1 Introduction
In some of the neurological diseases one or more brain components and substructures are affected. Some of these effects deform the shape of those components in one of the brain hemisphere or both. Therefore, unilateral and/or bilateral shape analysis of brain structures could lead to a better understanding of the abnormalities. There are evidences in the literature RefJ ; Ref2 ; Ref3 ; Ref4 showing that the brain structures deform when some diseases such as schizophrenia, Alzheimer’s, Parkinson, and epilepsy occur. So using precise representation is helpful to parameterize the deformation that will useful to diagnosis of the mentioned illnesses automatically. Hippocampus is a brain structure that belongs to the limbic system and is located in the medial temporal lobe. Hippocampus plays an important role in the formation of declarative, emotional, and longterm memories as well as language processing Ref5 . Hippocampus is one of the main targets of deformation in the brain in some disorders such as temporal lobe epilepsy. So, 3D representation and analysis of this structure could be useful in prognosis and diagnosis of those types of diseases. To characterize the deformation of the brain structures volumetric Ref6 or shape analysis is used. Volumetric analysis is easy to interpret deformation but shape analysis is more common because of its ability in geometric and morphometric features description. There are two groups of approaches used for shape representation; explicit and implicit shape representation. In explicit representation, the shape is illustrated as a parametric form and in explicit representation it represents as the level set of a scalar function. There are some examples of shape analysis methods in the literature in which either implicit or explicit shape representation is used. Statistical analysis of the anatomical shape deformations, which occur in epilepsy and other neurological disorders, require both global and local parameter based characterization of the anatomical shape that is deformed. Size and volume based analysis is the most popular method to achieve the parameterization of the shape deformationsRef8 ; Ref9 . Many methods are proposed to analysis hippocampus are needed complicated preprocessing task. Most of them are reviewed inRef8 ; Ref9 ; Ref10 ; Ref11 . Since many of the shapebased features can illustrated hippocampus deformation and employ for detection. This features are straightforward and avoided complicated pre processing steps. This method utilized spherical harmonics (SPHARM) that is a powerful mathematically tool that used for representation and analysis of 3D closed surfaces Ref15 . SPHARM can be utilized in 3D representation and analysis of the brain structures. However, in contrast to previous studies on hippocampus shape analysis using SPHARMBased shape descriptor, this work has been focused on the detection of asymmetry in hippocampus shape using SPHARM coefficients as the shape features. A sub set of the coefficients that were able to detect the asymmetry in left and right hippocampus was used to classify the images into either normal or epilepsy patient groups. Moreover, the complexity of this method is less compared to the other presented methods. As a result, in this paper, Combination of shapebased features and spherical harmonics, this proposed method can analysis hippocampus shape and diagnosis temporal lobe epilepsy in MR images.
2 Prevoius Works
Shen et al. Ref7 have reported about deformation of the hippocampus in epilepsy patients. The deformation of hippocampus has led the research groups to study the shape of the hippocampus for diagnosis purposes. Gerig et al. Ref8 to analysis hippocampal shape use volume measurments and shapebased features. Shapebased feature that is used is based on Mean Square Distance (MSD) between left and right hippocampus surface shapes. Then Support Vector Machines (SVM) is employed to classify, also leave oneout cross validation is applied to evaluation performance. Csernansky et al. Ref9 used registration to compare the hippocampal volume and shape characteristics in patient and control subjects. Computing transformation vector from the points on the hippocampus surface represents shape. Kodipaka et al. Ref10 used the Kernel Fisher Discriminant (KFD) algorithm for shapebased classification of hippocampal shapes. In this method, some landmarks manually placed on the hippocampus surface by an expert that indicated boundaries of the shape, then feature extraction is done by fitting a model to the landmarks using a deformable pedal surface, see detail about a deformable pedal surface in Ref13 . The result of this fitting process is a smooth surface that used in similarity alignment followed by a levelset nonrigid registration that describe in Ref14 . The output of this registration is local deformation between left and right hippocampus that used as input to the Kernel Fisher classifier. Esmaeilzadeh et al. employed spherical harmonics (SPHARM) to analysis hippocampus shape Ref11 . This method alignment hippocampus left and right to each other the extract features based on spherical harmonic confidents, finally using SVM for calssification. Shishegar et al. have used Laplace Beltrami operator for TLE diagnosis Ref12
. This method proposed a feature vector that described size and shape based on Laplace Beltrami eigenvalues and eigenfunctions.Some of pervious methods need registration as pre processing step
Ref6 ; Ref7 ; Ref8 . Also in Ref10 using shape alone could not capture shape differences. Furthermore the most of reviewed methods is complicated and need preprocessing step. In this paper, a new method is presented that using straightforward shapebased features for detection and omitted complicated preprocessing step such as registration.3 The Proposed Method
The purposed method has two parts for feature extraction. In each part using statistical ttest most significant features is selected. Then a main feature vector is formed. Finally SVM is employed for classification hippocampus shapes. Fig.1 illustrate block diagram of the purposed method.
