axons from almost every sensory 
 system connect here as the last site before the information reaches the cerebral cortex . information received from the diverse brain regions is passed on to the cortex through 
 the thalamus . anatomically , thalamus is the largest , most internal structures of 
 the diencephalon consisting of dual lobe masses of gray matter . it is located 
 at the rostral end of the mid brain on each side of the third ventricle . motor nuclei of the thalamus receive signals from 
 the striatum , cerebellum , project into the motor , and premotor areas 
 of the cerebral cortex . the thalamus play a major role in the regulation of 
 consciousness , alertness , arousal , and attention and is thus considered part of the limbic 
 system . thalamus and thalamus nuclei segmentation have become more and more 
 essential for a wide range of clinical and research applications . for example , 
 thalamus changes in terms of volume and intensity are involved in a large number 
 of diseases , such as schizophrenia , parkinson s disease , and multiple sclerosis . conventional imaging modalities such as computerized tomography ( ct ) or 
 magnetic resonance imaging ( mri ) however , do not provide the necessary image 
 contrast to differentiate the individual thalamic nuclei . on the other hand , a new noninvasive imaging modality diffusion tensor magnetic resonance imaging 
 ( dt - mri ) can relate the image intensities to the relative mobility of tissue 
 water molecules . in dt - mri , a tensor describing local water diffusion is 
 calculated for each voxel from measurements of diffusion in several directions . since water diffusion along neural fiber tracts of the brain is highly anisotropic , 
 dt - mri had been used to study the brain connectivity by extracting the fiber 
 tracts from the brain white matter . most recently , researchers have started to 
 use dt - mri for segmentation purposes . wiegell   et al . 
  among of the first to segment thalamic nuclei directly from the dt - mri data by using a k - means algorithm . behrens et al .   proposed an algorithm to identify the thalamic nuclei by mapping the connections between the thalamus and the cortex . jonasson et al .   presented a method for segmenting the thalamus and its subnuclei by propagating a set of coupled level sets through a region - based force defined from the similarity measure between the most representative tensor of each level sets and its neighboring voxels . in this paper , we propose a semi - automatic thalamus and thalamus nuclei segmentation algorithm from diffusion tensor magnetic resonance imaging ( dt - mri ) based on the mean - shift algorithm . comparing with existing thalamus segmentation algorithms which are based on k - means algorithm   or use k - means as an initialization , our mean - shift - based algorithm is more flexible and adaptive . furthermore , the single parameter in the mean - shift - based algorithm supports hierarchical clustering naturally . we will briefly review the background on diffusion tensor magnetic resonance imaging and mean shift 
 clustering in section 2 . the main algorithm for thalamus and thalamus nuclei segmentation will be described in section 3 . diffusion tensor magnetic resonance imaging ( dt - mri ) is a recent mr imaging modality . in diffusion tensor mri , a tensor describing local water diffusion the geometric nature of the diffusion tensors can quantitatively characterize the local structure in tissues such as bone , muscles , and white matter of the brain . a good review on dt - mri can be found in [ 1 , 6 ] . in general , the symmetric 3 by 3 diffusion tensor matrix d has 
 six degrees of freedom ( number of independent coefficients in a matrix representation ) . 
 to estimate the tensor , then , at least six measurements ( taken from different 
 noncollinear gradient directions ) are needed , in addition to the baseline image data s0 . 
 thus for each slice in the data set , seven images need to be collected 
 with different diffusion weightings and gradient directions . let s0 represents the signal intensity in the absence of a diffusion - sensitizing field gradient and sk the signal intensity in 
 the presence of gradient gk=(gkx , gky , gkz),k=1,  ,6 . the equation for the loss in signal intensity due to diffusion is given by the 
 stejskal - tanner formula : ( 1)ln(sk)=ln(s0)22(3)gktdgk , 
 where  is the gyromagnetic 
 ratio of hydrogen h ( protons ) ,  is the duration of the diffusion sensitizing gradient pulses , and  is the time between the centers of the two pulses . the tensor d can 
 then be computed by solving this system of six equations ( 1 ) . instead of assuming a fixed number of clusters as is common with other clustering methods ( e.g. k - means ) , mean shift extracts the modes of the 
 density function . we will review briefly the mean shift algorithm in the following . 
