High-Dimensional Spatial Standardization Algorithm for Diffusion Tensor Image Registration

Tao Guo, Quan Wang, Yi Wang and Kun Xie

(1.School of Computer and Science, Xidian University, Xi’an 710071, China; 2.School of Electronics and Information, Northwestern Polytechnical University, Xi’an 710072, China)

Abstract: Three high dimensional spatial standardization algorithms are used for diffusion tensor image (DTI) registration, and seven kinds of methods are used to evaluate their performances. Firstly, the template used in this paper was obtained by spatial transformation of 16 subjects by means of tensor-based standardization. Then, high dimensional standardization algorithms for diffusion tensor images, including fractional anisotropy (FA) based diffeomorphic registration algorithm, FA based elastic registration algorithm and tensor-based registration algorithm, were performed. Finally, 7 kinds of evaluation methods, including normalized standard deviation, dyadic coherence, diffusion cross-correlation, overlap of eigenvalue-eigenvector pairs, Euclidean distance of diffusion tensor, and Euclidean distance of the deviatoric tensor and deviatoric of tensors, were used to qualitatively compare and summarize the above standardization algorithms. Experimental results revealed that the high-dimensional tensor-based standardization algorithms perform well and can maintain the consistency of anatomical structures.

Key words: diffusion tensor imaging; high dimensional; spatial standardization; registration; template; evaluation

Diffusion tensor imaging (DTI)[1]is a new technique in magnetic resonance imaging (MRI), which has developed rapidly in recent years by measuring the diffusion characteristics of water molecules. There is a large number of fiber bundles in the white matter, and the diffusion of the water molecules occurring in the direction parallel to the fiber is faster than diffusion in the direction perpendicular to the fiber, demonstrating a significant anisotropy. The microstructural characteristics of water molecules in white matter can be reflected by measuring the diffusion characteristics (anisotropic diffusion, diffusion direction, average diffusion). DTI has become a very important tool for understanding the difference in white matter between people of different ages or between normal people and those with brain diseases[2]. Since the scalar diffusion coefficient cannot completely simulate anisotropic diffusion, the diffusion tensor is introduced (denoted by the symbol D). It is a 3×3 symmetric positive definite matrix, composed of nine elements, containing structural and directional information. The maximum eigenvalue of the diffusion tensor D corresponds to longitudinal diffusion and the mean of the other two eigenvalues represents radial diffusion. The directional information provided by eigenvectors is widely used for the estimation and reconstruction of the white matter fiber bundles[3-4]. Accompanied by its extensive application in neurophysiology, neuroanatomy, neurosurgery, amyotrophic lateral sclerosis[2], Alzhei-mer’s disease[5], Krabbe disease[6], multiple sclerosis[7]and other studies, DTI registration has become one of the hotspots in the field of image processing and analysis at home and abroad.

Image registration is used to find a mapping relationship between two images and their corresponding points. Starting in the 1970s, shortly after the emergence of digital images, the international community began a wide range of image registration research. Ordinary medical image registration allows the corresponding points of two images to reach the same spatial position and anatomical structure by spatial transformation, which belongs to the scalar image registration; its registration process is essentially a multi-parameter optimization problem. DTI registration is also an optimization problem and necessitates agreement between spatial positions and anatomical structures of corresponding points, which also involves spatial transformations, neighborhood interpolations, optimization algorithms and so on. But diffusion tensor image registration is different in that it must ensure that the direction of the tensor (i.e., the direction of the fiber) before and after the image transformation is consistent with the anatomical structure, involving tensor redirection[8]. See Ref.[9] for a principle introduction of DTI registration and the comparison of its advantages and disadvantages DTI.

In recent years, some scholars have found that spatial standardization plays an important role in the study of the difference in white matter in DTI. It can accurately map the diffusion tensor characteristics that play a very important role in auxiliary detection and diagnosis of neurological disorders, white matter lesions and other diseases. Spatial standardization of DIT has been achieved in most clinical studies using low-dimensional registration algorithms, and most clinical studies have chosen to use their structural images (e.g. T1, T2) or scalar image (e.g. Fractional Anisotropy, FA)[10]. Although the low-dimensional registration algorithm is simple and fast, it does not make full use of the directional information of the tensor, especially the directional information of the fiber. There is data loss and the registration accuracy is not ideal. Moreover, it cannot simulate morphological differences between the complex brains of different subjects[11]. DTI reflects the spatial compositions of organs and the water molecule exchanges in the pathological state. Thus, DTI-based registration is an ideal choice for understanding and analyzing white matter structures.

