A Locally Linear Least Squares Method for Simultaneously Smoothing

Journal of Medical and Biological Engineering, 33(3): 275-284

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A Locally Linear Least Squares Method for Simultaneously Smoothing DWI Data and Estimating Diffusion Tensors Xiaozheng Liu1,2,3,4

Wei Liu1,2,3

Yongdi Zhou1 1

Guang Yang2

Bradley S. Peterson3

Weidong Chen5

Junming Zhu6

Dongrong Xu3,*

Key Laboratory of Brain Functional Genomics, Ministry of Education and Shanghai Key Laboratory of Brain Functional Genomics, East China Normal University, Shanghai 20062, China 2 Shanghai Key Laboratory of Magnetic Resonance, East China Normal University, Shanghai 20062, China 3 MRI Unit, Department of Psychiatry, Columbia University and New York State Psychiatric Institute, New York 10032, USA 4 Center for Cognitive and Brain Disorders, Hangzhou Normal University, Hangzhou 310015, China 5 Qiushi Academy for Advanced Studies, Zhejiang University, Hangzhou 310027, China 6 Neurosurgery Department, The Second Affiliated Hospital, Zhejiang University, Hangzhou 310027, China Received 1 May 2012; Accepted 23 Nov 2012; doi: 10.5405/jmbe.1174

Abstract Magnetic resonance diffusion-weighted imaging (MR-DWI) data usually contain a great deal of noise and a significant number of outlier data points that can undermine the accurate estimation of diffusion tensors (DTs). Raw MR-DWI data therefore usually must undergo substantial preprocessing prior to tensor estimation. This study proposes an approach for the reconstruction of DT fields from MR-DWI data that combines into a single step the regularization of raw MR-DWI data and the optimized estimation of DT fields. The approach uses locally weighted linear least squares (LWLLS) estimation to correlate information within the local neighborhood of each voxel. It incorporates into the linear least squares (LLS) framework a bilateral filter which assigns different weights to neighbor voxels according to their intensities and relative distance. This method efficiently smoothes the MR-DWI data and estimates optimal tensors simultaneously. The performance of the proposed method was compared to that of traditional LLS estimation of tensors using both simulated and real-world human MR-DWI data. Both the simulated and real-world datasets demonstrated that the proposed method greatly outperforms the conventional LLS method and that the simultaneous smoothing of MR-DWI data and tensor estimation performs as well as the separate and sequential execution of these procedures. Keywords: Diffusion-weighted imaging data, Diffusion tensor, Locally weighted linear least squares

1. Introduction Diffusion tensor imaging (DTI) is a non-invasive in vivo imaging technique that uses magnetic resonance (MR) to infer the directional anisotropy of water diffusion in living tissues [1-3]. It is a powerful tool for the in vivo study of tissue microstructure, and particularly for studying white matter connectivity in the human brain. The anisotropic diffusivity of water molecules can be mathematically represented by a diffusion tensor (DT), a 3 × 3 symmetric matrix D that is positive definite. The eigenvector corresponding to the largest eigenvalue of the diffusion tensor at each voxel is presumed to represent the largest distance over which water diffuses, which is also regarded as the dominant axis of the underlying fiber bundles that traverse the voxel. The diffusion-weighted images * Corresponding author: Dong-Rong Xu Tel: +1-212-5435495; Fax: +1-212-5430522 E-mail: [email protected]

(DWIs) that constitute the raw DTI data from which tensors are ultimately estimated, however, contain substantial amounts of noise from multiple sources, including thermal and stochastic processes, motion artifacts, partial volume effects, and eddycurrent induced distortions [4,5]. Signal-to-noise ratios are usually well below 20 [4]. These noise sources and statistical outliers in the raw DTI data can seriously bias the estimation of tensor fields. Therefore, raw DWI data usually undergo extensive and careful preprocessing so as to reduce or eliminate these noise sources and their deleterious effects on subsequent tensor estimation. A number of approaches have been developed to estimate DTs accurately. Based on when and where in the processing protocol the noise in DWI data are addressed [4-7], the methods for reducing noise can be classified into three general categories: (1) DWI data regularized prior to tensor estimation [8-10] usually involves data smoothing, outlier removal, and distortion correction; (2) noise within DWI data that are assumed to follow a Rician distribution are usually addressed at

