J Med Syst (2014) 38:20 DOI 10.1007/s10916-014-0020-6

TRANSACTIONAL PROCESSING SYSTEMS

Classification of Normal and Diseased Liver Shapes based on Spherical Harmonics Coefficients Farshid Babapour Mofrad · Reza Aghaeizadeh Zoroofi · Ali Abbaspour Tehrani-Fard · Shahram Akhlaghpoor · Yoshinobu Sato

Received: 14 December 2013 / Accepted: 26 February 2014 / Published online: 24 April 2014 © Springer Science+Business Media New York 2014

Abstract Liver-shape analysis and quantification is still an open research subject. Quantitative assessment of the liver is of clinical importance in various procedures such as diagnosis, treatment planning, and monitoring. Liver-shape classification is of clinical importance for corresponding intra-subject and inter-subject studies. In this research, we propose a novel technique for the liver-shape classification based on Spherical Harmonics (SH) coefficients. The proposed liver-shape classification algorithm consists of the following steps: (a) Preprocessing, including mesh generation and simplification, point-set matching, and surface to template alignment; (b) Liver-shape parameterization, including surface normalization, SH expansion followed by parameter space registration; (c) Feature selection and This article is part of the Topical Collection on Transactional Processing Systems F. B. Mofrad () Faculty of Engineering, Science and Research Branch, Islamic Azad University (IAU), Tehran, 14515-775, Iran e-mail: [email protected] R. A. Zoroofi Control and Intelligent Processing Center of Excellence, School of Electrical and Computer Engineering, College of Engineering, University of Tehran, North Kargar Ave., Tehran 14395-515, Iran A. A. Tehrani-Fard Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran

classification, including frequency based feature selection, feature space reduction by Principal Component Analysis (PCA), and classification. The above multi-step approach is novel in the sense that registration and feature selection for liver-shape classification is proposed and implemented and validated for the normal and diseases liver in the SH domain. Various groups of SH features after applying conventional PCA and/or ordered by p-value PCA are employed in two classifiers including Support Vector Machine (SVM) and k-Nearest Neighbor (k-NN) in the presence of 101 liver data sets. Results show that the proposed specific features combined with classifiers outperform existing liver-shape classification techniques that employ liver surface information in the spatial domain. In the available data sets, the proposed method can successful classify normal and diseased livers with a correct classification rate of above 90 %. The performed result in average is higher than conventional liver-shape classification method. Several standard metrics such as Leave-one-out cross-validation and Receiver Operating Characteristic (ROC) analysis are employed in the experiments and confirm the effectiveness of the proposed liver-shape classification with respect to conventional techniques. Keywords Spherical Harmonics · SH-based shape approximation · Liver-shape · Shape classification · Frequency based feature selection

S. Akhlaghpoor Departments of Interventional Radiology, Sina Hospital, Tehran University of Medical Sciences, Tehran, Iran

Introduction

Y. Sato Department of Radiology, Osaka University, Graduate School of Medicine, 2-2-D11 Yamadaoka, Suita City, Osaka 565-0871, Japan

The liver is the largest organ in the abdomen and might be affected by various diseases. Recently, computer assisted liver-image analysis is getting more clinical attention day by day [1, 2]. In this regard, liver-shape analysis is of particular

