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Hindawi Journal of Healthcare Engineering Volume 2022, Article ID 5654424, 21 pages https://doi.org/10.1155/2022/5654424 Research Article Efficient Johnson-S Mixture Model for Segmentation of CT Liver Image 1 2 Yueqin Dun and Yu Kong School of Electrical Engineering, University of Jinan, Jinan, Shandong, China Department of Medical Imaging, Shandong Medical College, Jinan, Shandong, China Correspondence should be addressed to Yueqin Dun; dunyq828@163.com and Yu Kong; kongy@sdmcjn.edu.cn Received 18 August 2021; Revised 7 February 2022; Accepted 9 March 2022; Published 14 April 2022 Academic Editor: Jinshan Tang Copyright © 2022 Yueqin Dun and Yu Kong. /is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. To overcome the problem that the traditional Gaussian mixture model (GMM) cannot well describe the skewness distribution of the gray-level histogram of a liver CT slice, we propose a novel segmentation method for liver CT images by introducing the Johnson-SB mixture model (J MM). /e Johnson-SB model not only has a ﬂexible asymmetrical distribution but also covers a SB variety of other distributions as well. In this article, the parameter optimization formulas for J MM were derived by employing SB the expectation-maximization (EM) algorithm and maximum likelihood. /e implementation process of the J MM-based SB segmentation algorithm is provided in detail. To make better use of the skewness of Johnson-SB and improve the segmentation accuracy, we devise an idea to divide the histogram into two parts and calculate the segmentation threshold for each part, respectively, which is called J MM-TDH. By analyzing and comparing the segmentation thresholds with diﬀerent cluster SB numbers, it is illustrated that the segmentation threshold of J MM-TDH will tend to be stable with the increasing of cluster SB number, while that of GMM is sensitive to diﬀerent cluster numbers. /e proposed J MM-TDH is applied to segment four SB randomly obtained abdominal CT image sequences, and the segmentation results and robustness have been compared between J MM-TDH and GMM. It is veriﬁed that J MM-TDH has preferable segmentation results and better robustness than GMM for SB SB the segmentation of liver CT images. In practice, CT slices are grayscale images. /e pixels in 1. Introduction the image reﬂect the X-ray absorption coeﬃcient of the Liver cancer is a common malignant neoplasm worldwide, corresponding voxels. Black areas represent low absorption and the incidence of primary liver cancer is still on the rise at areas, that is, low-density tissues and organs of human body, the global level [1]. /e accurate understanding of the shape such as lungs; white areas represent high-absorption areas, of the liver, the location and size of lesions in the liver tissue, that is, high-density body parts, such as bones. /e ab- and the relationship between the liver and surrounding sorption value in CT images is given in Hounsﬁeld units (HUs). Compared with ordinary X-ray images, CT images blood vessels can help doctors to develop more eﬀective treatment options. In addition, accurate liver segmentation have a higher density resolution. /erefore, CT images can is also conducive to three-dimensional reconstruction for better show organs composed of soft tissue, such as lung, liver and virtual surgery. At present, manual delineation of liver, gallbladder, pancreas, and pelvis, and can well dem- each slice by experts is still the standard clinical practice for onstrate pathological changes in the tomography image. At liver demarcation [2]. Because the segmentation of organs present, the original pixel size of the abdominal CT image is and lesions has to be carried out layer by layer in CTslices, it 512 × 512, and the HU value of the liver varies widely. For is pretty cumbersome and time-consuming for doctors or example, a healthy liver has smooth contours and uniform experts to do this repetitive work. density, and the absorption values are 60 ± 6 HU (or 2 Journal of Healthcare Engineering histograms of Figures 2(a)∼2(c)). Although the peak shape 64 ± 5 HU). However, when the liver has about 80% stea- tosis, the absorption values will be reduced to about − 50 HU. of gray-level histogram has symmetry to some extent, its asymmetry is also very obvious, which is diﬀerent from the In contrast, the density of the liver parenchyma will increase due to the accumulation of iron in patients with hemo- symmetric characteristic of Gaussian distribution and the chromatosis, whose CT scan showed that the liver paren- asymmetric characteristic of the exponential, Rayleigh dis- chyma was clearly dense and bright, with an absorption tribution, etc. /erefore, it is very diﬃcult to ﬁt the peak value as high as +140 HU (so-called white liver) [3]. /ere is shape accurately with any single distribution shown in no doubt that it is quite diﬃcult to accurately segment the Figure 1. liver within such a large gray range. To solve the problem, many researchers have focused on forming a mixture model using distribution functions that To solve the above problems, many scholars have pro- posed a variety of liver CT image segmentation methods can better ﬁt the shape of a single peak. /e research ideas mainly concentrate on the following three kinds of mixture [4–7]. Some of these algorithms require human-computer interaction, such as active contour [8] and Livewire (in- models. /e ﬁrst kind of mixture model is forming an asym- telligent scissors) [9, 10]. Some are semi-automatic, for example, graph-cut [11] and region growth [12, 13] methods. metric generalized Gaussian distribution (AGGD) by in- Some other methods focusing on fully automatic segmen- troducing shape parameters or functions into the tation include statistical shape models (SSMs) [14, 15] and generalized normal distribution to describe the skew thresholding algorithm [16]. Neural network algorithms characteristic [28, 29], so that it can describe not only a aiming to achieve automatic feature extraction have also symmetric distribution but also an asymmetric distribution. been applied to the segmentation of medical images in recent However, the expression of the AGGD is complicated by embedding the gamma function. In the case of using the EM years [17, 18]. /e ultimate common goal of diﬀerent methods is to segment images accurately and automatically, algorithm