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the time of tensor estimation or when tensors, which must be positive definite, are examined [11,12]; (3) DWI that are already reconstructed into DT fields can undergo denoising, smoothing, and non-positive definite regularization [13-18]. Application of noise reduction techniques in the first and the third of these categories of data processing require a sequential combination of two unrelated techniques. Reducing noise in the second of these categories, at the time of tensor estimation, is much more efficient than the sequential application of multiple techniques because it involves a seamless coupling of two techniques, both noise reduction and tensor estimation, into a single technique. The present study develops this kind of combined application of techniques. The most prevalent method for tensor estimation is the linear least squares (LLS) approach [3]. The LLS method, however, has two major limitations: 1) it assumes that signal variability in DWI data is affected only by thermal noise, and it therefore does not account for signal perturbations that originate from other sources of system error, such as motion and distortion. The LLS method thus seriously underestimates the influences of outliers in the dataset. 2) It also does not address the problem of ill-conditioned tensors, because the LLS method includes in the estimation of tensors all outliers, no matter how large a bias they may cause in that estimation. This method therefore can also produce tensors that are non-positive definite, particularly in regions where DWI data are misaligned or that contain substantial partial-volume effects, such as at interfaces of different tissues. Many methods have been proposed to remedy these difficulties associated with the use of LLS in accurately estimating tensors from DWI data [18-23]. Considering a likelihood distribution of MRI data that is Rician, not Gaussian, Landman et al. [11] used a maximum Rician likelihood to estimate DTs. This method, however, requires the estimation of many parameters, such as noise variance and the initial value of a tensor for tensor estimation. It also uses an algorithm called simplex to determine the probability of maximum likelihood, and therefore it must iterate the estimation at each voxel many times to achieve an optimal solution, which renders this method excessively timeconsuming. Moreover, a small bias in the initial values of these parameters will greatly influence the final estimates. This method is therefore not practical for general use. Anderson et al. [12] proposed a maximum posteriori method with a Rician likelihood estimate for DTs. The maximum posteriori strategy has several advantages over the maximum likelihood method for tensor estimation. Methods based on probability estimation are highly complex, however, and they require extensive iteration and are thus computationally demanding, even though they cannot guarantee that tensors will be positive definite. The present study proposes an approach for the estimation of DT fields that uses a locally weighted LLS (LWLLS) method for smoothing raw DWI data using local information within the neighborhood simultaneously while estimating DTs. In a recent conference report [24], we extended the LWLLS model developed in this work for a DTI model to a Q-ball imaging model and applied it to process high-angularresolution diffusion imaging (HARDI) data and to estimate the

orientation distribution functions that are crucial for tractography in regions of crossing fibers in HARDI using a truncated series of spherical harmonics. The novelty of the proposed formulation is the combined application of these two complicated steps into one efficient computation. This method applies linear regression to the processing of noisy DWI data and therefore does not need to configure initial values for estimating tensors. Experiments using simulated and real-world human data are used to demonstrate that the proposed method outperforms the conventional LLS method in noise reduction and tensor estimation and that the simultaneous smoothing of DWI data and tensor estimation performs as well as the sequential execution of these two procedures.

2. Materials and methods 2.1 Log-linear least squares DWI signals Sk (k = 1,….n) from observation are related to a tensor through the Stejskal-Tanner equation [[1]]: T

S k ( X )  S0 ( X )e  bg k D ( X ) g k

(1)

where k denotes the k-th gradient direction; S0 is the baseline signal without the application of any gradient; b is a signal attenuation constant (known as the b-value) that depends on the strength of the applied gradient, the duration of its application, and the interval between the two polarized gradients or reverse radio-frequency pulses; gk is a unit vector of the k-th DW gradient direction; and D is the diffusion tensor, a symmetric positive definite 3 × 3 matrix:

 Dxx Dxy Dxz    D   Dxy D yy D yz  D D D   xz yz zz 

(2)

The number of DW directions and b-values are determined by experimental design and are always known at the time of data acquisition. When a set of Sk along at least six noncollinear (and non-coplanar) gradient directions and at least one baseline image are acquired, the tensor can be linearly estimated using Eq. (1) and a logarithmic algorithm (below). Bias is always present in the estimation because of the noise effect. The estimation essentially minimizes the error Euclidean norm between Sk and S0 ( X )e  bg D ( X ) g , as expressed by the following term: T k

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min  Sk ( X )  S0 ( X )ebgk D ( X ) gk D

k 1

T

k

2 2

(3)

where N is the number of gradient directions. Solving Eq. (3) for the minima yields: The diffusion tensor D can thus be estimated using Eq. (4) and a multivariate regression of log-linear least squares [[3],25]. However, the log-linear least squares (Eq. (3)) considers only the cases of one individual voxel and is easily affected by the outliers and noise in the dataset.