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interest. One major step in liver-shape analysis is liver-shape classification. The liver is a non-rigid object. It is under the influence of various organic motions such as cardiac, aortic, respiratory and gastro-intestinal motions. The growth, disorder and disease may affect geometrical features of the liver. The goal of this research is to propose a feasible way in terms of quality and speed for liver-shape classification applicable for abdominal images acquired from multi-slice CT images. And comparison of the liver-shape classification based on SH coefficients as a feature vectors versus the control points in space domain. Spherical Harmonics (SH) [3] is a mathematical method for shape representation of 3D objects. This technique was widely employed in different researches for shape description, classification and recovery of medical volumetric data [4–12]. In another research [4], SH was utilized in five randomly selected datasets of Magnetic resonance (MR) brain tumors images and showed that 3D shapes of tumors could be classified by this technique. Registration and modeling with SH were among other widely used applications. In other quantitative procedures, the SH was applied in registration and modeling [5–7], evolution [7], segmentation [8], phantom development for internal dosimetry [9], MRI [10] and clinical shape reconstruction [11]. In this research, the features of liver surface shape were extracted by utilizing the SH components [13]. Moreover, unlike previous works such as [12], in our approach the PCA is applied to SH coefficients in a selected order of liver-shapes. The rest of this paper is organized as follows. We first explain the available datasets, the imaging procedures and a priori assumptions. We then generally describe the proposed techniques. Next, we give a brief mathematical description about the SH and explain how we employ the SH in the liver-shape discrimination by demonstrating typical examples on several data sets with various liver shapes in normal and diseased cases. We then explain about the preprocessing, feature extractions, classifications and validation procedures in details. Results of applying the proposed techniques are quantitatively reported and compared with a conventional method in the presence of 101 selected datasets. We also express the advantages and unique features of the current approach and clarify our achievements. The paper ends with concluding remarks and describing the remaining works and future challenges.

Materials and methods

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between 102 to 159. Only male datasets in this research were selected. The cases were classified into two classes, i.e. (a) normal and (b) diseased (liver cirrhosis/chronic hepatitis). Total number of liver datasets was 154 cases. Then, 101 male datasets including 52 normal controls and 49 diseased cases were selected in the research. Ages of the subjects were between 50–75 years. Each slice was manually segmented by a liver specialist. Typical images are shown in Fig. 1. Figure 2 illustrates typical images of 3D representation. Datasets associated with normal and diseased livers are shown in (a) and (b), respectively. To our knowledge, discrimination of normal and diseased livers from the liver shape was not adequately addressed in the previous works and there is no prominent visual clue that can be generalized for notifying all normal and/or diseased cases. The challenge in this research is to automatically classify the normal and diseased livers based on the features extracted from their surfaces data by Spherical Harmonics (a preliminary work). Proposed methods In this research, a multi-step approach including preprocessing, shape parametrization, feature selection and classification for discriminating normal and diseased livers based on their surface shapes is introduced. Flowchart of the proposed approach is shown in Fig. 3. As seen, the method starts with pre-processing. In this step, number of liver surface points are reduced and fixed by mesh simplification. Then one of the datasets as reference is selected. Next, we heuristically specify six landmarks in the reference liver shape and determine the corresponding points in the remaining datasets by previous works [14–16]. The method then align the reference and other datasets by a rigid registration technique [15, 16]. In the next step, a non-rigid registration between the reference and the rest of datasets is perform. In this case, the surface points of liver in the spatial domain are mapped into a unit sphere for 3D sorting and normalization. Next the SH coefficients for the normalized liver shape points is calculated. The method then employ a non-rigid registration approach that minimizes the distance between the SH coefficient of the reference and other liver datasets. In the next step, we employ the SH coefficients and propose applying feature vectors of different orders in the operations. The size of each feature space is then reduced by a PCA and re-ordered by p-value of a t-test. We finally employ two well known classifiers for discriminating the normal and diseased livers from each other.

Datasets Preprocessing The datasets were acquired by a 4-channel multi detector CT scanner (LightSpeed QX/I, GE Healthcare) at Osaka University Hospital. Number of the slices for datasets were

Mesh generation and simplification As explained in section “Proposed methods”, liver boundary in CT images was

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Fig. 1 Datasets. a and b are associated with the slices of two typical datasets of abdominal CT images corresponding to normal and diseased livers. Segmented liver tissues associated with (a) and (b) are respectively shown in (c) and (d)

