the segmented data (labeled data) that minimize a certain energy function, which integrates the LCG model and the MGRF model. To validate the accuracy of our algorithm, a special 3D geometrical phantom motivated by statistical analysis of the LDCT data is designed. Experiments on both phantom and 3D LDCT data sets show that the proposed segmentation approach is more accurate than other known alternatives. 1 Introduction Isolating the lung from its surrounding anatomical structures is a crucial step in many studies such as detection and quantification of interstitial disease, and the detection and/or characterization of lung cancer nodules. For more details, see Sluimer s et al. survey paper [1]. But CT lung density depends on many factors such as image acquisition protocol, subject tissue volume, volume air, and physical material properties of the lung parenchyma. These factors make lung segmentation based on threshold technique difficult. So, developing new accurate algorithms with no human interaction, which depends on gray level difference between lung and its background, is important to precisely segment the lung. The literature is rich with approaches of lung segmentation in CT images. Hu et al. [2], proposed an optimal gray level thresholding technique which is G. Bebis et al. (Eds.): ISVC 2008, Part I, LNCS 5358, pp , c Springer-Verlag Berlin Heidelberg 2008

2 Automatic Lung Segmentation of Volumetric Low-Dose CT Scans 259 used to select a threshold value based on the unique characteristics of the data set. In [3], Brown et al. integrated region growing and morphological operations with anatomical knowledge expert to automatically segment lung volume. A segmentation-by-registration scheme was proposed by Sluimer et al. [4] for automated segmentation of the pathological lung in CT. In that scheme, a scan with normal lungs is registered to a scan containing pathology. When the resulting transformation is applied to a mask of the normal lungs, a segmentation is found for the pathological lungs. Although shape-based, or Atlas-based (e.g.[5]), segmentation overcomes the problem of gray level inhomogeneities, the creation of a 3D shape model of the lung is not an easy task in the 3D case. Also these techniques need a registration step. Conventional methods that perform lung segmentation in CT depend on a large contrast in Hounsfield units between the lung and surrounding tissues. Although these methods accurately segment normal lung tissues from LDCT, they tend to fail in case of gray level inhomogeneity, which results form the abnormal lung tissues. The main advantage of our proposed segmentation approach over the conventional techniques is that it is based on modeling both the intensity distribution and spatial interaction between the voxels in order to overcome any region inhomogeneity existing in the lung region. Moreover, the proposed segmentation algorithm is fast which makes it suitable for clinical applications. Recently, graph cuts has been used as an interactive N-D image segmentations tool (For more details see [6]). Many studies used graph cuts for lung segmentation. Boykov and Jolly in [7] introduced an interactive segmentation framework. In that work, the user must identify some voxels as object and others as background seeds. Then graph cut approach is used to find the optimal cut that completely separates the object seeds from the background seeds. To overcome the time complexity and memory overhead of the approach in [6] for high resolution data, Lombaert et al. [8] performed graph cuts on a low-resolution image/volume and propagated the solution to the next higher resolution level by only computing the graph cuts at that level in a narrow band surrounding the projected foreground/background interface. Although the results of these approaches looked promising, manual interaction was still required. Interactive segmentation imposes some topological constraints reflecting certain high-level contextual information about the object. However, it depends on the user input. The user inputs have to be accurately positioned. Otherwise the segmentation results are changed. Chen et al. [9] used morphological operations and graph cuts to segment the lung from radiographic images automatically. In that work, the authors initialized an outer boundary for each lung region by shrinking 10 pixels from the boundaries of both vertical halves of an image. As in our case, this method does not work in axial CT slices, where there is a lung part in the middle of the image. Inner boundaries were obtained by dilating the regional minimum. However, due to the inhomogeneity in the given image, there were many regional minimums so they selected a regional minimum based on a threshold. Then the authors in [9] used graph cuts to find the boundaries of each lung region between its inner and outer boundaries. The data penalty and