Local Regression for Diffusion Tensor Estimation

Dxx Dyy

 ( g x2 g y2 g z2 2 g x g y 2 g x g z 2 g y g z )1    Dzz           Dxy ln( sN / s0 )  ( g x2 g y2 g z2 2 g x g y 2 g x g z 2 g y g z ) N   D xz ln( s1 / s0 )

(4)

Dyz

2.2 Locally weighted linear least squares To incorporate a smoothing operator into the estimation, as is done in the preprocessing of raw DWI data, this calculation should include a dependency or correlation measure for the current voxel that depends on values in immediately adjacent voxels. Data in a small window of the immediate neighborhood is therefore needed to account for the local dependency of signals. Different weights should be used for voxels with different signal intensities and relative distances in this window. If M neighboring voxels are considered, a locally weighted least squares method can be developed. To simplify the expression, we consider a two-dimensional (2D) case in a 3 × 3 neighborhood, with M = 9: 1

N

min   wki  h , j  h ( X ) S ki  h , j  h ( X )  S0i , j ( X )e  bgk D ( X ) gk D

T

h 1 k 1

2 2

(5)

where wk is the weighting factor, i and j are the shoulder marks denoting the current voxel being processed, and h is used for navigating all the voxels in the neighborhood. Solving Eq. (5) for the minima on D using the least squares method, D can be calculated as follows: 1

1

1

D  ( B  B ) ( B  )Y T

T

(6)

Seeking the minima of Eq. (5) will yield an equation with the measures from the voxels in the neighborhood incorporated. Because the neighborhood window contains M voxels, Y is a MN × 1 vector, the elements of which are ln(Ski h, j h / S i, j ) and B is an MN × 6 matrix, whose rows are ( g x2 g y2 g z2 2 g x g y 2 g x g z 2 g y g z ) N . D is still a 6 × 1 vector, (Dxx, Dyy, Dzz, Dxy, Dxz, Dyz)T and 1 is a diagonal matrix with dimensions MN × MN (N is the number of gradient directions) whose elements are the weighing factors wki-h,j-h. Note that LWLLS (Eq. (5)) performs regression in the neighborhood based on weights that differ for differing voxels, whereas weights for the same voxel always remain the same, i.e., w1i h, j  h  w2i  h, j h    wNi h, j h . ln( s1( i 1, j 1) / S 0i , j )  ln( s N( i 1, j 1) / S 0i , j )

Y=

 ln( s1( i  1. j 1) / S 0i , j )  ln( s N( i  1, j  1) / S 0( i , j )

(7)

 ( g x2 g y2 g z2      ( g 2 g 2 g 2  x y z B     2 2 2 ( g x g y g z      ( g 2 g 2 g 2  x y z

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       2 g x g y 2 g x g z 2 g y g z ) (Ni , j )         ( i 1, j 1) 2 g x g y 2 g x g z 2 g y g z )1        2 g x g y 2 g x g z 2 g y g z ) (Ni 1, j 1)  2 g x g y 2 g x g z 2 g y g z )1(i , j )

(8)

Dxx(i , j ) Dyy(i , j )

D=

Dzz(i , j )

(9)

Dxy(i , j ) Dxz(i , j ) Dyz(i , j )

 w1i 1, j 1 0  0          0 wNi 1. j 1  0     1          w1i 1, j 1 0  0        i 1, j 1 0  wN  0 

(10)

Differentiating Eq. (5) on tensor D, it is found that Eq. (6) is a self-adaptive, local linear filter known as the NadarayaWatson estimator (NWE) [27,28]. This estimator has the following form: 1