 for a complete description of mean shift given an arbitrary set of n points =x1,  ,xn in the d - dimensional 
 euclidean space rd . the multivariate kernel density estimate obtained with kernel k(x ) and window 
 radius h , computed 
 in the point x , 
 is defined as ( 2)f^(x)=1nhdi=1nk(xxih ) , 
 where k(x ) is the spherically symmetric kernel function satisfying ( 3)k(x)0 ,  rdk(x)dx=1 , 
 and h is 
 a smoothing parameter called the bandwidth . we can further define a profile function k(x ) for the kernel 
 function k(x ) of ( 2 ) such that 
 ( 4)k(x)=ck , d  k(x2 ) , 
 where ck , d is the normalized constant . the density estimator of ( 2 ) can then be rewritten as 
 ( 5)f^(x)=ck , dnhdi=1n  k(xxih2 ) . assume now that we are interested in subdividing scattered data  into a set of clusters . it is natural to consider the points where f^ defined by ( 5 ) 
 have local maxima as centers of the clusters . the simplest method to 
 find the local maxima of the f^ is to compute 
 the gradient of f^ and use a hill - climbing process to map each input 
 point to its local maxima ( i.e. , mode ) defined 
 by f^. 
 these resulting modes can then be used to select cluster shapes using basins of 
 attraction , and can have very nontrivial shapes unlike k - means clustering where 
 points are simply assigned to the nearest cluster 
 center . the single bandwidth parameter h allows 
 the number of clusters to be chosen in terms of a length scale in the input point 
 space . from ( 5 ) we can compute the gradient of f^ : ( 6)f^(x)=2ck , dnhd+2i=1ng(xxih2)m(x ) , 
 where g(x)=k(x ) . m(x ) is the 
 mean shift vector and is given by ( 7)m(x)=i=1n  xig((xxi)/h2)i=1n  g((xxi)/h2)x , 
 for example , the difference between the weighted mean , using the kernel g for weights , 
 and x , 
 the center of the kernel ( window ) . the general mean shift clustering procedure consists of 
 the following two steps : 
 initialize : y0=x;update by hill climbing : yj+1=yj+m(yj )   until convergence . in this section , we will describe our framework for thalamus and thalamus 
 nuclei segmentation from dt - mri data based on the previously described 
 mean - shift algorithm . there are two different domains of similarity : 
 spatial and tensor in the dt - mri image data , for example , each point xi in the 
 data set  of the joint spatial - tensor domain has two components of different nature , xi=(xis , xir ) , where xis is the spatial 
 component , and xir is the tensor component . the mean shift algorithm can be employed by extending the 
 density estimator of ( 5 ) with the following separable kernels : ( 8)f^(x)=ci=1nks(xsxishs2)kr(xrxirhr2 ) , 
 here ks is the kernel profile in the spatial domain with bandwidth parameter hs , kr is the kernel profile in the tensor domain with bandwidth parameter hr , and c is the 
 corresponding normalization constant . as suggested by , an epanechnikov kernel with 
 profile ke(x ) , ( 9)ke(x)={1-xif 0x1,0otherwise , 
 or a ( truncated ) normal kernel with profile kn(x)=exp((-1/2)x),x>0 , always 
 provides satisfactory performance , so the user only has to set the bandwidth parameter h=(hs , hr ) , 
 which , by controlling the size of the kernel , determines the resolution of the mode 
 selection . for dt - mri data , we have tried both kernels and found no significant 
 difference . in our experiment the distance metric in the spatial domain is the euclidian distance 
 metric , the distance metric in the tensor domain is the following frobenius 
 norm : 
 ( 10)x1rx2r=trace ( 
 ( dx1dx2 ) 
 ( dx1dx2)t ) , 
 dx1,dx2 are the tensor 
 matrices at points x1 ,  x2 , 
 respectively . to validate our method , we have applied our method on some synthetic data . for 
 comparison purpose a slice of the synthetic 
 tensor field is shown in figure 6 . in figure 7 , the regions have been segmented on the 
 synthetic field without any noise added . 
 in figure 8 , noise is added into the same 
 synthetic field at signal - to - noise - ratio ( snr ) level 32 . 
 in both of these two examples , our 
 algorithm obtained much better results than both the k - means algorithm and the 
 method of . in this section , we will show some of the experimental results of thalamus 
 and thalamus nuclei segmentation from dt - mri data . the initial thalamus segmentation is conducted 
 interactively , for example , after applying the mean shift algorithm with a bigger 
 bandwidth parameter , the user needs to identify the pair of symmetric clusters ( left 
 thalamus and right thalamus , marked by circles in figure 1(a ) for illustration ) 
 from other clusters . since the thalamus is bounded by relatively homogenous 
 structures such as the fiber tract and cerebrospinal fluid ( csf ) , this step can be 
 done quite easily ( figure 1(b ) ) . 