for maximum-likelihood estimation (MLE), all AGGD parameters in the mixture model are represented by but this goal is still a bottleneck problem in liver CT image segmentation, due to the complexity of abdominal CT highly nonlinear equations, which makes the numerical solutions cumbersome and sensitive to initial EM values images and the diﬀerences between diﬀerent liver mor- phologies. In this study, we aim to study on the ﬁnite [30]. mixture model (FMM), one kind of threshold segmentation Combining Gauss with other distributions to form a new algorithm, to improve the segmentation accuracy of liver CT model is the second idea. Wilson selected two Gaussian images, and try hard to segment automatically at the same distributions and one uniform distribution to ﬁt the low- time. gray and high-gray regions of the brain MRA data histo- According to the idea of the threshold segmentation gram, respectively [31]. Hassouna proposed a linear com- bination of a ﬁnite mixture model using one Rayleigh algorithm, if the grayscale threshold of the liver in the CT slice can be accurately determined, it is possible to realize the distribution and two Gaussian distributions [32]. Hence, for diﬀerent problems, people need to determine in advance liver segmentation automatically. In 1893, Karl Pearson made an experiment using the method of moments to ﬁt a which existing models can be used to form a new probability mixture of two normal components to the crabs’ data, which distribution model, and the number of each distribution also proved the FMM could improve the accuracy of clustering needs to be determined in advance. /erefore, this kind of [19]. Since then, FMM was adopted to improve the accuracy model is not ﬂexible, and it is also diﬃcult to realize seg- of threshold segmentation methods. In 1972, Chow and mentation automatically. Kaneko applied FMM in medical images to segment the left /e third way is to combine the components of the ventricle from cine angiograms with two Gaussian distri- mixed model with the non-Gaussian distribution. Lee and butions [20]. McLachlan introduced a ﬁnite mixture of canonical fun- damental deviation t (CFUST) distributions for asymmetric /ere are two core points in FMM, one is the selection of the probability density function of the mixed components and possibly long-tailed clusters [33]. Seﬁdpour and Bou- guila proposed and investigated the segmentation of spatial and the other is the parameter estimation of the mixture model. /e most common mixed component probability color images using the Dirichlet and Beta-Liouville distri- distribution used in FMM is the Gaussian distribution, butions [34]. /e normal inverse Gaussian distribution because in many cases there is a normal distribution in (NIG) is chosen by Karlis and Santourian to deal with univariate and multivariate data. /erefore, the Gaussian skewed subpopulations [35]. Franczak et al. studied the mixture model (GMM) has been widely used in the seg- asymmetric Laplace distribution (ALD) for clustering and mentation of the images [21–24]. In addition to the Gaussian classiﬁcation [36]. NIG and ALD belong to the family of generalized hyperbolic (GH) distributions designed by distribution, gamma distribution, Student’s t distribution, exponential distribution, and Rayleigh distribution also Barndorﬀ-Nielsen [37]. Browne and McNicholas extended a special case for the generalized hyperbolic distribution [38]. commonly appear in FMM [25–27], and their probability density diagrams with diﬀerent parameters are shown in Wraith and Forbes studied the properties of these distri- butions in multiscale and their application in multivariate Figure 1, respectively. /e upper, middle, and lower slices of the liver CT image clustering [39]. Although there are many combination sequence are shown in Figures 2(a)∼2(c)), respectively. methods of non-Gaussian distribution, only a few methods Figures 2(d)∼2(f) give the corresponding gray-level are used for the segmentation in medical images, especially Journal of Healthcare Engineering 3 μ = 127, x = 127, α = 127, 0.1 0.06 0.04 0.4 B = 10 σ = 10 v = 2 β = 1 0.04 µ = 10 α = 255, x = 255, v = 20 μ = 0, α = 1, 0.03 μ = 255, 0.3 0.04 β = 1 σ = 10 β = 20 σ = 10 x = 0, 0.05 0.02 0.02 µ = 20 0.2 v = 0.2 0.02 B = 50 0.01 0.1 B = 100 µ = 30 0 0 0 0 100 200 300 0 100 200 300 0 100 200 300 0 100 200 300 0 100 200 300 (a) (b) (c) (d) (e) Figure 1: Probability density of normal, gamma, exponential, Student’s t, and Rayleigh distributions. (a) μ: mean, σ: standard deviation. (b) α: shape parameter, β: scale parameter. (c) μ: mean. (d) x: position parameter, ]: degrees of freedom. (e) B: scale parameter. (a) (b) (c) 3500 2500 2000 2000 1600 1500 1200 1000 800 500 400 0 0 0 0 50 100 150 200 250 0 50 100 150 200 250 0 50 100 150 200 250 (d) (e) (f) Figure 2: Upper, middle, and lower slices of the liver CT image and their histograms. (a) /e upper slice. (b) /e middle slice. (c) /e lower slice. (d) Gray-level histogram of (a). (e) Gray-level histogram of (b). (f) Gray-level histogram of (c). in liver CT images. /erefore, the study on the segmentation are provided in Section 4. Finally, conclusions are drawn of liver CT images still needs further research. in Section 5. Considering the asymmetric skew characteristic of the gray histogram of the liver CT image, and inspired by the 2. Finite Johnson-S Mixture Model skew characteristic of Johnson-S distribution, we pro- pose a novel mixture model with the combination of In 1949, Johnson deduced a curve system called the Johnson Johnson-S distribution to segment the liver CT image. In system, which contains Johnson-S , Johnson-S , and B B L the following paper, the Johnson-S mixture model and Johnson-S . /e symbol S means “log-normal system,” S B U L B the optimized parameters with the EM algorithm are means “bounded system,” and S means “unbounded sys- introduced ﬁrstly in Section 2. Secondly, Section 3 gives tem” [40]. /e Johnson system can closely approximate the implementation details of the segmentation algorithm many continuous distributions by one of the three func- based on the Johnson-S mixture model, and the eﬀects of tional forms, so it is very ﬂexible to ﬁt variety curves. Many cluster number on the segmentation threshold are also of the commonly used continuous distributions, such as analyzed and compared between GMM and Johnson-S normal, log-normal, gamma, beta, and exponential, are mixture model in this section. /en, the segmentation special cases of the Johnson system; therefore, it has more experimental results of four randomly obtained abdom- advantages to ﬁt curves with the Johnson system than any inal CT image sequences from diﬀerent image databases other single distribution [41]. 