S 0i , j exp(  BD ) 

w

h 1

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1

S ki  h , j  h

(11)

w

h 1

ih, j h k

Both Eqs. (6) and (11) are obtained by differentiating Eq. (5) on diffusion tensor D, but they are expressed in different forms. A comparison with Eq. (11) indicates that estimating tensors using Eq. (6) in one step is equal to performing the two tasks of smoothing the DWI data Si,j (the central voxel in the neighborhood) and estimating the tensors using LLS. The LWLLS uses one simple linear regression to smooth the DWI data and estimate tensors simultaneously. 2.3 Weighting function i, j

The weighting coefficients wk assigned to every voxel in the neighborhood window should have the following characteristics: (1) the DWI signals are weighted equally for the same voxels but differently for differing voxels; and (2) the difference in weightings should reflect the differing correlations of the given voxel in the neighborhood and the central voxel for which the local window is defined. A Gaussian kernel is the most commonly used weighting function for DTI processing, which provides a weighting structure based on the position of the neighbor voxels. The weightings are usually calculated using a single measure or value, such as the Euclidean distance from the central voxel. To

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take into account the neighborhood dependency and accommodate the requirement of incorporating both the regularization of imaging data and tensor estimation, a nonlinear bilateral filter originally developed by Tomasi and Manduchi is used [26]. This filter can simultaneously consider the Euclidean distance of both image intensity (in the DWI data) and the spatial locations of two voxels. The bilateral filer employed as a weighting function in Eqs. (6), (10), and (11) is defined as: wi , j  exp( 

d 2 (Vi , V j )



2 d

) exp( 

d 2(Xi, X j )

 r2

)

(12)

where d2(Vi,Vj) is the Euclidean distance between the physical locations of voxel Vi and voxel Vj; d2(Xi,Xj) is the Euclidean distance between log(Xi) and log(Xj); and Xi and Xj are N × 1 vectors of DWI data, where N is the number of gradient directions. σd is the geometric spread in the domain and σr is the photometric spread in the image. Increasing σr and σd will consider dependency in a wider neighborhood, and consequently the local details over the neighborhood will be weakened. In contrast, a decrease in σr and σd will lessen the dependency within the neighborhood and therefore will preserve more local details. 2.4 Experimental design Both synthesized and real-world datasets were used to validate the effectiveness of the proposed method for optimized tensor estimation. In the first experiment, synthesized datasets were used to evaluate the theoretical soundness and validity of the proposed model. In the second experiment, real-world datasets from a healthy human subject were used to assess the performance of the model in practice by comparing it with the conventional method for DTI data processing and tensor estimation. Following a well-received schema for simulating DT data [20,29], anisotropic tensor fields on a 3D lattice with dimensions of 32 × 32 × 7 were synthesized for the first experiment. The tensor field consisted of four homogeneous regions representing in white matter two underlying fiber pathways with differing fractional anisotropy (FA) values and spatial orientations, using the following values for baseline measure S0 and tensor D: (1st and 3rd quadrants) S0 = 100.00, D = 0.001 × [0.9697 1.7513 0.8423 0.0 0.0 0.0]; (2nd and 4th quadrants) S0 = 83.3, D = 0.001 × [1.5559 1.1651 0.8423 0.3384 0.0 0.0]. The tensor D is denoted as [Dxx, Dyy, Dzz, Dxy, Dxz, Dyz]. The DWI data S was then generated for every voxel using the Stejskal-Tanner equation (Eq. (1)) along 25 non-collinear directions adopted from the HD platform of the GE Signa EXCITE 3.0T MRI scanner. Rician noise was superimposed onto S and noisy DWI data S was obtained using the formula: S ( X )  ( S ( X )  nr ) 2  ni2