manually determined by a clinical expert. The boundary points and a Delaunay mesh generation technique were employed [17] to generate and simplify liver surface points. In Delaunay triangulation approach, a surface is divided into a set of triangles so that each triangle shares its edges with other triangles. By Delaunay technique, experimentally a number of 2000 as surface points was selected for all liver datasets in this research. Initial and simplified meshes for a typical liver dataset is shown in Fig. 4. Automatic landmark detection In this step, six landmark points in a liver surface were selected as follows: By the available segmented liver boundaries in a liver dataset, we determine the axial, coronal and sagittal planes with maximum liver areas. The cross sections of these three orthogonal planes are associated with six surface points of the liver. These points are selected as landmarks of the reference liver. Then the method developed by Chui et al. [15] was employed to estimate these six reference points in other liver surfaces. In the next step, as we employ these points only for liver surface alignment not for non-rigid registration, finding the exact locations of these points is not critical. Liver surface alignment In this step, a rigid registration between the reference liver and other livers by using the

points estimated in section “Mesh generation and simplification” was performed. In this regard, the rotation and translation parameters of livers were estimated in the datasets with respect to the reference liver by a quaternion-based algorithm [16] as follows: 1 qi − Rpi − T 2 n n

f (R, T ) =

(1)

i=1

where P = {p1 , p2 , .., pn } and Q = {q1 , q2 , .., qn } denotes landmarks before and after applying point matching, respectively. And also R, T denotes rotation and translation, respectively. After this step all shapes go as near as possible to the model shape [8]. In section “Liver shape parameterization”, we employ the aligned livers and perform point based surface matching. Liver shape parameterization Surface parameterization A typical liver surface consists of lots control points called vertices. Sorting these control points in an organized manner is required in our approach. However, sorting 3D points in Cartesian domain is a difficult task. For this reason, in this step, a continuous and uniform mapping between the surface of a liver and a unit sphere [12] was generated. This is equivalent by solving a

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constrained optimization problem [18]. The result is assigning each point of the liver surface to a unique (θ, φ) on the surface of the unit sphere as follows: ⎛ ⎞ x(θ, φ) S(θ, φ) = ⎝ y(θ, φ) ⎠ (2) z(θ, φ) Spherical harmonics expansion Spherical Harmonics (SH) are the basis functions of degree l and order m which are defined as follows [3]:  2l + 1 (l − m)! m m Yl (θ, φ) = . P (cos θ).eimφ (3) 4π (l + m)! l Yl−m (θ, φ) = (−1)m Ylm∗ (θ, φ)

(4)

In above equation, Ylm ,−l ≤ m ≤ l is a spherical harmonic function of degree l and order m and Ylm∗ denotes the complex conjugate of Ylm . The term Plm is an associated Legendre polynomials function. θ is regarded as the polar (co-latitudinal) coordinate with θ ∈ [0, π], and φ as the azimuthal (longitudinal) coordinate with φ ∈ [0, 2π). Plm (ω) =

m l m+l  (−1)m  2 2 d 2 1 − ω ω − 1 2l l! dωm+l

(5)

To represent a surface by the SH, we determine the coordinate’s values of the surface, i.e., S(θ, φ) = (x(θ, φ), y(θ, φ), z(θ, φ))T as follows: S(θ, φ) =

∞  l 

clm Ylm (θ, φ)

(6)

l=0 m=−l

 T m , cm , cm The coefficients clm = cxl are usually comyl zl plex numbers. And they can be estimated by solving one set of linear equation in a least square. Each object surface can be representing by these coefficients. The approximated

Fig. 2 Datasets. a normal and b diseased liver shapes

Segmented datasets

Pre-processing

Liver shape parameterization

Feature selection and classification

Mesh generation and simplification

Surface parameterization (Normalization)

SH based feature selection

Automatic landmark detection

Spherical Harmonics expansion

Liver surface alignment

Parameter space registration

Classification

Liver shape-based classified

Fig. 3 Outline of the high level proposed classification algorithm for liver

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Fig. 4 Liver surface point generation by a Delaunay triangulation. Generated liver meshes before and after simplification are shown in (a) and (b), respectively

surface will be reconstructed with each degree of Lmax in the following formula: ∧

S (θ, φ) =

L max 

l 

clm Ylm (θ, φ)

(7)