4 Journal of Healthcare Engineering 2.1. Johnson-S Distribution. Johnson-S is one of the three B B p(x|Θ) � Φ p x|θ , (6) distributions of the Johnson system. It corresponds to the k k k k�1 distribution of a continuous random variable x in which a particular transformation is applied to obtain a normal where x⊂ X is the pixel grayscale value in the CT image, distribution. /e transformation is as follows: θ � (δ , c ), p (x|θ ) is the density of the kth component, k k k k k and Θ � (Φ , . . ., Φ ; θ , . . ., θ ) is the vector of parameters. x − ξ 1 k 1 k z � c + δln , (1) Note that p (x| Θ) in Eqn (3) deﬁnes a density that is called a λ + ξ − x K-component ﬁnite mixture density. Here, the weight of the kth mixing proportion Φ satisﬁes the following relations: where x is a given continuous random variable. In this study, x is the grayscale value of the pixel in the CT image. x∈ (ξ, 0 ≤ Φ ≤ 1 k � 1, 2, 3, . . . , K, (7) ξ + λ), ξ � min (x), λ � max (x) − min (x), and c and δ are shape parameters, δ > 0, c ∈ (− ∞, +∞). Z is a standard and normal random variable, and its probability density function is as follows: Φ � 1. (8) 1 2 − z /2 k�1 √�� � p(z) � .e . (2) 2π According to Eqn (3), the probability density of a Johnson-S mixture model is deﬁned as follows: We write x − ξ x − ξ z � c + δf(y), y � , and f(y) � ln , p(x|Θ) � Φ p x|δ , c . (9) k k k k λ + ξ − x λ + ξ − x k�1 (3) Here, where z is the inverse function of y. δ λ k − 1/2[c+δln(x− ξ/λ+ξ− x)] According to the transformations of continuous random √�� � p x|δ , c � . .e , k k k 2π (x − ξ)(λ + ξ − x) variable, (10) − 1/2[c+δf(y)] ′ √�� � ′ p(y) � δf (y)p(z) � .f (y).e . (4) 2π is the probability density of a random variable X for a Johnson-S distribution with the parameters δ and c . k k /en, the probability density function (PDF) with regard to x is as follows: 2.2.2. Optimizing Parameters with EM Algorithm. /e δ λ − 1/2[c+δln(x− ξ/λ+ξ− x)] vector of parameters Θ typically introduced by the log- √�� � p(x) � . .e . (5) 2π (x − ξ)(λ + ξ − x) likelihood function is deﬁned as follows: N N K Figure 3 shows some typical probability density function L(Θ) � lnp(X|Θ) � lnp x |Θ � ln Φ p x |δ , c , i k k i k k curves of Johnson-S with diﬀerent parameters. /e range of i�1 i�1 k�1 horizontal coordinates is from 0 to 255, and the vertical N K coordinates are the corresponding probability density 2 δ λ k − 1/2[c+δln(x− ξ/λ+ξ− x)] √�� � � ln Φ . .e . function values. Figure 3(a) presents the curves with dif- x − ξ λ + ξ − x 2π i i i�1 k�1 ferent δ and c � 0, and it is a normal distribution in the (11) middle. Figures 3(b)∼3(e) present the curves with diﬀerent δ and c � 0, respectively, and they have better skew charac- Here, x is the ith discrete grayscale value and N is the teristics, especially at the two ends of the abscissa. It can be number of discrete dots of CT image. seen that c controls the position of the function, and the /e detailed derivation of the parameters given in distributions of the function are normal distribution, neg- equations (6)–(8) is presented in Appendix A. atively skewed distribution, and positively skewed distri- bution when c � 0, c > 0, and c < 0, respectively. 3. Segmentation Algorithm Based on J MM SB 3.1. Implementation Details. /ere are mainly four steps to 2.2. Johnson-S Mixture Model (J MM) and Optimizing implement the segmentation algorithm based on J MM, B SB SB Parameters and the details of each step in Algorithm 2 are introduced as follows. 2.2.1. Johnson-S Mixture Model (J MM). /e ﬁnite mix- B SB ture model (FMM) refers to the linear superposition of distribution functions of the same type but with diﬀerent 3.1.1. Obtaining the Approximate Gray Value (LV_A). parameters. In the discrete case, the probability density /e Hounsﬁeld unit (HU) value of liver tissue varies from function of a ﬁnite mixture distribution can be expressed as a patient to patient. In addition, the X-ray tube of the CT p-dimensional random vector X [42]. machine will age with longtime use, which results in the Journal of Healthcare Engineering 5 0.15 0.04 –3 δ = 2.6, γ = 0 δ = 1, γ = -2 ×10 δ = 1, γ = 2 δ = 3, γ = 10 δ = 3, γ = -10 0.015 0.02 8 δ = 0.5, γ = 0.1 δ = 0.5, γ = -0.1 0.03 δ = 1.8, γ = 0 0.1 0.015 0.01 δ=0.1, γ=-5 δ=0.1, γ=5 0.02 δ = 1, γ = 1 δ = 1, γ = –1 6 0.01 δ = 3, γ = –6 δ = 3, γ = 6 0.005 0.05 0.005 δ = 1, γ = 0 0.01 δ = 3, γ = 2 δ = 3, γ = –2 0 0 0 0 0 100 200 0 100 200 300 0 100 200 300 0 100 200 0 100 200 (a) (b) (c) (d) (e) Figure 3: Probability density function curves with diﬀerent parameters. Input: initial values of θ � (Φ , δ , c ) (k � 1, 2, 3, . . ., K) k k k k Output: ﬁnal converged values of θ � (Φ , δ , c ) k k k k (1) Give the initial values of θ � (Φ , δ , c ) k k k k (2) while θ is not converged do (3) E-step: calculate the possibility of each xi coming from the kth submodel, based on the current parameters θ . P � P(Φ |x , θ ) � k i,k k i k √�� � √�� � 2 2 − 1/2[c+δln(x− ξ/λ+ξ− x)] K − 1/2[c+δln(x− ξ/λ+ξ− x)] Φ .δ / 2π.λ/(x − ξ)(λ + ξ − x ).e / Φ .δ / 2π .λ/(x − ξ)(λ + ξ − x ).e . k k i i j�1 j j i i (4) M-step: optimize the model parameters Φ , δ , c of the new iteration by using maximum likelihood. k k k N K √�� � Q(θ ) � P ln Φ + ln δ + lnλ/ 2π (x − ξ)(λ + ξ − x ) − 1/2[c + δ ln(x − ξ/λ + ξ − x )] , k i,k k k i i k k i i i�1 k�1 N N c � − [P ln(x − ξ/λ + ξ − x )]/ P δ � − A/Bδ , k i,k i i i,k k k i�1 i�1 2 2 2 2 δ � B/ [P ln (x − ξ/λ + ξ − x )] − A /B � B/C − A /B, k i,k i i i�1 Φ � P /N � B/N k i,k i�1 N N N where A � [P ln(x − ξ/λ + ξ − x )], B � P , C � [P ln (x − ξ/λ + ξ − x )]. i�1 i,k i i i�1 i,k i�1 i,k i i (5) end while (6) return θ � (Φ , δ , c ) k k k k ALGORITHM 1: EM algorithm for JSBMM. (1) Obtain an approximate grayscale value of the liver (LV_A) of a given CT image sequence. (2) for each slice in the CT image sequence do (3) Find the maximum grayscale value of the liver (LV_M) near the LV_A in the gray-level histogram of the CT slice. (4) At the LV_M point, divide the gray-level histogram into left and right parts. Fit the left and right histograms by using