(13)

where nr(0, σn) and ni(0, σn) are Gaussian white noise with a standard deviation of σn [12]. A b-value of 1000 s/mm2 was adopted. To quantitatively compare the performance of our model with those of the others, different levels of noise were

superimposed onto the simulated data, varying the standard deviation. The second experiment used real brain data from a healthy human volunteer. This experiment was approved by the local institutional review board (IRB) and informed consent was obtained in writing from the volunteer. The DTI data was obtained using a 3.0T Siemens scanner (Magnetom Trio, Siemens Medical Solutions, Erlangen, Germany) with an 8-element head coil array. The imaging matrix was 128 × 128 at 50 slice locations with no gaps and a resolution of 1.7969 × 1.7969 × 2.8 mm3. The data were acquired with one baseline measure and then along 12 gradient directions. The b-value was 1000 s/mm2. The echo time (TE), repetition time (TR), and number of excitations (NEX) were 114 ms, 7800 ms, and 2, respectively. For comparison with other methods on real data, a reference dataset of relatively better quality was constructed by averaging 12 replications [10,29]. This work was approved by the local IRB and written consent was obtained from the participants. The performance of the proposed model was compared with that of three other standard algorithms. Because the proposed model incorporates into a single step the smoothing of DWI data using a bilateral filter [26] and the estimation of tensors using LLS [3], tensors were also estimated using LLS applied both to unprocessed DWI data and to tensor fields smoothed using the bilateral filter. In addition, a two-step model, which used a non-local mean filter [10] to smooth DWI data and then estimated the tensor field using LLS, was used for comparison. Thus, four algorithms were compared: (A) LLS method using Eq. (3); (B) 2-step method, with DWI data smoothing using the non-local mean filter [10] and then tensor estimation using LLS in Eq. (3); (C) proposed model that unifies DWI smoothing and tensor estimation, as defined in Eq. (5), and using the bilateral filter in Eq. (12); (D) a method that employs two distinct steps for tensor estimation using LLS in Eq. (3) and data smoothing with the bilateral filter of Eq. (12). In the experiments, σd was set equal to the noise derivation σn and σr was set equal to 10*σn for methods C and D. For method B, the window sizes were empirically defined to be 5 × 5 × 5 pixels and a similarity window of size 2 × 2 × 2 pixels was used for synthetic data, whereas a search window of size 11 × 11 × 11 pixels and a similarity window of size 5 × 5 × 5 pixels were used for real brain data [10]. The tensor fields generated by approaches (A), (B), (C), and (D) and their corresponding FA images were evaluated in both experiments using the root mean square error (RMSE) : RMSE 



iZ i 1

 , D )2 /  d (D i i

(14)

where Z is the total number of tensors in the field, D is the  is the tensor known tensors (the “ground truth”), and D  , D ) is the square estimated using each of the approaches. d ( D i of the Euclidean distance between the 6 pairs of components of the two tensors, which is regarded as a stringent measure of small differences. Similarly, the RMSE was measured for the derived FA values as: i

Local Regression for Diffusion Tensor Estimation



iZ i 1

 i , FA ) 2 / Z d ( FA i

279

error in most cases (Table 1). A quantitative evaluation of the RMSE for both the tensors and the tensor-derived FA values at differing levels of noise indicates that the proposed model (method C) performed consistently and significantly better than did the conventional LLS method, but similar to the two-step methods (Fig. 2). In most cases, the errors generated from method B were slightly higher than those generated by the proposed method and method D (Table 1 and Fig. 2).

(15)

 is for the where FA is for the known tensors, and FA estimated tensors. In the second experiment using human data, the bias of the principal direction (PD) was measured from the reference data within a selected region of interest (ROI) in the estimated DT field for a sensitive comparison of the performances of the methods. The ROI was randomly selected on a slice of the corpus callosum (CC) because fiber tracts are well organized in this structure and so noise there should be minimal.

In the second experiment, the generated FA maps of the estimated tensor fields show that the proposed model recovered the brain structure well (Fig. 3). Compared with the FA map generated using the conventional LLS method, noise was largely absent in the maps generated using the proposed method. Moreover, the boundary information along the edges of white and grey matter was also well preserved. Maps generated using the proposed method (Fig. 3d) were similar in quality to those generated using two-step method D (Fig. 3e), and were smoother than those generated using two-step method B (Fig. 3c). The LLS method generated the worst results (Fig. 3b), as expected. In white matter regions that are well-known for complex structures (Fig. 4), the proposed method produced clearly visible structural details (Fig. 4e), whereas the results generated using the LLS method contained heavy noise, which blurred all the critical boundaries necessary to tell apart the structures in whiter matter (Fig. 4c).