3.1 Materials
Our dataset contained 25 T1weighted coronal brain MR images (MRI) from Henry Ford hospital. MRI data is very useful for imaging in different applications such as breast cancer Ref28 and sarcomas Ref29 . 20 of the images were acquired from temporal lobe epilepsy subjects with medically intractable seizures and the remaining five were from normal subjects used as control dataset. Ten images out of 25 (all from epilepsy patients) were acquired with 1.5 Tesla MRI scanners and the remaining ones (15 images from epilepsy patients and all the normal cases) were from 3.0 Tesla MRI scanners. Slice thickness of all the images was 2.0 mm and the 2015,inplane resolution of the pixels varied from to . For each image one expert provided a manual segmentation, then the segmentation labels were reviewed and confirmed by two more experts.
3.2 Feature extraction based on spherical harmonics
This part consists of pre processing steps that explained below.
3.3 Surface meshing
To generate a surface mesh for hippocampus shape, first, 3D manual segmentation was applied. For the further shape analysis and also for the classification of the hippocampus structure, it was necessary to have a mathematical representation of the 3D structure. One of the most common and simplest representations for explicit representation of the 3D surfaces is triangulation. In this method the shape is represented as a set of 3D discrete points that connected via triangular mesh. The marching cube algorithm Ref26 is a popular triangulation method applied to a 3D surface to obtain triangle surface mesh. After applying marching cube algorithm, the 3D shape is represented by coordinates of triangle mesh vertices as:
(1) 
Where n is the number of vertices.
3.4 Spherical parametrization
First step in SPHARM representation is mapping the 3D surface to unit sphere under bijective mapping with lower distortion in area and topology called spherical parameterization Ref25 . This mapping is applied to a 3D surface represented as a triangular mesh. After mapping the surface mesh on the unit sphere each vertices can be represented in spherical polar coordinate with two parameters, the inclination angle and the azimuth angle .
In this paper, for spherical parametrization we use, CALD algorithm that proposed by Shen and Makedon Ref26 . This algorithm consists of two steps: first, initial parametrization for triangular mesh Ref25 and second, local smoothing and global smoothing parametrization improve the quality of the parametrization. The goal of local smoothing step is minimization of the area distortion at a local sub mesh, this goal is achieved by solving a linear system and controlling its worst length distortion simultaneously. The global smoothing step as the second step of the algorithm compute the distribution of the surface distortions for all the mesh vertices, to equalize them over the complete sphere. In an overall view, the CALD algorithm merges the local and global methods together and executes each method alternately until a best parametrization is achieved. Fig.2 shows the left part of a hippocampus mapped to the unit sphere through this 3D mapping approach.
3.5 SPHARM analysis
The SPHARM is a powerful mathematically tool that used for representation and analysis of 3D closed surface. The overall SPHARM analysis includes three main steps:

Surface meshing

Surface parametrization

Spherical expansion
The first two steps have been explained in section II.C. When the 3D surface is represented via two parameters (the inclination angle and the azimuth angle ) the surface can be expanded into a complete set of SPHARM basis functions .