 in this example , the bandwidth parameter is set as h=(hs , hr)=(7,13 ) . the initially segmented thalamus will then serve as the mask for the subsequent thalamus 
 nuclei segmentation , which will be conducted automatically with a smaller bandwidth 
 parameter . the parameter chosen will determine the scale of features detected , so 
 different values may be desired based on the data set quality , features of interest , and so forth . figure 2 shows the thalamus nuclei segmentation results of the left thalamus . to illustrate 
 the hierarchical nature of the mean - shift - based algorithm , we fixed the spatial bandwidth hs as 7 , and gradually reduced 
 the tensor bandwidth hr from 13 ( figure 2(a ) ) to 11 ( figure 2(b ) ) , 10.5 ( figure 2(c ) ) , 10 ( figure 2(d ) ) , and 8 
 ( figure 2(e ) ) . by gradually reducing the bandwidth , more and more detailed nuclei 
 structures can be seen . we conducted similar hierarchical segmentation for the right thalamus 
 as well and is shown in figure 3 . we again fixed the spatial bandwidth hs as 7 , and gradually reduced 
 the tensor bandwidth hr from 13 ( figure 3(a ) ) to 11 ( figure 3(b ) ) , and 7 ( figure 3(c ) ) . however , 
 quite different settings of the bandwidth parameter need to be chosen h=(hs , hr)=(1,1.5 ) to 
 obtain the desired final thalamus nuclei segmentation ( figure 3(d ) ) . we do not know the 
 exact reason of this ( the different parameter setting for the left and right thalamus ) , one 
 of the possible reasons might be the artifact during the image acquisition 
 process ( e.g. , the image slice is not totally orthogonal across the thalamus ) , 
 or the nuclei structures between the left and the right 
 thalamus are not symmetrical indeed . more experiments and 
 research are certainly needed in the future to answer these questions . nonetheless , if 
 we compare figure 2(e ) and figure 3(d ) with figure 4 , which is the histological atlas of the 
 human thalamus with nuclei outlined by black lines , we can see they are very close . finally , a 3d rendering of the thalamus nuclei segmentation result is shown in 
 figure 5 . the main contribution of the paper is the application of the powerful mean shift clustering algorithm for thalamus segmentation from the dt - mri data . comparing with 
 existing thalamus segmentation algorithms ( [ 2 , 4 ] ) which are based on k - means 
 algorithm , our mean - shift - based algorithm has several potential advantages . ( 1 ) since 
 the mean shift algorithm is based on nonparametric density estimation , it does not 
 assume the data is always gaussian , hence it is more generic and flexible . ( 2 ) 
 unlike k - means algorithm , the mean - shift algorithm does not assume a fixed 
 number of clusters , hence it is more adaptive to the diversity of the dataset . ( 3 ) 
 there is only one parameter in the mean - shift - based algorithm , the bandwidth 
 parameter , which controls the scale of the features detected . ( 4 ) and by setting 
 the bandwidth parameter from large to small , mean shift naturally supports 
 hierarchical clustering , as shown in this paper on thalamus and thalamus nuclei 
 segmentation . there are two main directions to further improve the thalamus segmentation results : 
 ( 1 ) currently , the thalamus segmentation is conducted semi - automatically , that is , 
 the user has to pick the pair of distinct clusters ( left and right parts of the 
 thalamus ) from other neighboring clusters such as the fiber tracts and csf . although this is quite easy to do , it would be even better if the thalamus can be 
 automatically segmented ; moreover , a postprocessing active - contour - based diffusion 
 step ( as is done in ) might be able to further smooth the nuclei boundary 
 obtained from the clustering algorithm . ( 2 ) we would like to work closely with 
 domain specialists such as neurobiologists to verify and validate the segmentation 
 results , and to identify the thalamus nuclei structures . the collaboration with 
 domain specialists will also help us to choose the best bandwidth parameter for 
 the mean shift algorithm to create clinically most meaningful segmentation 
 results .
we propose a semi - automatic thalamus and thalamus nuclei segmentation algorithm from diffusion tensor magnetic resonance imaging ( dt - mri ) based on the mean - shift algorithm . 
 comparing with existing thalamus segmentation algorithms which are mainly based on k - means algorithm , our mean - shift based algorithm is more 
 flexible and adaptive . it does not assume a gaussian distribution or a fixed number of clusters 
 . 
 furthermore , the single parameter in the mean - shift based algorithm supports hierarchical clustering naturally .