3. Results In the first experiment, the PD vectors of the tensor fields computed from the original and the restored diffusion tensor fields using the four methods show that the proposed model estimated the tensors better than did the conventional LLS method (Fig. 1). Noise caused the conventional LLS method to estimate tensors poorly at numerous locations (Fig. 1b), whereas the proposed model recovered the correct tensor fields robustly (Fig. 1d). In addition, the accuracy of tensor estimation for the proposed one-step method for smoothing and tensor estimation was similar to that of methods B and D, which applied these two processing steps separately (Figs. 1c and 1e). The proposed method obtained the smallest angular

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(d)

(e)

Figure 1. Typical image slices of the results for the synthetic data with noise with a standard deviation of 5.0. PD maps computed using (a) reference tensor field, (b) linear regression with LLS (method A), (c) 2-step method which separately smoothes the DWIs and then estimates the tensor field (method B), (d) proposed unified algorithm (method C), and (e) method D, which separately estimates the tensor filed and then smoothes the tensor field. Table 1. Mean and standard deviation of PD orientation bias of the synthetic tensor fields with various levels of noise (standard derivation). Angular bias is given for PD orientation. The best results are shown in bold. SD of noise LLS method (A) 2-step method (B) Proposed method (C) 2-step method (D)

5 14.97 ± 10.90 6.13 ± 3.93 5.37 ± 4.63 5.42 ± 4.74

10 31.12 ± 21.01 13.12 ± 11.19 11.15 ± 7.61 11.22 ± 7.70

15 40.75 ± 23.16 19.76 ± 16.40 19.65 ± 15.45 19.77 ± 15.61

20 44.57 ± 23.11 24.36 ± 18.63 26.50 ± 19.00 26.62 ± 19.04

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(a)

(b)

Figure 2. RMSE of the synthetic tensor fields with various levels of noise (standard derivation σn). (a) RMSE of the tensor data and (b) RMSE of the derived FA values. (Please see online e-version for color panels)

bias of the PD also shows that the proposed method generally outperformed the other methods, with less orientation bias and smaller standard error of bias (Table 2). In addition, the PD fields generated using the latter three methods were smoother than the maps from the LLS method (Fig. 5) in regions where the local PDs were well organized.

An assessment of RMSE for tensors and FA values (Table 2) in the second experiment using a real-world dataset shows that the accuracy of the proposed method approximates that of the two-step methods (the proposed method performed better than other methods with smaller RMSE for FA values), and that the accuracy of methods B, C, and D was generally superior to that of method A. An evaluation of the orientation

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Figure 3. FA images of the real datasets used in the second experiment. FA images of (a) reference dataset with12 averages of the raw data and tensor fields estimated using (b) LLS method A, (c) two-step method B, (d) proposed unified method C, σd = 15，σr = 150, and (e) two-step method D, σd = 15，σr = 150. (f)-(j) Color-encoded images of PDs for the tensors corresponding to panels (a)-(e). (See color version online)

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Figure 4. Enlarged ROIs of color FA images of the real datasets used in the second experiment. (a) Location of the selected ROI (red rectangle) known for complex white matter structure, shown on the color FA image of the reference dataset with12 averages. (b) Amplified view of the ROI in (a). Results generated using (c) LLS (method A), (d) two-step approach (method B: smoothing + LLS), (e) proposed LWLLS ( method C), σd = 15，σr = 150, and (f) two-step method (method D: LLS + bilateral filter), σd = 15，σr = 150. (See color version online)

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Table 2. Quantitative comparisons of the estimated diffusion tensors in the second experiment using human data. RMSE values are given for FA values and tensors. Angular bias is given for PD orientation. The best results are shown in bold.

LLS method (A) 2-step method (B) Proposed method (C) 2-step method (D)

PD orientation bias (mean + standard deviation) 10.91 ± 10.55 8.49 ± 9.01 6.42 ± 8.88 6.61 ± 8.77

RMSE of tensor field ( × 10-3) 0.6906 0.6489 0.5637 0.5285

RMSE of FA 0.0979 0.0819 0.0749 0.0786

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Figure 5. PD fields. (a) Selected ROI on the corpus callosum. (b) PD field of the reference dataset with NEX = 12. PD field of the tensor field estimated using (c) LLS method A, (d) two-step method B, (e) proposed unified method C, and (f) two-step method D.