(2) 
where
and
and is the associated Legendre polynomials defined by the differential equation as follows:
(3) 
The surface is independently decomposed through SPHARM as:
(4) 
(5) 
(6) 
These terms can be combined into a single function:
(7) 
Where
(8) 
And
(9) 
The coefficients
are computed using leastsquares estimations. According to equation (7) these coefficients are determined up to userdesired maximum degree
.The original surface is indicated as and is an estimate for the coefficients which are Featureselection obtained by solving previous equation follow manner:
(10) 
As claimed by equation (7), after coefficients are estimated, the 3D surface can be reconstructed. Fig.3 illustrates reconstruct shape in four different degrees using higher degrees yield to more details in the reconstructed shape.
3.6 Featureselection
The real parts of SPHARM coefficients were used as features in this work. For each subject hippocampus has two parts; left and right. For each part the coefficients were computed up to degree 15. So the length of feature vector is for each side. To make the coefficients comparable across left and right hippocampus, we needed to normalize them by cancelling out the translational and rotational misalignments. Rotation was achieved by alignment of reconstructed shape based on first degree coefficients only (i.e. an ellipse). We used the alignment algorithm that used in Ref26 . For that purpose we aligned the shortest and the longest axis of the ellipse along xaxis and zaxis, respectively. We also translated the shapes to the origin of the coordinate system by ignoring the coefficients of degree 0. Since we used the ratio of left and right coefficients for each hippocampus, no scaling adjustment was required. After normalizing the coefficients, we calculated sum of the power of each coefficients for all the degrees as shown in equation (11).
(11) 
This step yielded to a feature vector of elements. Since our aim was to detect hippocampal asymmetry we calculated the ratio of logarithm of left coefficients to logarithm of right coefficients. We used logarithm to makes the features of different orders more comparable.(Fig.4)
Since only some of the listed features carried applicable information for the classification of shape (significantly differentiate between normal and abnormal shapes), we employed a statistical ttest to identify and select the most discriminative features. Fig.5 shows a schematic diagram of the feature selection algorithm. We run ttest on the each element (feature) of features vector to compare the element values of patients and normal subjects in a training set. For two sets of samples (in this paper one set is the feature values for normal cases and the second set is the feature values for the epilepsy cases) with averages of M1 and M2 the pvalue indicates the probability that the observed difference between M1 and M2 is due to chance under the null hypothesis that M1 and M2 are the same. Therefore, a lower pvalue shows statistically stronger difference that corresponds to a more significant feature. We obtained the associated pvalue for each feature and selected those features with pvalues of less than 0.05 as significantly discriminative features. Our hypothesis was that the significant features could help better in classification of hippocampus into healthy and epileptic classes. To select the best features that are able to differentiate between normal and epileptic subjects, first we randomly selected a subset of 12 images from our data set to form our training set. The training set contained both normal (two images) and epileptic (ten images) subjects. The remaining images were used as the test dataset in the classification explained in the next subsection. We extracted all SPHARM coefficients then created the feature vectors as described; we used ttest to select those features in which the value of the feature between normal and patient cases was detected significantly different (
).Feature  pvalue 

Maximum shape diameter  0.7187 
Shape volume  ** 
Surface area  0.7690 
CompactnessRef26  0.7054 
Mesh size  0.0015** 
Firstorder border moments Ref25 
** 
Thirdorder border momentsRef25  0.5088 
Circumscribed sphere volume to  0.6028 
shape volume ratio  
Curvature  0.075* 
3.7 Feature extarction based on shapebased features
In this section we first introduced all the features we used to study shape characteristics. Table 1 shows a list of shape and size features we have used in this project to detect asymmetry between right and left hippocampus. Fig.6 shows a general block diagram of our feature extraction method. Following we described these features.
3.7.1 Maximum shape diameter
The maximum possible Euclidian distance between two points of the surface of the shape. Since epilepsy disease could deform the surface of the hippocampus Ref25 it might changes maximum diameter of the shape.
3.7.2 Shape volume
To measure the shape volume of right and left hippocampus we multiply number of voxels by the volume of a single voxel. Surface area: To measure the surface area we calculated the summation of the outer surfaces of all the voxels that formed the hippocampus surface.