√ distance MSD between SH coefficients based on Euler angles (α, β, γ ) was minimized: MSD =

l=0 m=−l

Lmax  l

m

1  m

c

mod,l − ct em,l 4π

(8)

l=0 m=−l

A larger Lmax (higher degree) is associated with a more accurate approximated surface. Figure 5 illustrates the liver after approximation by the SH for different values of Lmax . In this research, based on our previous work [8], a value of Lmax = 15 in the experiments was selected.

m where cmod,l , ctmem,l are coefficients in each (m, l) of model and template, respectively. Given coefficients cln before rotation, the method can calculate the coefficients clm (α, β, γ ) after rotation with Euler angles (α, β, γ ) [19] as follows:

Parameter space registration Although the liver surface points are sorted by the technique employed in section “Mesh generation and simplification”, the correspondence between surface points in different livers are not yet achieved. First the SH coefficients of the reference and a selected liver in the dataset were calculated. then a cost optimization procedure was employed so that the difference between the SH coefficients of the reference and selected liver is minimized. This is equivalent to displacement of the control points on the surface of the selected liver so that the distance between corresponding liver points in two surfaces are globally minimized. The details of the implemented technique [6, 7] is as follows. Root mean square

clm (α, β, γ ) =

l 

l Dmn (α, β, γ ) cln

(9)

n=−l

where l l Dmn (α, β, γ ) = e−iγ n .dmn (β).e−iαm

l dmn (β) =

(10)

min(l+n,l−m)

t t =max(0,n−m) (−1)



(l + n)!(l − n)!(l + m)!(l − m)! (l + n − t)!(l − m − t)!(t + m − n)!t! β (2l+n−m−2t ) β (2l+m−n) × cos sin 2 2

×

(11)

By the above equations and minimizing the metric defined in Eq. 8, the surface control points of the reference and selected liver are efficiently registered. Hence, resultant surface points of livers in different cases can be employed for feature extraction and classification in the next steps. (L max =1)

(L max =10)

( L max =5)

( L max =15)

Fig. 5 A typical example of liver surface reconstructions using SH coefficients for Lmax = 1, 5, 10 and 15. See section “Spherical harmonics expansion” for details

Feature selection and classification SH based feature selection In this research, SH coefficients for feature extraction of liver surface points were employed. In this case, different orders of Spherical Harmonics coefficients were utilized. The first 16 spherical harmonic coefficients (real and imaginary pairs) for X, Y and Z coordinates are typically shown in Fig. 6. Referring to Fig. 5, we can see that lower orders are associated with low frequency components of a liver surface. Feature extraction based on SH coefficients was investigated in the previous

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Fig. 6 Typical real and imaginary pairs of the first 16 SH coefficients for X, Y and Z coordinates, are respectively shown in (a) and (b)

works [12]. To our knowledge, surface feature extraction for different orders of SH coefficients was not studied for the liver. Although the impact of selecting a special order as feature set in liver-shape classification is not available. Moreover, feature extraction for liver surface control points was previously performed only in the spatial domain [2]. In our approach, however, we proposed a novel feature extraction and concrete framework for liver shape based classification that was not available in the previous works. We regard such an approach as conventional technique for liver-shape classification. This approach is denoted as a in the following lines. For SH based feature extraction, seven features groups denoted as b to h were proposed as follows: a. b. c. d. e. f. g. h.

Surface control points (CP) in the spatial domain. Order-1: (18-coefficients) Order-2: (30-coefficients) Order-3: (42-coefficients) Order- 0, 1: (24-coefficients) Order- 0, 1, 2: (54-coefficients) Order- 0, 1, 2, 3: (96-coefficients) Order- 0∼15: (1536-coefficients)

As an example, referring to Eq. 6 for Order-2, there are five complex coefficients, i.e., −2, −1, 0, 1, and 2 for each of X, Y and Z components. Hence, they are associated with 30 coefficients including 15 real and imaginary pairs. These features are employed as feature set of c in the operations. Typical SH coefficients for X,Y, and Z corresponding to feature set g is shown in Fig. 6. In the same manner as of conventional approach [2, 12], a Principal Component Analysis (PCA) (www. mathworks.com) [20] was employed to reduce the dimension of the SH feature sets in the operations.