Algorithm 1 to determine the grayscale segmentation points of the liver. (5) Binarize the liver section according to the grayscale segmentation points solved in step 4, and then obtain the ﬁnal liver segmentation image after mathematical morphology processing. (6) end for ALGORITHM 2: Segmentation algorithm based on J MM. SB intensity decrease in the X-ray source. /erefore, the liver usually takes no more than 20 seconds [43]. /erefore, in the HU values of diﬀerent CT image sequences are usually same CT image sequence, the range of gray values between diﬀerent. Clearly, it is not possible to identify all liver scan all liver slices varied a little. It is feasible to obtain the liver sequences with the same HU range value. HU value of this sequence using the slice image with the We can make full use of the continuity of CTscan section largest liver area. images to solve this problem. CT scans require patients to Figure 4(a) shows the frontal outline of the liver, and hold their breath during data collection, and this process Figures 4(b)∼4(d) give the upper, middle, and lower slice 6 Journal of Healthcare Engineering (a) (b) (c) (d) (e) Figure 4: Frontal outline and the upper, middle, and lower sections of the liver CT scan sequence. 2500 2000 1500 178 0 0 0 50 100 0 100 200 (a) (b) (c) (d) Figure 5: Slices of 1/3 of the liver and the corresponding LV_A. (a) A slice of 1/3 of the liver in the ﬁrst sequence. (b) /e LV_A of (a) is 131. (c) A slice of 1/3 of the liver in the second sequence. (d) /e LV_A of (b) is 178. images of the liver CT scan sequence, respectively. It can be approximate grayscale value of the liver (LV_A) in seen that the liver is almost completely surrounded by the Figures 5(a) and 5(c), respectively. It should be noted that ribs, and the liver image at about 1/3 of the entire sequence is the maximum statistical number of the grayscale value 0 the largest one [44]. For example, if a complete sequence of should be omitted, although it is more than 2000, because it CTscans of the liver has 90 slices, the liver slice near the 30th means the black zone and is useless for the segmentation of slice will be more signiﬁcant than others. the liver. Figures 5(a) and 5(c) show two liver slices taken from two diﬀerent CT scan sequences, which locate at 1/3 of each 3.1.2. Finding the Maximum Gray Value (LV_M). sequence. Grayscale-level statistics were performed at about 1/4 of the body section, shown as the dotted line in Although liver HU values do not change signiﬁcantly during the scan process, there also exist some slight Figures 5(a) and 5(c). /e approximate grayscale value of the changes. To segment each slice accurately, it is necessary liver (LV_A) can be easily calculated after removing the grayscale values of black and white zones. Figures 5(b) and to further locate the maximum grayscale value of the liver (LV_M) close to the LV_A on the gray-level histogram of 5(d) are the corresponding gray-level histograms with a sampling width of 10 pixels for the positions shown as the each slice. For instance, Figure 6(a) shows the slice of the upper part dotted lines in Figures 5(a) and 5(c), respectively. Here, the horizontal coordinates are the grayscale values, and the of the liver CT sequence shown in Figure 5(c). Figure 6(c) displays the slice of the lower part in the same sequence. vertical coordinates are the statistical numbers of the cor- responding grayscale values. /e grayscale values with Figure 6(b) gives the gray-level histogram of Figure 6(a), and the gray-level histogram of Figure 6(c) is shown in maximum statistical numbers are 131 and 178 as shown in Figures 5(b) and 5(d), respectively, which are taken as the Figure 6(d). Because the value of LV_A shown in Figure 5(d) Journal of Healthcare Engineering 7 0 0 0 100 200 0 100 200 (a) (b) (c) (d) Figure 6: Maximum gray value of the liver (LV_M) in CTslice image. (a) /e upper section. (b) LV_M of Figure 6(a), (c) /e lower section. (d) LV_M of Figure 6(c). is 178, the corresponding LV_M values of Figures 6(a) and are three basic morphological set transformations. /ese 6(c) are 179 and 176, as shown in Figures 6(b) and 6(d), transformations involve the interaction between image and respectively. Here, it needs to be stated that we should structuring element [45]. Figure 7(c) is obtained by binar- choose the peak value closest to 178 as LV_M, because 178 is izing Figure 6(c). /e ﬁnal boundary of the liver segmen- determined in Figure 5(d). tation can be drawn as shown in Figure 7(d) after the procession of ﬁlling, erosion, and expansion algorithms in mathematical morphology. 3.1.3. Determining the Segmentation Points. /e position of peak and skewness characteristics of the gray-level histo- gram of liver slices are uncertain and nonuniqueness, so it is 3.2. Eﬀect of Cluster Number on the Segmentation Cresholds. diﬃcult to take full advantage of the J MM in describing A key point of FMM is how to select the cluster numbers to SB skewness if J MM is applied directly to ﬁt the whole gray- realize a better ﬁtting, and underﬁtting or overﬁtting may SB level histogram. To make full use of the advantages of occur if the cluster number is selected inappropriately [46]. Johnson-S in describing the skew characteristics, we di- In this study, the cluster number means the number of vided the gray-level histogram into the left and right parts at curves in a cluster used to ﬁt the gray-level histogram. To analyze the eﬀect of diﬀerent cluster numbers on segmen- LV_M to ensure that the gray level of the liver is just at the boundary of the histogram, to improve the Johnson-S tation thresholds, we choose two diﬀerent slices to be seg- segmentation accuracy. mented by GMM and J MM in the following Section 3.2.1 SB /e LV_M of Figure 6(c) is 176. At the LV_M point, the and Section 3.2.2, respectively. /e initial location of the gray-level histogram shown in Figure 6(d) is divided into the grayscale value is another consideration to aﬀect the seg- left part and the right part, and the grayscale values of the mentation threshold. In this study, the initial locations are segmentation points are obtained using J MM to ﬁt the evenly arranged within the range of the grayscale values of SB gray-level histograms. Figures 7(a) and 7(b) give the ﬁtting the entire image. results of the left part and the right part, respectively. /e grayscale value of the segmentation