3.7.3 3D Compactness
Compactness is a unit less shape feature that describes how closely packed the shape is. In the other hand, it shows the roughness of a shape surface to its volume Ref26 . Sphere has the smallest value of 3D compactness i.e. about 113. As the sphere deforms towards a more complicated shape, compactness becomes larger and larger. 3D compactness (C) calculated as ratio of the cubed surface area (A) to the squared surface volume (V):
(12) 
3.7.4 Mesh Size
In this paper we measured the number of vertices in the shape surface mesh using marching cube meshing algorithm Ref26 . The marching cubes algorithm is a 3D isosurface representation technique for generating mesh for a 3D surface.
3.7.5 Moments
Moments are the statistical property of the shape that are evolved from the moment concept in physics. There are two types of moments, region moments and boundary moments. In this work, we use boundary moments Ref25 . The pth moment is defined as:
(13) 
where d(i) is Euclidian distance between ith voxels and the centroid, N is number of voxels. The pth central moment defined as:
(14) 
In this paper we use loworder moments such as M1 and M3 that are defined as below:
(15) 
and
(16) 
where
(17) 
3.7.6 Circumscribed sphere volume to shape volume ratio
This is the ratio of the volume of the circumscribed sphere centred at the shape centroid to the volume of the shape.
3.7.7 Curvature
We defined the curvature of the shape as follows:
(18) 
Where and are the Euclidian distances between centroid of the shape and two opposite tips of the shape that are located in the maximum distances to the centroid and : is the Euclidian distance between the two tips of the shape. To analysis the shape and size of the left and right hippocampus, we extracted all the features for left and right hippocampus, separately. Then we measured the ratio of right feature values to left feature values as well as left feature values to right. We chose the ratio that is greater or equal to 1 as an indicator for hippocampal symmetry/asymmetry. In this part feature selection is done according to statistical feature selection test that describe in Feature Selection part.
3.8 Classification
Support vector machine (SVM) is a supervised learning models used for classification and regression analysis with works very well on similar studies
Ref27 ; Ref28 ; Ref29. Linear SVMs search for the optimal hyperplane that separate the dataset to groups, i.e. the hyperplane that makes the maximal margin between groups. For more information about SVM classifiers you can see
Ref26 . We used SVM classifier to classify each test image to normal or patient categories using the selected features (Fig.7). We used our test dataset to evaluate the performance of our classification algorithm.3.9 Evaluation metrics
To evaluate our detection algorithm performance we calculated specificity, sensitivity and accuracy metrics (equations 19 to 21) for the classifier results. These metrics have been largely used in medical image analysis Ref25 ; Ref26 .
(19) 
(20) 
(21) 
TP , TN , FP and FN are, respectively, true positive, true negative, false positive and false negative.
3.10 Experimental results
We run our algorithm on 25 segmentation labels of hippocampus on T1weighted MRI. We divided our image dataset to two subsets; training and test. Training images (12 images, 2 normal and 10 epileptic) were used for feature selection. Test image set consisted of 13 images; 3 normal and 10 epileptic images. All the implementations of this work were implemented in MATLAB platform and run on a Mac operating system.
3.10.1 Features selection
Feature selection consists of two parts. First, feature extraction based on spherical harmonics coefficients. Second, feature extraction based on shapebased features. In each part statistical ttest is used for feature selection. In both parts, the most significant features (associated with pvalue 0.05) is selected. Then combination two selected features create feature vector.
3.10.2 Classification
Leave–one–out cross validation methodology has been applied to divide the test image set to one test image and 12 training for SVM classifier. Having the 18 selected features used in our SVM classifier, the specificity, sensitivity, and accuracy of our detection algorithm were 84%, 100% and 80%, respectively.