Classification In this section, two known classifiers including k-nearest neighbor (k-NN) [20, 21] and Support Vector Machine (SVM) [22, 23] were employed in the operations. To our knowledge, liver-shape calcification based on the proposed SH feature sets (b to h) and the above classifiers was not proposed in the previous works. As noted earlier and shown in Fig. 2, in common clinical practice, a liver specialist reviews the liver images slice by slice and diseased liver classification is performed by a volume intensities investigation. The healthy and diseased livers of this study were classified in this conventional manner as well. In spite of the difficulty of liver shape classification for human eye, we follow a supervised approach to automatically differentiate healthy and diseased livers. our results show that this approach is effective and novel with respect to previous works. visual classification of livers corresponding to normal and diseased livers based on the livers surfaces shapes is very difficult. In this section, we attempt to automatically discriminate normal and diseased livers by employing the above feature sets and classifiers. We assess the feature sets in the spatial domain (denoted as a) and SH domain (denoted as b to h) and clarify the behavior of the proposed supervised learning scheme for different orders of SH coefficients corresponding to liver surface in classification procedures. To perform a supervised classification, we need to select a subset of the most useful features [12]. As noted earlier, the PCA to effectively reduce dimension of the feature space was applied in the operations. In addition, the method follow a previous approach and utilize a two-sample t-test [24] and calculates a p-value to rank the effective features as follows: mx −my T= 2 sx /Nx +s2y /Ny

(12)

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X: 3 Y: 0.6364

0.7 0.6

Correct Rate

Fig. 7 Typical results of liver-shape classification: a Feature set d employed in k-NN classifier and applied to normal datasets. As seen, the best classification result is associated with utilizing 3 features in the classifier. b Feature set e employed in SVM classifier and applied to diseased datasets. In this case, the best result with minimum features performed when we employed 6 features

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0.5 0.4 0.3 0.2 0.1 0

2

4

6

8

10

12

14

16

18

20

22

16

18

20

22

Number of Features

a

0.7

Correct Rate

0.6 0.5 0.4 0.3 0.2 0.1 0

2

4

6

8

10

12

14

Number of Features

b

Results As explained in section “Datasets”, out of 150 available liver datasets with a range of 102 to 159 slices per cases, 52 normal and 49 diseased liver male datasets were selected and leave-one-out cross-validation was employed in the experiments. In this case, 100 datasets1 were employed as 1 ((49

diseased + 52 normal) − 1)

training sets and the remaining dataset for test in classification procedures. As explained in section “Classification”, four classifiers including k-NN, SVM, k-NNp and SVMp

Mean correct classification rate(%)

where mx and my are sample means, sx and sy are sample standard deviations, and Nx and Ny are sample sizes. The pvalue is the probability, under the null hypothesis. A lower p-value is associated with more significant feature [12]. In this research, the features optimized by PCA and pvalue were employed in the k-NN and SVM classifiers. That is, four classification schemes are used in the operations. For better understanding, when p-value ordering applied, classifiers are denoted as k-NNp and SVMp, respectively.

85

80

75

70

65

60

a

b

c

d

e

f

g

h

Feature sets

Fig. 8 Results of liver-shape classification in the presence of different classifiers and different feature sets

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Table 1 Comparison between conventional (feature set a) and proposed method (feature set g:referring to Fig. 8) for all liver datasets

1 0.9

Conventional method

Proposed method

60.40 %{5}∗ a 81.19 %{19}a 53.47 %{28}a 82.18 %{19}a

77.23 %{17}g 90.09 %{14}g 76.23 %{26}g 92.08 %{8}g

0.8

As seen, the proposed feature set outperform the conventional technique * The number of features

Sensitivity

0.7

k-NN SVM k-NNp SVMp

0.6 0.5 0.4 0.3 0.2

K−NN(Traditional Method) SVM (Traditional Method)