3 260 A.M. Ali and A.A. Farag discontinuity penalty were chosen to be inversely proportional to the gray levels difference of the neighborhood pixels. This selection will be improper in the case of the axial CT lung slices, due to their gray level inhomogeneities. In this paper, we propose a novel automatic lung volume segmentation approach that uses graph cuts as a powerful optimization technique to get the optimal segmentation. Different from the previous graph cuts studies, in our segmentation approach, no user interaction is needed; instead, we use the volume gray level to initially pre-label the volume. To model the low level information in the CT lung volume, the gray level distribution of lung volume is approximated with a new Linear Combination of Gaussian (LCG) distributions with positive and negative components. Due to the closeness of the gray levels between the lung tissues and the chest tissues, we do not depend only on volume gray level, but we use the graph cuts approach to combine the volume gray level information and the spatial relationships between the region labels in order to preserve the details. Often, the potential of Potts model which describes the spatial interaction between the neighboring voxels is estimated using simple functions that are inversely proportional to the gray scale difference between the two voxels and their distance. Another contribution in this work is that the potentials of Potts model are estimated using a new analytical approach. After the lung volume is initially labeled we formulate a new energy function using both volume appearance models. This function is globally minimized using s/t graph cuts to get the final and optimal segmentation of lung. 2 Proposed Graph Cuts Segmentation Framework To segment a lung, we initially labeled the volume based on its gray level probabilistic model. Then we create a weighted undirected graph with vertices corresponding to the set of volume voxels P, and a set of edges connecting these vertices. Each edge is assigned a nonnegative weight. The graph also contains two special terminal vertices s (source) Lung, and t (sink) Chest and other tissues. Consider a neighborhood system in P, which is represented by a set N of all unordered pairs {p, q} of neighboring voxels in P. LetL the set of labels { 0, 1 }, correspond to lung and its background respectively. Labeling is a mapping from P to L, and we denote the set of labeling by f = {f 1,...,f p,...,f P }. In other words, the label f p, which is assigned to the voxel p P,segmentsit to lung or background. Now our goal is to find the optimal segmentation, best labeling f, by minimizing the following energy function: E(f) = D p (f p )+ V (f p,f q ), (1) p P {p,q} N where D p (f p ), measures how much assigning a label f p to voxel p disagrees with the voxel intensity, I p. D p (f p )= ln P(I p f p )isformulatedtorepresent the regional properties of segments, Sec.2.1. The second term is the pairwise interaction model which represents the penalty for the discontinuity between voxels p and q, Sec.2.2.

4 Automatic Lung Segmentation of Volumetric Low-Dose CT Scans Gray Level Probabilistic Model To initially label the lung volume and to compute the data penalty term D p (f p ), we use the modified EM [10] to approximate the gray level marginal density of each class f p, lung and background, using a LCG with C + f p negative components as follows: positive and C f p C + fp P (I p f p )= r=1 Cfp w + f ϕ(i p,r p θ + f ) p,r l=1 w f ϕ(i p,l p θ f p,l ), (2) where ϕ(. θ) is a Gaussian density with parameter θ (μ, σ 2 )withmeanμ and variance σ 2. w + f p,r means the rth positive weight in class f p and w f p,l means the l th negative weight in class f p. This weights have a restriction C+ fp r=1 w+ f p,r C fp l=1 w f p,l = 1. Fig. 1 illustrates an example for the lung gray level LCG model and its components Empirical Density Gaussian Components... Estimated Density (a) (b) (c) (d) (e) (f) Fig. 1. Example of a lung gray level LCG Model. (a) The empirical and initial estimated densities and the dominant components. (b) The scaled absolute deviations between the empirical and initial estimated densities. (c) Approximation error for the scaled absolute error as a function of the number of Gaussians, which is used to approximate the scaled absolute error in (b). (d) The components of the final LCG model. (e) The final LCG density approximation. (f) The LCG models of each class with the best separation threshold t = 109.

5 262 A.M. Ali and A.A. Farag 2.2 Spatial Interaction Model The homogenous isotropic pairwise interaction model which represents the penalty for the discontinuity between voxels p and q is defined as follows: { γ if fp f V (f p,f q )= q ;. (3) 0iff p = f q The simplest model of spatial interaction is the Markov Gibbs random field (MGRF) with the nearest 6-neighborhood. Therefore, for this specific model the Gibbs potential can be obtained analytically using the maximum likelihood estimator (MLE) for a generic MGRF [11]. So, the resulting approximate MLE of γ is: ) γ = (K K2 K 1 f neq(f). (4) where K = 2 is the number of classes in the volume and f neq (f) denotes the relative frequency of the not equal labels in the voxel pairs and it is defined as follows. f neq (f) = 1 δ(f p f q ), (5) T N {p,q} T N where the indicator function, δ(a) equals one when the condition A is true, and zero otherwise, T N = {{p, q} : p, q P; {p, q} N} is the family of the neighboring voxel pairs supporting the Gibbs potentials. 2.3 Graph Cuts Optimal Segmentation To segment a lung volume, instead of independently segmenting each 2D slice of the volume, we segment the 3D lung using a 3D graph (e.g. Fig. 2) where each vertex in this graph represents a voxel in the lung volume. Then we define the weight of each edge as shown in table 1. After that, we get the optimal segmentation surface between the lung and its background by finding the minimum cost cut on this graph. The minimum cost cut is computed exactly in polynomial time for two terminal graph cuts with positive edges weights via s/t Min-Cut/Max-Flow algorithm [12]. 3 Experiments and Discussion To assess the performance of the proposed segmentation framework, we demonstrate it on axial human chest slices obtained by spiral-scan low-dose computer tomography (LDCT), (the 8-mm-thick LDCT slices were reconstructed every 4 mm with the scanning pitch of 1.5 mm). Each volume is initially labeled using the gray level LCG model s threshold. However, due to the gray levels inhomogeneities, one can not precisely segment the lung using only this threshold as shown in Fig. 3 where, the misclassified voxels may include abnormal lung