3.10.3 Conclusion and future work
Our preparatory results show that combination SPHARM coefficients and shapebased features could be helpful to describe the hippocampus shape deformation and could be used in diagnosis of the temporal lobe epilepsy disease in MRI. Since the dataset we used in our study is formed based on some part of the dataset used in Ref10 ; Ref11 , not the whole, it is challenging to reach a fair comparison between presented results and ours. As an advantage, the complexity of our algorithm is less than the other presented algorithms. In contrast to other work, our method detects the asymmetry in hippocampus structure shape for diagnosis of epilepsy and the preliminary results showed that hippocampal asymmetry detection could be useful for diagnosis of the brain abnormalities such as temporal lobe epilepsy. However, small size of the dataset is a limitation in our work. As future work we are going to increase the number of images in the training and test dataset. We will also compare the results to other reported results in the literature using appropriate statistical tests to examine the significance of the results. We would like to improve the results of our work by adding more shapebased features that are able to significantly separate patients from control groups. It could be also considered to use and evaluate this algorithm for detecting other brain abnormalities.
References
 (1) Csernansky, John G and Joshi, Sarang and Wang, Lei and Haller, John W and Gado, Mokhtar and Miller, J Philip and Grenander, Ulf and Miller, Michael I, Hippocampal morphometry in schizophrenia by high dimensional brain mapping, Proceedings of the National Academy of Sciences, 95, 11406–11411 (1998)
 (2) Oppenheim, Catherine and Dormont, Didier and Biondi, Alessandra and Lehéricy, Stéphane and Hasboun, Dominique and Clémenceau, Stéphane and Baulac, Michel and Marsault, Claude, Loss of digitations of the hippocampal head on highresolution fast spinecho MR: a sign of mesial temporal sclerosis.,American journal of neuroradiology,19,457–463,(1998)
 (3) Convit, A and De Leon, MJ and Tarshish, C and De Santi, S and Tsui, W and Rusinek, H and George, A, Specific hippocampal volume reductions in individuals at risk for Alzheimer’s disease, Neurobiology of aging, 18, 131–138, (1997)

(4)
Dam, Agnete Mouritzen, Epilepsy and neuron loss in the hippocampus, Epilepsia, 21, 617–629, (1980)
 (5) Duvernoy, Henri M, The human hippocampus: functional anatomy, vascularization and serial sections with MRI, Springer Science & Business Media (2005),
 (6) Anstey, KJ and Maller, JJ, The role of volumetric MRI in understanding mild cognitive impairment and similar classifications, Aging & mental health, 7, 238–250, (2003),
 (7) Spherical mapping for processing of 3D closed surfaces, Shen, Li and Makedon, Fillia, Image and vision computing, 24, 743–761, (2006)
 (8) Gerig, Guido and Styner, Martin and Shenton, Martha E and Lieberman, Jeffrey A, Shape versus size: Improved understanding of the morphology of brain structures, Medical Image Computing and ComputerAssisted Intervention–MICCAI 2001, 24–32, Springer (2001),
 (9) Csernansky, John G and Joshi, Sarang and Wang, Lei and Haller, John W and Gado, Mokhtar and Miller, J Philip and Grenander, Ulf and Miller, Michael I, Hippocampal morphometry in schizophrenia by high dimensional brain mapping, Proceedings of the National Academy of Sciences, 95, 11406–11411, (1998)
 (10) Kodipaka, Santhosh and Vemuri, Baba C and Rangarajan, Anand and Leonard, Christiana Morison and Schmallfuss, I and Eisenschenk, S, Kernel fisher discriminant for shapebased classification in epilepsy, Medical Image Analysis, 11, 79–90, (2007)
 (11) EsmaeilZadeh, Mohammad and SoltanianZadeh, Hamid and JafariKhouzani, Kourosh, SPHARMbased shape analysis of hippocampus for lateralization in mesial temporal lobe epilepsy, Electrical Engineering (ICEE), 2010 18th Iranian Conference on, 39–44, IEEE (2010),