point of the left part is 3.2.1. Segmentation Cresholds of GMM with Diﬀerent 157, which is the intersection of the two rightmost ﬁtting Cluster Numbers. Table 1 gives the segmentation thresholds curves shown in Figure 7(a). /e grayscale value of the of the upper, middle, and lower slices from two diﬀerent CT segmentation point of the right part is 190, which is the image sequences. By analyzing the segmentation thresholds intersection of the two leftmost ﬁtting curves shown in with diﬀerent cluster numbers n, it can be found that the Figure 7(b). However, if there is a peak of one ﬁtting curve segmentation thresholds are changeable with the change in appearing between the intersection point of the other two n, while they do not tend to be stable with the increasing n. ﬁtting curves and the LV_M, the grayscale value of this peak In this study, n denotes the cluster numbers used to ﬁt the will be taken as the segmentation point, which can get better gray-level histogram, and n is the same as K in Algorithm 1. segmentation results. For example, the grayscale value of the To illustrate the eﬀect on the segmentation results with segmentation point of the right part should be 188, because diﬀerent cluster numbers, we give some segmentation results the grayscale value of the peak is 188, which appears before of the upper slices of the sequences S1 and S2. Here, Figure 8 the intersection of 190. /erefore, the range of the grayscale shows the threshold segmentation results of the upper slices value for liver segmentation in Figure 6(c) is set from 157 to with n � 6, 8, 10, 12, and 14 using GMM, respectively. /e 188, which is the range of the segmentation thresholds for ﬁrst row and the second row give the segmentation results Figure 6(c). and thresholds for the upper slice of the ﬁrst sequence S1. /e third row and the fourth row give the results and 3.1.4. Binarizing the Images and Processing with Mathe- thresholds for the upper slice of the second sequence S2. matical Morphology. Mathematical morphology is a tool for By comparing the segmentation results in the ﬁrst row, it extracting image components. Erosion, dilation, and ﬁlling is not diﬃcult to ﬁnd that the results of the ﬁrst three 8 Journal of Healthcare Engineering 0.06 0.02 0.04 20 36 43 47 57 67 83 97 111 125137147 0.01 0.02 0 50 100 150 180 200 220 240 (a) (b) (c) (d) Figure 7: Grayscale values of segmentation points and the binarized result. Table 1: Segmentation thresholds of two diﬀerent CT image sequences using GMM. Segmentation thresholds Cluster number (n) Slices of sequence S1 Slices of sequence S2 Upper Middle Lower Upper Middle Lower 6 153∼201 168∼196 158∼204 115∼149 117∼149 115∼147 7 157∼201 168∼195 162∼194 109∼150 117∼150 115∼148 8 153∼202 165∼197 157∼196 122∼148 121∼145 120∼144 9 175∼193 169∼197 175∼188 112∼149 117∼152 116∼148 10 154∼201 172∼190 158∼196 124∼150 122∼150 121∼145 11 173∼195 169∼193 168∼191 113∼148 118∼153 116∼149 12 174∼194 168∼194 169∼191 126∼150 124∼150 124∼147 13 177∼193 172∼184 168∼189 115∼147 117∼150 116∼149 14 173∼194 169∼190 166∼191 127∼150 124∼150 125∼146 15 173∼194 168∼193 168∼191 117∼144 118∼142 117∼145 columns with n � 6, 8, and 10 are similar, because the three slices of the ﬁrst liver CT sequence S1, and here, the seg- segmentation results include some other organs that do not mentation thresholds are obtained with J MM-TDH, which SB belong to the liver. Although the segmentation result in the means the gray-level histogram is divided into left and right fourth column with n � 12 only includes the liver, the seg- parts, and it is denoted as thresholds of dividing histogram mentation result is a little smaller liver than that of the ﬁfth (TDH). While thresholds of whole histogram (TWH) mean column with n � 14. /erefore, the segmentation result with that the segmentation thresholds are obtained with J MM SB n � 14 is relatively more reasonable by comparing with the under the whole gray-level histogram, and it is denoted as other four results for the ﬁrst sequence. J MM-TWH. /e segmentation results of J MM-TWH SB SB Comparing the results of row 3 for the second sequence, are not given in Figures 9∼11, and only the segmentation it is obvious that the results of the ﬁrst two columns with thresholds are summarized in the TWH column given in n � 6 and 8 are not correct, and the segmentation result of Table 2. /e segmentation results of J MM-TDH with n � 6, SB the third column with n � 10 is a little smoother than that of 8, 11, and 13 are shown in Figures 9(a)∼9(d), respectively. the fourth one with n � 12, while the segmentation result of Figures 9(e)∼9(n) show the segmentation thresholds of the the ﬁfth column with n � 14 turns to be rougher boundary left and right parts with n � 6∼15, respectively. due to overﬁtting. /us, the better segmentation result can Figures 10(a)∼10(n) and 11(a)∼11(n) give the corresponding be obtained when n is 10 for the second sequence. As can be segmentation results and thresholds of J MM-TDH for the SB seen from Figure 8, the segmentation results of GMM are middle and lower slices, respectively. sensitive to cluster numbers, and it is not easy for GMM to /e segmentation thresholds with n � 6∼15 for the up- use ﬁxed cluster number to get accurate segmentation re- per, middle, and lower slices are summarized in the cor- sults, which is a diﬃculty in the application of GMM, es- responding TDH column given in Table 2. It is obvious that pecially for a large number of images that need to be the segmentation thresholds almost tend to be stable when n segmented. is bigger than 12. While the segmentation thresholds in Here, it is needed to be stated that the red contours in the TWH column are also changeable, even n is bigger than 12, ﬁgures in Figure 8 indicate the error zones of the seg- which is similar to the trend of GMM. By analyzing and mentation. It is the same meaning in the following ﬁgures. comparing the range of segmentation thresholds between J MM-TDH and J MM-TWH with the same n, it can be SB SB seen that most of the upper limit values are close to each 3.2.2. Segmentation Cresholds of J MM with Diﬀerent other, but the lower limit values of J MM-TDH are smaller SB SB Cluster Numbers. Figures 9∼11 give the segmentation than that of J MM-TWH, and the diﬀerences are mostly SB thresholds and the results of the upper, middle, and lower around 10. Journal of Healthcare Engineering 9 