4. Discussion Many methods directly regularize (smooth) the tensor fields, ignoring the raw DWI data. Such methods are intrinsically inadequately designed, in that the uncorrected bias in DWI data is deleterious and propagates to the tensor fields, making the tensor fields impossible to be corrected, thereby further making future processing and analysis (e.g., fiber tracking) unreliable. Therefore, tensor fields should be reconstructed based on corrected DWI data. The proposed model does this and efficiently integrates the preprocessing into the procedure of tensor reconstruction. Although the current version of the proposed method still cannot guarantee the positive definitiveness of the estimated tensors, the smoothing procedure embedded in the algorithm for the estimation of tensors has minimized the possible number of invalid tensors. This work provides the mathematical basis for incorporating the smoothing of DWI data within a neighborhood into the procedure of tensor estimation; the method is being developed into an iterative approach that will guarantee the positive definitiveness of the estimated tensors. This approach will facilitate the estimation of positive definitive tensors, which is different from existing approaches that use computationally expensive yet ineffective methods of matrix decomposition. The proposed method is based on the classic DT model, which profiles a diffusion probability density function of water molecules using a diffusion tensor model with its principal

eigenvector inferring one single orientation of underlying local fiber tracts. Therefore, this work is not applicable for studying tissues of crossing fibers, which requires non-Gaussian diffusion to be characterized for reflecting local complex structures [30,31], although it can be extended to process data using models for crossing fibers [24]. Many works have been developed to resolve crossing fibers in diffusion tensor imaging [31], and extending the model reported in this work using advanced diffusion models (e.g., using HARDI data [24]) to study fiber crossing is one of the future steps we are taking. The estimation of the diffusion tensor using least squares (or weighted least squares) has been proved to be optimal for Rician data [7,32]. The proposed method is basically a least squares method, and combines a smoothing filter by neighborhood correlation. No particular statistical model but implicitly a Rician model [10,29] for processing DTI data is assumed [32,33]. Only the correlation between a voxel and its neighborhood voxels using the bilateral filter, which is a widely used smoothing filter, is considered. Moreover, when the signal-to-noise ratio (SNR) of an MR image is higher than 3.0, Rician noise reduce to and can be approximated very well by Gaussian noise [13]. The SNR of DTI data is usually relatively high, around 15.0 [4], and many classic filters can effectively reduce errors in DTI data. Therefore, the present work is free from this issue. Recent work has indicated that the noise model of real DTI data acquired from a Siemens MRI scanner using parallel imaging and GRAPPA reconstruction is no longer Rician, but

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may be something much more convolved [32,33]. In order to create an optimized golden standard by averaging repetitions, the bias of the real data should be considered. However, the proposed method is basically a post-processing algorithm and examining the source and mechanism of the noise will open up a new topic for discussion in depth, whereas the issue itself remains arguable [32,33]. Focusing on the developed model for raw DWI data using the DT model, we opted to process multi-channel DTI data following the prevailing convention [10,29].

5. Conclusion The drawbacks of the most commonly used LLS method for the estimation of tensors and the regularization (smoothing) of steps, which are necessary for correctly estimating tensors, were analyzed. The LLS method was generalized to a locally weighted approach that combines the smoothing of the raw DWI data in a neighborhood with the estimation of diffusion tensors. The weighting factors were shown to be a NadarayaWatson estimator and were developed into a bilateral filter. Because this bilateral filter embedded in the model is an anisotropic weighting strategy and simultaneously uses the spatial distance and the difference of voxel intensities, the proposed model can well preserve the information of the boundary in the tensor fields, producing clear but smooth tensor fields with sharp edges. The model was developed to use a bilateral filter as the weighting function to combine the approach with a smoothing function. Experiments using both simulated and real datasets demonstrated that this method performs exceptionally well.

[6] [7]

[8]

[9] [10]

[11] [12] [13] [14] [15] [16] [17] [18]

Acknowledgments This work was supported in part by NIMH grant K0274677, NIBIB grant 1R03EB008235-01A1, Shanghai Commission of Science and Technology grant #10440710200, a NASARD grant, a grant from China NSF (approval #  91232701), and a grant from the East China Normal University (ECNU) School of Psychology and Cognitive Science. This work was also supported in part by the Large Instrument Open Foundation of ECNU.

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