 (12) Shishegar, Rosita and SoltanianZadeh, Hamid and Moghadasi, Seyed Reza, Hippocampal shape analysis in epilepsy using LaplaceBeltrami spectrum, Electrical Engineering (ICEE), 2011 19th Iranian Conference on, 1–5, IEEE (2011)
 (13) Vemuri, Baba C and Guo, Yanlin, Snake pedals: compact and versatile geometric models with physicsbased control, Pattern Analysis and Machine Intelligence, IEEE Transactions on, 22, 445–459, (2000),
 (14) Vemuri, BC and Ye, J and Chen, Y and Leonard, CM, A levelset based approach to image registration, Mathematical Methods in Biomedical Image Analysis, 2000. Proceedings. IEEE Workshop on, 86–93, IEEE (2000),
 (15) Gerig, Guido and Styner, Martin and Jones, D and Weinberger, Daniel and Lieberman, Jeffrey, Shape analysis of brain ventricles using spharm, Mathematical Methods in Biomedical Image Analysis, 2001. MMBIA 2001. IEEE Workshop on, 171–178, IEEE (2001),
 (16) Chaudhury, Baishali and Zhou, Mu and Farhidzadeh, Hamidreza and Goldgof, Dmitry B and Hall, Lawrence O and Gatenby, Robert A and Gillies, Robert J and Weinfurtner, Robert J and Drukteinis, Jennifer S, Predicting Ki67% expression from DCEMR images of breast tumors using textural kinetic features in tumor habitats, SPIE Medical Imaging, 9785, 97850T–97850T7, (2016),
 (17) Farhidzadeh, Hamidreza and Chaudhury, Baishali and Zhou, Mu and Goldgof, Dmitry B and Hall, Lawrence O and Gatenby, Robert A and Gillies, Robert J and Raghavan, Meera, Prediction of treatment outcome in soft tissue sarcoma based on radiologically defined habitats, SPIE Medical Imaging, 9414, 94141U–94141U5, (2015),
 (18) Lorensen, William E and Cline, Harvey E, Marching cubes: A high resolution 3D surface construction algorithm, ACM siggraph computer graphics, 21, 163–169, (1987),

(19)
Brechbühler, Ch and Gerig, Guido and Kübler, Olaf, Parametrization of closed surfaces for 3D shape description, Computer vision and image understanding, 61, 154–170, (1995),
 (20) Shen, Li and Makedon, Fillia, Spherical mapping for processing of 3D closed surfaces, Image and vision computing, 24, 743–761, (2006),
 (21) Andersen, Per and Morris, Richard and Amaral, David and Bliss, Tim and O’Keefe, John, The hippocampus book, Oxford University Press, USA (2006),

(22)
Bribiesca, Ernesto, An easy measure of compactness for 2D and 3D shapes, Pattern Recognition, 41, 543–554, (2008),
 (23) Shen, Liang and Rangayyan, Rangaraj M and Desautels, JE Leo, Application of shape analysis to mammographic calcifications, Medical Imaging, IEEE Transactions on, 13, 263–274, (1994),
 (24) Schölkopf, B and Akritas, Michael G and Politis, Dimitris N, An introduction to support vector machines, (2003)

(25)
Zhu, Wen and Zeng, Nancy and Wang, Ning and others, Sensitivity, specificity, accuracy, associated confidence interval and ROC analysis with practical SAS® implementations, NESUG proceedings: health care and life sciences, Baltimore, Maryland, 1–9, (2010)
 (26) Nascimento, Jacinto C and Marques, Jorge S, Adaptive snakes using the EM algorithm, Image Processing, IEEE Transactions on, 14, 1678–1686, (2005),
 (27) Farhidzadeh, Hamidreza and Goldgof, Dmitry B and Hall, Lawrence O and Gatenby, Robert A and Gillies, Robert J and Raghavan, Meera, Texture Feature Analysis to Predict Metastatic and Necrotic Soft Tissue Sarcomas, Systems, Man, and Cybernetics (SMC), 2015 IEEE International Conference on, 2798–2802, IEEE, (2015),
 (28) Farhidzadeh, Hamidreza and Kim, Joo Y and Scott, Jacob G and Goldgof, Dmitry B and Hall, Lawrence O and Harrison, Louis B, Classification of Progression Free Survival with Nasopharyngeal Carcinoma Tumors, SPIE Medical Imaging, 9785, 97851I–97851I7, (2016),
 (29) Farhidzadeh, Hamidreza and Zhou, Mu and Goldgof, Dmitry B and Hall, Lawrence O and Raghavan, Meera and Gatenby, Robert A, Prediction of treatment response and metastatic disease in soft tissue sarcoma, SPIE Medical Imaging, 9035, 903518–9035186, (2014),
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