Threshold:174~194 Threshold:173~194 Threshold:153~201 Threshold:153~202 Threshold:154~201 0.015 0.02 0.015 0.015 0.015 173 7 26 44 6381 107 127150 199 227246 2941 73 118 10 14 68 94 123 154 201 236 153 201 174 194 0.015 14 45 59 81 105126 150 228 246 0.01 0.01 0.01 0.01 37 81 119 153 202 231 0.01 0.005 0.005 0.005 0.005 0.005 0 0 0 0 0 0 100 200 0 100 200 0 100 200 0 100 200 0 100 200 Threshold:122~148 Threshold:115~149 Threshold:124~150 Threshold:126~150 Threshold:127~150 0.03 23 0.03 0.03 124 0.03 126 150 0.03 33 67 115 149 23 33 150 10 33 63 80 174 198 222245 241 122 148 65 177 213 245 2535 63 90 98 105 152 200 243 127 150 0.02 0.02 0.02 0.02 0.02 10 35 67 87 107 171187208230247 0.01 0.01 0.01 0.01 0.01 0 0 0 0 0 100 200 0 100 200 0 100 200 0 100 200 0 100 200 (a) (b) (c) (d) (e) Figure 8: Segmentation results and thresholds of GMM with diﬀerent cluster numbers. (a) n � 6. (b) n � 8. (c) n � 10. (d) n � 12. (e) n � 14. (a) (b) (c) (d) 0.02 0.02 0.08 194 204 220 233 245 194 203 215 227 239 248 0.015 0.06 0.015 0.06 25 41 72 107 147 37 42 49 59 85 122 143 0.01 0.04 0.01 0.04 0.005 0.02 0.005 0.02 0 0 0 0 0 50 100 150 200 220 240 260 0 50 100 150 200 220 240 (e) (f) Figure 9: Continued. 10 Journal of Healthcare Engineering 0.02 0.08 0.02 194 194 199 206 215 224 232 240 248 201 209 219 230 240 248 0.015 0.06 0.015 0.06 25 44 62 82 115 159 25 44 61 81 104 125 147 160 0.01 0.04 0.01 0.04 0.005 0.02 0.005 0.02 0 0 0 0 0 50 100 150 200 220 240 260 0 50 100 150 200 220 240 (g) (h) 0.02 0.02 194 194 197 203 209 216 223 230 237 242 248 198 204 212 220 228 235 241 248 0.015 0.06 0.015 0.06 38 46 19 63 81 101 119 135 147 161 0.01 0.04 0.01 0.04 23 42 56 75 91 112 131 147 160 0.005 0.02 0.005 0.02 0 0 0 0 0 50 100 150 200 220 240 260 0 50 100 150 200 220 240 260 (i) (j) -3 ×10 0.02 194 196 197 202 207 213 220 226 232 238 243 249 201 205 211 217 223 229 233 239 244 249 0.06 15 0.015 0.06 139 149 19 35 47 61 77 92 109 163 18 34 47 58 71 84 100 115 129 151 0.01 0.04 0.04 0.005 0.02 0.02 0 0 0 0 50 100 150 200 220 240 260 0 50 100 150 200 220 240 260 (k) (l) -3 -3 ×10 ×10 194 196 200 214 220 225 230 235 239 195 228 237 244 204 209 244 249 199 203 207 212 217 222 232 240 249 15 0.06 15 0.06 44 112 156 17 32 48 61 75 85 99 135 10 48 0.04 10 0.04 18 33 56 119 154 164 65 79 92 107 132 5 0.02 5 0.02 0 0 0 0 0 50 100 150 200 220 240 260 0 50 100 150 200 220 240 260 (m) (n) Figure 9: Segmentation thresholds of JSBMM-TDH with diﬀerent cluster numbers for the upper slice. (a) n � 6. (b) n � 8. (c) n � 11. (d) n � 13. (e) n � 6. (f) n � 7. (g) n � 8. (h) n � 9. (i) n � 10. (j) n � 11. (k) n � 12. (l) n � 13. (m) n � 14. (n) n � 15. (a) (b) (c) (d) Figure 10: Continued. Journal of Healthcare Engineering 11 0.03 0.08 0.03 162 199 0.08 143 14 48 99 147 153 62 48 50 80 117 0.06 0.02 0.02 0.06 0.04 0.04 0.01 0.01 0.02 0.02 0 0 0 0 0 50 100 150 200 220 240 0 50 100 150 200 220 240 (e) (f) 0.03 0.03 0.08 17 47 60 83 110 146 20 47 105 125 147 60 82 0.08 0.02 0.06 0.02 0.06 0.04 0.04 0.01 0.01 0.02 0.02 0 0 0 0 0 50 100 150 200 220 240 0 50 100 150 200 220 240 (g) (h) 15 35 47 61 81 100 116 132 147 20 46 4957 72 91 112 130 147 161 195 194 0.08 0.08 0.02 0.02 0.06 0.06 0.04 0.04 0.01 0.01 0.02 0.02 0 0 0 0 0 50 100 150 200 220 240 0 50 100 150 200 220 240 (i) (j) 17 33 47 60 76 91 109 123136148 18 33 47 68 69 84 99 113 128139149 162 194 0.08 0.08 0.02 0.02 0.06 0.06 0.04 0.04 0.01 0.01 0.02 0.02 0 0 0 0 0 50 100 150 200 220 240 0 50 100 150 200 220 240 (k) (l) 17 32 595664 79 91 106117 131142 17 32 41 57 60 73 85 97 111 122 133 144151 44 150 162 0.08 0.08 0.02 0.02 0.06 0.06 0.04 0.04 0.01 0.01 0.02 0.02 0 0 0 0 0 50 100 150 200 220 240 0 50 100 150 200 220 240 (m) (n) Figure 10: Segmentation thresholds of JSBMM-TDH with diﬀerent cluster numbers for the middle slice. (a) n � 6. (b) n � 8. (c) n � 11. (d) n � 13. (e) n � 6. (f) n � 7. (g) n � 8. (h) n � 9. (i) n � 10. (j) n � 11. (k) n � 12. (l) n � 13. (m) n � 14. (n) n � 15. Analyzing and comparing the segmentation results of sequence S1 shown in Table 1. By analyzing and comparing (a)∼(d) shown in Figures 9∼11, respectively, the better the range of segmentation thresholds between J MM-TWH SB segmentation results can be obtained for the upper, and GMM with the same n, it is found that most of the upper middle, and lower slices when n is 13. /erefore, con- and lower limit values are close to each other, which means sidering the trend of segmentation thresholds given in the segmentation results of J MM-TWH are similar to that SB Table 2, we determine to take n � 13 as the ﬁxed cluster of GMM. /erefore, the segmentation results of J MM- SB number for J MM-TDH to segment diﬀerent CT image TDH are both better than the results of J MM-TWH and SB SB sequences in the following segmentation experiments in GMM. Section 4. Table 3 gives the segmentation calculation time with As a matter of convenience in comparison, Table 2 also diﬀerent cluster numbers as shown in Figures 9–11. /e gives the segmentation thresholds of GMM for liver CT segmentation time is less aﬀected by the diﬀerent cluster 12 Journal of Healthcare Engineering (a) (b) (c) (d) 0.06 0.05 200 217 230 248 0.02 0.02 0.04 40 46 60 86 120 0.04 58 67 93 126 0.015 0.03 0.01 0.02 0.01 0.02 189 0.005 0.01 0 0 0 0 0 50 100 150 180 200 220 240 0 50 100 150 180 200 220 240 (e) (f) 0.05 195 0.05 0.02 0.02 20 45 60 85 119 42 79 0.04 21 45 60 101 122 0.04 0.03 0.03 0.01 0.01 0.02 0.02 0.01 0.01 0 0 0 0 0 50 100 150 180 200 220 240 0 50 100 150 180 200 220 240 (g) (h) 0.05 0.05 0.02 0.02 0.04 0.04 24 43 62 81 98 116 130 145 21 414857 68 88 108 156 126 146 0.03 0.03 0.01 0.01 0.02 0.02 0.01 0.01 0 0 0 0 0 50 100 150 180 200 220 240 0 50 100 150 180 200 220 240 (i) (j) 0.06 0.05 0.02 0.02 21 4046 60 73 89 105 121 134146 0.04 0.04 19 354447 57 67 83 97 111 125137 0.03 0.01 0.01 0.02 0.02 189 188 0.01 0 0 0 0 0 50 100 150 180 200 220 240 0 50 100 150 180 200 220 240 (k) (l) 189 188 0.05 0.05 0.02 0.02 0.04 0.04 44 120 141 149 18 30 60 71 83 96 108 130 158 0.03 158 45 18 3244 5054 63 77 89 103 116128139148 0.03 0.01 0.01 0.02 0.02 0.01 0.01 0 0 0 0 0 50 100 150 180 200 220 240 0 50 100 150 180 200 220 240 (m) (n) Figure 11: Segmentation thresholds of JSBMM-TDH with diﬀerent cluster numbers for the lower slice. (a) n � 6. (b) n � 8. (c) n � 11. (d) n � 13. (e) n � 6. (f) n � 7. (g) n � 8. (h) n � 9. (i) n � 10. (j) n � 11. (k) n � 12. (l) n � 13. (m) n � 