0.1

K−NN(Proposed Method) SVMp(Proposed Method)

p

0

were utilized for evaluation. Also the performance of the feature sets of a to h were evaluated in the experiments. Correct Classification rate was performed by the following scheme: CCR=

Nc Nt

where Nc is the times that input classified correct and Nt is the total number of tests. As an example, for the feature set d, the mentioned value for k-NN classifier was 63.64 % in the presence of normal datasets. This value was obtained based on the maximum point of the curve drawn in Fig. 7a. In this case, the number of features for feature set d was changed from 2 to 100 features and the best value was found when we employed 3 features(in different cases the optimum number of features was between 2 to 100 features). Another typical result associated with using feature set e for SVM classifier and diseased datasets is shown in Fig. 7b. In this case, the maximum value was associated with 50 % arising from 6 features. Mean of best classification results of feature sets of a to h in the presence of four classifiers for all (normal + diseased) patients are shown in Fig. 8. In addition, the highest achievement of each classification technique for conventional technique, i.e., employing feature set a and proposed SH based technique, i.e. employing the best feature set g (according to Fig. 8) is summarized in Table 1. As seen, best result of proposed feature sets is better than conventional approach for the same classifier. This evidence clearly confirms that the proposed technique based on SH outperforms the conventional technique.

0

0.2

0.4

0.6

0.8

1

1−Specificity Fig. 9 Typical ROC analysis on all datasets for evaluation of the proposed technique with respect to conventional technique in the presence of two known classifiers. As seen, the proposed technique in both cases is of higher performance

In another evaluation, the quantitative impact of using the proposed features with respect to conventional scheme was mentioned. The results is shown in Table 2. As seen, the proposed technique improves the classification result in all cases. As highlighted in Table 2, the proposed SH based classification technique is 18.22 % better than conventional technique in average sense. Figure 9 is associated with typical ROC analysis [25, 26] of all liver datasets for the proposed and conventional techniques in the presence of two classifiers. In this case, a leave-one-out cross-validation and two classifiers, i.e., the k-NN and SVMp for evaluation were employed. As seen in both classifiers, the proposed technique outperform the conventional technique. As shown in Table 1, best classification results of the proposed method was achieved by employing the feature set g and the SVMp classifier. Refering to liver datasets in Fig. 2a and b it is clear that visual classification of normal and diseased livers is very difficult. The proposed technique in spite of the failed cases might have a positive impact in encouraging the clinician to seriously consider image assisted analysis in their future clinical practices. The aim of this paper is to classify normal and diseased livers automatically, the method uses manually segmented liver region as an input. When using the method in the clinical environment, the input of the method will be liver region segmented automatically or semi-automatically.

Table 2 Increase percentage and average of increase percentage for each classifier and each dataset in the proposed method

Conclusion All

k-NN

SVM

k-NNp

SVMp

Average

20.48 %

10.84 %

29.85 %

12.05 %

18.22 %

In this study, a total solution for classification of liver CT datasets based on liver shapes was proposed. The proposed

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techniques consisted of several steps including preprocessing, liver shape parameterization, feature extraction and classification. By preprocessing and liver shape parameterization, we normalized and registered livers of different subjects with respect to each other. In this case, correspondences among liver surface points in different datasets were achieved. For feature extraction of liver surface, a technique by performing different orders of the Spherical Harmonics coefficients was proposed. The proposed and conventional features, i.e., liver surface points in the presence of different classes of normal, diseased, and all liver datasets were evaluated by employing four classifiers. The impact of proposed features and employed classifiers for different datasets were reported and discussed in details. As shown, the proposed SH based feature extraction technique outperformed the conventional technique in all cases. As the future step of this research, the plan is to extend the proposed shape based classification to discriminate various classes of diseased livers and to see how these correspond to biological changes in the liver i.e. what is the cause of the abnormal shapes corresponding to these features. Applying the proposed techniques to other 3D human organs for shape classification is another future challenge.

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Classification of normal and diseased liver shapes based on Spherical Harmonics coefficients.

Liver-shape analysis and quantification is still an open research subject. Quantitative assessment of the liver is of clinical importance in various p...
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