6 Automatic Lung Segmentation of Volumetric Low-Dose CT Scans 263 Table 1. Graph Edges Weights Edge Weight for γ f {p, q} p f q 0 f p = f q {s, p} ln[p (I p 1 )] p P {p, t} ln[p (I p 0 )] p P Fig. 2. Example of graph that used in Lung Volume Segmentation. Note: Terminals should be connected to all voxels but for illustration issue we did not do this. Fig. 3. Samples of segmented lung slices using LCG model s threshold. (Error shown in red). tissues. After computing the initially labeled volume, the potential, which indicates the spatial interaction between its voxels, is computed using this labeled volume by Eq.(4). After that, we construct a graph for the given volume using the 6-neighborhood system (e.g. Fig. 2). Then the s/t graph cuts approach gives the minimum of the binary energy Eq.(1), which corresponds to optimal segmentation. The segmentation errors are evaluated, with respect to the ground truth produced by an expert (a radiologist). Fig. 4 shows samples of segmented slices for different subjects as well as their segmented 3D lung volumes. Table 2. Accuracy and time performance of our segmentation on 7 data sets in comparison to ICM and IT. Average volume 256x256x77. Algorithm Our ICM IT Minimum error, % Maximum error, % Mean error, % Standard deviation,% Significance, P Average time, sec

7 264 A.M. Ali and A.A. Farag (a) 2.08% (b) 2.21% (c) 2.17% (d) 1.95% Fig. 4. Samples of segmented lung slices using the proposed algorithm,(error are shown in red). Corresponding 3D lung volumes (Error are shown in green). Evaluation: To evaluate the results we calculate the percentage segmentation error as follows: 100 N umber of misclassif ied voxels. (6) error% = N umber of lung volume voxels We ran the proposed approach on 23 data sets. The statistical analysis of seven of them, for which we have their ground truths (radiologist segmentation),

8 Automatic Lung Segmentation of Volumetric Low-Dose CT Scans 265 are shown in Table 2. For comparison, the statistical analysis of Iterative Conditional Modes (ICM) [13] technique results, and the Iterative Threshold (IT) [2] approach results are also shown. The unpaired t-test is used to show that the differences in the mean errors between the proposed segmentation, and (ICM/and IT) are statistically significant (the two-tailed value P is less than ). The main problem in the segmentation of ICM and IT is that the misclassified voxels include abnormal lung tissues (lung cancer), bronchi and bronchioles as shown in Fig. 5. These tissues are important if lung segmentation is a pre-step in a detection of lung nodules system. The motivation behind our segmentation is to exclude such errors as far as possible. All algorithms are run on a PC 3Ghz Pentium 4, and 2GB RAM. All implementations are in C++. (Subj1) (Subj2) (a) (b) (c) Fig. 5. Examples of segmented lung slices that have nodules (bounded by yellow circle). (a) IT and (b) ICM approaches misclassified these parts as chest tissues (error is shown in red). However, (c) proposed algorithm correctly classified them as a lung. 4 Validation Due to the hand shaking errors, it is difficult to get accurate ground truth from manual segmentation. To assess the robustness of the proposed approach, we have created a 3D geometric lung phantom (256x256x81). To cerate this phantom, we started with a segmented (by a radiologist) lung volume (lung regions, arteries, veins, bronchi, and bronchioles). Then lung and its background signals of the phantom are generated according to the distributions P (I 0), and P (I 1), respectively, in Fig. 1(f) using the inverse mapping approach [10]. Fig. 6 shows some slices of the lung volume phantom. The error 0.71% between our results and ground truth confirms the high accuracy of the proposed segmentation framework. Fig. 7 shows proposed approach segmentation as well as the ICM segmentation of the phantom volume. As expected, in our approach the

9 266 A.M. Ali and A.A. Farag (a) (b) Fig. 6. Slices from the synthetic volume Fig. 7. Lung phantom segmentation results (a) The proposed algorithm 0.71%, and (b) The ICM technique 2.31% misclassified voxels are located at the boundary, and the misclassified voxels in ICM result lose abnormal lung tissues. 5 Conclusion In this paper, we have presented a novel framework for automatic lung volume segmentation using the graph cuts approach. Our proposed method addresses the interactive techniques limitation. We initially pre-label the volume using its gray level information. The gray level distribution of lung volume is approximated with a LCG distributions with positive and negative components. A MGRF model is used to describe the spatial interaction between the lung voxels. A new analytical approach to estimate 3D spatial interaction potentials for the MGRF model is presented. Finally, an energy function using the previous models is formulated, and is globally minimized using graph cuts. Experimental results show that the developed technique gives promising accurate results compared to other known algorithms. References 1. Sluimer, I., Schilham, A., Prokop, M., van Ginneken, B.: Computer analysis of computed tomography scans of the lung: a survey. IEEE Transactions on Medical Imaging 25, (2006)

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