14. (n) n � 15. Journal of Healthcare Engineering 13 Table 2: Segmentation thresholds of S1 with diﬀerent segmentation methods. /e upper slice /e middle slice /e lower slice n J MM J MM J MM SB SB SB GMM GMM (Figure 8) GMM (Figure 8) (Figure 8) TDH (Figure 9) TWH TDH (Figure 10) TWH TDH (Figure 11) TWH 6 143∼194 157∼194 153∼201 166∼200 167∼195 168∼196 149∼189 160∼194 158∼204 7 147∼194 156∼198 157∼201 162∼199 171∼190 168∼195 148∼195 161∼193 162∼194 8 159∼194 164∼193 153∼202 162∼198 168∼195 165∼197 151∼195 165∼192 157∼196 9 160∼194 176∼192 175∼193 162∼196 168∼196 169∼197 153∼196 170∼190 175∼188 10 160∼194 176∼192 154∼201 161∼195 173∼190 172∼190 156∼196 171∼187 158∼196 11 161∼194 175∼192 173∼195 162∼194 170∼187 169∼193 156∼190 168∼189 168∼191 12 163∼193 175∼193 174∼194 162∼194 171∼188 168∼194 157∼189 170∼189 169∼191 13 164∼ 194 176∼193 177∼193 161∼193 171∼189 172∼184 157∼ 188 171∼189 168∼189 14 164∼ 194 177∼192 173∼194 161∼193 171∼185 169∼190 158∼ 188 169∼186 166∼191 15 164∼ 194 172∼185 173∼194 162∼ 193 171∼186 168∼193 158∼ 188 170∼188 168∼191 It is obvious that the segmentation thresholds almost tend to be stable when n is bigger than 12, which is shown as the bold values shown in TDH columns. Table 3: Segmentation calculation time for the J MM with TDH for the upper, middle, and lower slices when n � 6∼15, respectively (time SB unit is second). n 6 7 8 9 10 11 12 13 14 15 Upper (Figure 9) 2.62 11.7 5.71 3.05 3.30 5.60 3.01 4.89 5.15 1.99 Middle (Figure 10) 5.09 6.26 5.51 4.58 4.97 4.46 4.03 3.64 3.94 3.64 Lower (Figure 11) 2.41 7.57 4.38 2.59 3.09 2.95 2.56 4.10 5.41 3.47 numbers, and it almost tends to decrease for the middle slice best one chosen from the segmentation results with n � 6∼15 of Figure 10 with a large liver area. In this study, only the for each slice for the sequences S2, S3, S4, and S5, respec- parameters Φ, δ, and c are used to calculate the threshold tively, while all segmentation results of J MM-TDH are SB value of analytic solution, so the segmentation speed is quite ﬁtted with the same cluster number n � 13. However, most fast, and the eﬀect of segmentation time with diﬀerent segmentation results of J MM-TDH are better than those of SB cluster numbers can be ignored. GMM, and the other few results are similar to each other. On the whole, it is illustrated that not only the segmentation results of J MM-TDH are better than those of GMM, but SB 4. Segmentation Experimental Results also the cluster numbers can be ﬁxed at n � 13 without worrying about overﬁtting. /is is good for the imple- /e J MM-TDH was applied to segment four randomly SB obtained abdominal CT image sequences from diﬀerent mentation of automatic segmentation of liver CT images. image databases [47,48], and the segmentation thresholds /e Jaccard index and Dice coeﬃcient are common and results of J MM-TDH are compared with that of indexes for quantitative evaluation of image segmentation. SB GMM, as shown in Figures 12∼14, which are the upper, /e Jaccard index is a statistic used for comparing the similarity of sample sets. /e Dice coeﬃcient is another middle, and lower slices of the four diﬀerent CT image sequences, respectively. In these ﬁgures, the ﬁrst row shows similarity measure index. /e Jaccard index and Dice co- eﬃcient are calculated to quantitatively evaluate the seg- the original image from four diﬀerent CT image sequences, the second row gives the threshold segmentation results of mentation results of GMM and J MM-TDH, respectively. SB /e values of the Jaccard index and Dice coeﬃcient are given GMM, and the third row is the binary images obtained according to the segmentation results of GMM. /e fourth in Table 4, and the values in the column of diﬀerence mean the diﬀerence value between J MM-TDH and GMM, that row shows the binary images obtained depending on the SB segmentation results of J MM-TDH. /e ﬁfth row is the is, the Jaccard index (or Dice coeﬃcient) of J MM-TDH SB SB minus that of GMM. /e last row gives the average value. segmentation results of GMM, and the sixth row gives the results of J MM-TDH. /e seventh row and the last row /e maximum value of diﬀerence column in the Jaccard SB provide the left and right segmentation thresholds of index is 0.1987, and the average value of this column is J MM-TDH, respectively. 0.0691. /e maximum value of diﬀerence column in the Dice SB Table 4 gives the comparison of segmentation thresholds coeﬃcient is 0.1863, and the average value of this column is 0.048. By comprehensive quantitative comparison, it is and quantitative evaluation for Figures 12–14, respectively. Here, it needs to be stated that the segmentation results of found that the segmentation results of J MM-TDH are SB better than that of GMM. GMM are ﬁtted with diﬀerent cluster numbers, which is the 14 Journal of Healthcare Engineering 0.03 0.02 4 33 56 80 105 130 158196 226247 0.01 152 169 126 150 0.02 10 32 59 86 114 136 192 229246 12 40 59 81 105 174 196 226 248 0.015 127 143 0.008 15 31 58 81 105 170 194 219 244 0.02 0.006 0.01 0.01 0.004 0.01 0.005 0.002 0 0 0 0 100 200 0 100 200 0 100 200 0 100 200 0.04 0.03 11 37 52 67 84 103 121 138 154 170 174 181 29 39 49 60 71 82 92 102 111 0.01 0.025 7 0.03 17 3346 49 61 74 87 100112123133 110 133 0.02 0.008 0.02 12 25 32 37 46 58 71 81 91 101 110 0.015 0.006 0.02 0.01 0.004 0.01 0.01 0.005 0.002 0 0 0 0 0 50 100 0 50 100 150 0 50 100 0 50 100 150 200 0.1 159166 177 188 197 200217220234 242 0.1 0.06 0.06 152 177182 189 196 203 212 220 227233239 246 212 0.08 0.04 0.04 0.06 0.05 0.04 0.02 0.02 0.02 0 0 0 0 150 200 250 160 180 200 220 240 150 200 250 210 220 230 240 250 Figure 12: Binary images, segmentation thresholds, and results of J MM-TDH and GMM for the upper slice. SB Journal of Healthcare Engineering 15 –3 x 10 152 173 188 221 0.025 93037 54 80 96 124 206 227247 0.02 33 57 81 104 130 157 178 247 0.02 130 145 123 154 18 33 58 87 113 166 194 224 247 0.02 10 14 41 57 79 104 174 207 227 246 0.015 0.01 0.01 0.01 0.005 0 0 0 0 100 200 0 100 200 0 100 200 0 100 200 0.03 0.03 0.02 37 52 67 84 104 122 138 154168 20 37 50 61 74 84 100 112 123 135 0.03 0.015 13 23 32 41 51 61 73 83 93102 110 114 0.02 0.02 0.02 0.01 0.01 0.01 0.01 117 0.005 0 0 0 0 0 50 100 0 50 100 0 50 100 150 200 0 50 100 150 0.1 0.1 0.1 166 170 180 188 200 209218 226 234 243 154162168 178 189 199 208 218227233241 0.06 0.08 0.06 0.04 0.05 0.05 0.04 0.02 0.02 0 0 0 0 150 200 250 160 180 200 220 