Image registration is a very important platform for spatial standardization. Spatial standardization is mainly used to eliminate individual differences by transforming different subjects to the same space in order to meet the corresponding structure, then measure and analyze the subjects in the same space. High dimensional spatial standardization can make full use of tensor data to improve the accuracy of registration. Moreover, Ref. [2] has shown that the high dimensional standardization algorithm affects the white matter analysis of different groups, and the tensor can improve the precision of white matter fiber bundle registration rather than the scalar derived from the tensor. At present, DTI registration is mostly based on voxel analysis. Spatial standardization is an important factor based on voxel analysis. The quality of spatial standardization determines the precision of white matter fiber bundle registration. The high dimensional standardization methods of high dimensional registration algorithms improve the quality of standardization. The difference in the shape of the tensor will confuse the difference in anisotropy (e.g. FA) in some way, whereas the low-dimensional standardization methods do not exclude the difference in shape, which is not conducive to the diagnosis and treatment of the disease. The high dimensional standardization methods can significantly reduce the interference of the anisotropy analysis resulting from differences in shape, and can more extensively describe the size of the white matter structure as well as the difference in organ structure. This indicates that the high-dimensional spatial standardization methods are superior to the low-dimensional spatial standardization methods. So far, high-dimensional standardization methods based on high-dimensional registration algorithms have been very rarely clinically applied in white matter studies, and a typical case is the asymmetric analysis of white matter[12].

Based on the above, we will continue to study high-dimensional space standardization for DTI registration. The results of the study will be important for the study of brain development, the diagnosis of white matter lesions, the analysis of internal brain structure, the provision of anatomical information for the brain, computer-aided diagnosis, and the assessment and treatment of disease development. In addition, it is also important for the study of non-human primates (such as macaques)[13].

1 Method

The standardization algorithm mainly involves[14-15]: a transformation model (including regularization kernel), similarity measure, and optimization criterion. In general, image standardization is a process in which an optimal transformation space is found, and the voxels of the source image are mapped to those of the target image to depict the similarity process through means of a specific objective function. Ref.[16] shows that spatial standardization is the process of registering subjects and templates, thus transforming the subjects’ data into a template space. Thus, the subjects will be transformed into the same space and their structures will be linked. This allows for the difference analysis in white matter meaningful.

The high dimensional standardization algorithm is of great benefit when comparing and integrating different research objects[17], detecting white matter differences[6]and analyzing fiber bundles[18]. Three representative methods that can be used today include[2]: ① the low-dimensional standardized method with FA; ② the high-dimensional standardized method with FA; ③ the high-dimensional standardization method with full tensor. These methods can detect differences in white matter structure between a patient and a healthy person. For example, in amyotrophic lateral sclerosis (ALS) patients and normal people, Zhang et al. have demonstrated that DTI’s high-dimensional spatial standardization improves the detection of white matter differences[2]. Adluru et al. studied the brain template of DTI by using the full-tensor high-dimensional standardization algorithm on imaging data retrieved from macaques[13]. Spatial standardization is important for the assessment and treatment of disease development. Keihaninejad et al. have studied such matters in Alzheimer’s disease (AD)[5].

1.1 High dimensional standardization algorithm

Fig.1 Flowchart of high-dimensional FA based standardization

The high-dimensional standardization algorithm is divided into two categories: FA-based high-dimensional standardized algorithm and tensor-based high-dimensional standardized algorithm. The processes of the specific methods are shown in Fig.1 and Fig.2 respectively. Registration is a platform for achieving spatial standardization. The difficulty in using the DTI registration algorithm is that the direction of the tensor needs to be corrected during the image deformation to ensure consistency with the anatomical structure. At present, the spatial transformations mainly involve rigid transformation, affine transformation and diffeomorphic transformation. The rigid transformation is only related to the rotation of the tensor. For the affine transformation and the diffeomorphic transformation, the method of reorienting the tensor primarily involves the preservation of the principle directions (PPD), which mainly depends on the direction of the principle eigenvector and finite strain (FS)[19], both of which are rotationally invariant. Ref. [19] has shown that PPD is more accurate than the FS for tensor redirection. Ref. [2] also shows that FS is mainly used in the estimation of spatial transforms, while PPD is mainly used for tensor transformations. In this paper, PPD was used as the tensor reorientation method in high dimensional FA based standardization algorithm.