240 150 200 250 210 220 230 240 250 Figure 13: Binary images, segmentation thresholds, and results of J MM-TDH and GMM for the middle slice. SB 16 Journal of Healthcare Engineering 0.03 –3 0.03 x 10 0.02 158 170 8 3 36 53 74 110 138 173 200 227 243 192 223 125 151 100 126 34 60 83 104 127 157 177 246 0.015 0.02 15 41 58 81 105 173 200 221 245 13 31 54 79 136140 169 201 232 249 0.02 0.01 0.01 0.01 5 0.005 0 0 0 100 200 0 100 200 0 100 200 0 100 200 0.02 0.03 0.03 0.02 125 113 0.015 12 34 47 62 76 93 109 125 140154166 19 37 50 61 75 89 102 114125 136 149 14 25 31 38 47 59 71 81 90 101 113 6 22 29 39 48 58 68 78 88 96 106 0.02 0.02 0.01 0.01 0.01 0.01 0.005 0 0 0 0 0 50 100 0 50 100 150 0 50 100 0 50 100 150 200 0.06 179184 191 198 205 213 220 228233 240 247 0.08 175 155164 174 184 196 206 216 225233 243 206 0.06 0.06 0.06 0.04 0.04 0.04 0.04 0.02 0.02 0.02 0.02 0 0 0 150 200 250 160 180 200 220 240 150 200 250 210 220 230 240 250 Figure 14: Binary images, segmentation thresholds, and results of J MM-TDH and GMM for the lower slice. SB Journal of Healthcare Engineering 17 Table 4: Comparison of segmentation thresholds and quantitative evaluation for Figures 12–14. Segmentation thresholds Quantitative evaluation Jaccard index Dice’s coeﬃcient Liver slice position Sequence JSBMM- GMM JSBMM- JSBMM- TDH(n � 13) GMM Diﬀerence GMM Diﬀerence TDH TDH 126∼150 S2 126∼146 0.9063 0.8854 − 0.0209 0.9509 0.9392 − 0.0117 (n � 12) 152∼169 S3 146∼171 0.8793 0.9483 0.069 0.9358 0.9735 0.0377 Upper slice (n � 12) (Figure 12) 127∼143 S4 122∼152 0.5955 0.7608 0.1653 0.7465 0.8641 0.1176 (n � 12) 196∼226 S5 181∼212 0.5818 0.7805 0.1987 0.7356 0.8767 0.1411 (n � 12) 123∼154 S2 117∼144 0.9173 0.9298 0.0125 0.9569 0.9636 0.0067 (n � 12) 152∼173 S3 149∼177 0.9549 0.948 − 0.0069 0.9769 0.9733 − 0.0036 Middle slice (n � 13) (Figure 13) 130∼145 S4 108∼151 0.6523 0.9527 0.3004 0.7895 0.9758 0.1863 (n � 12) 188∼221 S5 172∼213 0.9744 0.9744 0.0 0.9501 0.9501 0.0 (n � 11) 125∼151 S2 125∼140 0.8852 0.8794 − 0.0058 0.9391 0.9358 − 0.0033 (n � 12) 158∼170 S3 149∼175 0.8234 0.904 0.0806 0.9031 0.9496 0.0465 Lower slice (n � 14) (Figure 14) 100∼126 S4 113∼153 0.8715 0.8777 0.0062 0.9314 0.9349 0.0035 (n � 13) 192∼223 S5 169∼212 0.9422 0.9719 0.0297 0.8907 0.9454 0.0547 (n � 11) Average value 0.832 0.9011 0.0691 0.8922 0.9402 0.048 /e website addresses of the data sets S1, S2, S3, S4, and with those of GMM. Analyzing the segmentation results and S5 are provided in Appendix B. quantitative evaluations, it is further illustrated that J MM- SB TDH does not have the overﬁtting phenomenon with the increase in cluster number, which veriﬁes that J MM-TDH SB 5. Conclusions has preferable segmentation results and better robustness than GMM. J MM-TDH makes it possible to realize the /e J MM-TDH with ﬂexibly skewed characteristics pro- SB SB automatic segmentation of live CT image due to the ro- posed in this study is suitable for ﬁtting the skewness dis- bustness with the ﬁxed cluster number. /e J MM-TDH tribution of the gray-level histogram of liver CT images. /e SB can be used not only for liver CT image segmentation, but parameter optimization algorithm employing EM and the also for other CT image segmentation as well. implementation process of the segmentation algorithm has been given in detail. /e eﬀects of cluster number on seg- mentation threshold were discussed and compared for Appendix GMM, J MM-TWH, and J MM-TDH, respectively. It is SB SB shown that the J MM-TDH threshold will tend to be stable SB at cluster number 13, while the threshold of GMM and the threshold of J MM-TWH are similar and sensitive to M-step updates equations (6)∼(8) for solving the JSBMM SB diﬀerent cluster numbers. /e proposed J MM-TDH with parameters presented in Section 2.2.2, which are derived by SB cluster number 13 is applied to segment four random CT maximizing the complete data log-likelihood Q with respect image sequences, and the segmentation results are compared to each model parameter as follows: 18 Journal of Healthcare Engineering N K ⎧ ⎨ ⎫ ⎬ λ 1 x − ξ √�� � Q θ � P lnΦ + lnδ + ln − c + δ ln . (A.1) k i,k k k k k ⎩ ⎭ 2π x − ξ λ + ξ − x 2 λ + ξ − x i i i i�1 k�1 N N N ⎧ ⎨ ⎫ ⎬ zQ x − ξ x − ξ i i � P − c + δ ln � − c P + δ P ln . (A.2) i,k k k k i,k k i,k ⎩ ⎭ zc λ + ξ − x λ + ξ − x k i i i�1 i�1 i�1 P ln x − ξ/λ + ξ − x i�1 i,k i i (A.4) Let c � − δ � − δ , k k k P i�1 i,k zQ � 0. (A.3) where zc /en, N N x − ξ A � P ln , B � P . (A.5) i,k i,k λ + ξ − x i�1 i�1 zQ 1 x − ξ x − ξ i i � P − c + δ ln ln . (A.6) i,k k k zδ δ λ + ξ − x λ + ξ − x k k i i i�1 (A.7) ∵c � − δ . k k zQ 1 A x − ξ x − ξ i i ∴ � P − − δ + δ ln ln i,k k k zδ δ B λ + ξ − x λ + ξ − x k k i i i�1 (A.8) N 2 1 − − A/B + ln x − ξ/λ + ξ − x δ ln x − ξ/λ + ξ − x i i k i i � P . i,k i�1 Let /en, zQ � 0. (A.9) zδ A x − ξ x − ξ i i P 1 − − + ln δ ln � 0. (A.10) i,k B λ + ξ − x λ + ξ − x i i i�1 A x − ξ x − ξ i i P 1 − − + ln δ ln � 0 i,k B λ + ξ − x λ + ξ − x i i i�1 N N A x − ξ x − ξ i i ⇒δ P − + ln ln � P i,k i,k B λ + ξ − x λ + ξ − x i i i�1 i�1 P 2 i�1 i,k (A.11) ⇒δ � P − A/B + ln x − ξ/λ + ξ − x ln x − ξ/λ + ξ − x i�1 i,k i i i i P i,k i�1 N N 2 − A/B P ln x − ξ/λ + ξ − x + P ln x − ξ/λ + ξ − x i,k i i i,k i i i�1 i�1 B B � � . N 2 2 2 P ln x − ξ/λ + ξ − x − A /B C − A /B i�1 i,k i i Journal of Healthcare Engineering 19 where BC − A > 0 is proved, because N 2 x − ξ B B 2 i 2 C � P ln , δ � � . i,k k 2 2 λ + ξ − x C − A /B BC − A i�1 (A.12) N N N x − ξ x − ξ i i A � P ln B � P C � P ln . (A.13) i,k i,k i,k λ + ξ − x λ + ξ − x i i i�1 i�1 i�1 Let /en, x − ξ ln � X . (A.14) λ + ξ − x N N N 2 2 ⎡ ⎣ ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ BC − A � P P X − P X i,k i,k i i,k i i�1 i�1 i�1 N N− 1 N N N− 1 N 2 2 2 2 2 2 ⎢ ⎥ ⎢ ⎥ ⎡ ⎢ ⎤ ⎥ ⎡ ⎢ ⎤ ⎥ ⎣ ⎦ ⎣ ⎦ � P X + P P X + X − P X + P P 2X X (A.15) i,k j,k i,k j,k i j i,k i i j i,k i i�1 i�1 j�i+1 i�1 i�1 j�i+1 N− 1 N � P P X − X . i,k j,k i j i�1 j�i+1 ∴BC − A > 0. (A.16) We have Φ is solved by applying the method of Lagrange multipliers as follows: Φ � 1. (A.17) k�1 zQ 1 � P + α i,k zΦ Φ k k i�1 N N K N K K N zQ 1 1 1 1 1 (A.18) � 0 ⇒ P � − α⇒ P � − Φ ⇒ P � − Φ ⇒ P � − 1⇒α � − i,k i,k k i,k k i,k zΦ Φ α α α N k k i�1 i�1 i�1 i�1 k�1 k�1 k�1 P i�1 i,k Φ � � . 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Journal of Healthcare Engineering – Hindawi Publishing Corporation
Published: Apr 14, 2022
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