Fig.2 Flowchart of high-dimensional tensor based standardization

The high-dimensional FA-based standardization algorithm using the symmetric image normalization (SyN) algorithm in the Advanced Standardization Tool (ANTS) is a diffeomorphic type transformation. Ref. [20] has shown that SyN is a highly accurate, voxel-based standardized method of 14 standardized methods with high consistency. The high dimensional tensor-based standardization algorithm uses DTI-TK[21-22]. Ref.[6] has proved that the DTI-TK algorithm has higher accuracy than other algorithms. The difference between these two types of high-dimensional algorithms is that the tensor reorientation in the high-dimensional FA algorithm is after standardization, while tensor reorientation in the high-dimensional tensor-based standardization algorithm is during the registration process.

Diffeomorphic transformation can obtain better shape changes and retain the topology that establishes the basis for comparing different subjects. The study of brain transformation in cell structure also shows that the brain holds the cell layout[23], which further enhances the diffeomorphic mapping in brain research applications. Ref. [24] has also shown that diffeomorphic transformation is superior to greedy and exponential transformation models. The main features of ANTS include diffeomorphic standardization algorithms. ANTS mainly employs SyN, using a cross correlation method and symmetric diffeomorphic mapping[25]with the Euler-Lagrangian equation used in optimization. The elastic standardization method (referred to as elastic) performs the cross correlation as a similarity measure based on the elastic deformation model. ANTS supports standard multi-dimensional image analysis for both large and small deformations[24,26].

DTI-TK is nonparametric diffeomorphic mapping image registration algorithm. DTI-TK introduces the tensor reorientation method, FS, into the transformation estimation process. The image is divided into four regions, then the affine transformation is carried out in each region. The conjugate gradient algorithm is used to optimize the objective function of tensor redirection. The entire tensor is considered for similarity measures. The tensor reorientation is clearly optimized. FS is used in the registration process to adjust the tensor, then the PPD method is used in the final step.

1.2 Template

Spatial standardization based on voxel analysis requires a template so that the other images are spatially normalized. Ref.[27] shows that a high-quality template can improve the quality of space standardization. Here, the template is chosen based on the object’s data through an unbiased iterative calculation (with DTI-TK). Refs.[6,28] have proved that DTI-TK can produce templates with higher accuracy. For the method of calculating the template, Ref.[22] compared the high dimensional method that clearly optimizes the tensor orientation, i.e. the large deformation diffeomorphic registration method (LDDRM) for vector fields and the piecewise affine deformation registration algorithm (referred to as DTI-TK), which shows that DTI-TK is a high-performance DTI spatial standardization algorithm. Ref.[13] also proves that DTI-TK is the most advanced DTI space standardization and Atlas creation tool. The diffusion features can be grasped in an unbiased manner while the anatomical shape features are maintained. A tensor-based registration algorithm with nonparametric, highly deformed, diffeomorphic registration was used in this paper. The direction of the local fiber is incorporated into the algorithm, and the tensor is regarded as a whole for the similarity measure, which is very effective for the spatial standardization of the tensor orientation and the morphology of fiber bundles.

In this paper, 16 subjects were used to perform the experiment. The data pertaining to the 16 subjects were downloaded from the DTI-TK

website (http:∥www.nitrc.org/pro-jects/dtitk/). The flow chart for DTI-TK which was used in obtaining the template can be seen in Fig.3. The tensor of the template obtained by DTI-TK and its FA images, as well as corresponding images of one subject are shown in Fig.4 in the axial, coronal and sagittal direction, No. 32, 64, and 64 slices respectively. Color-oriented tensors are in the above row; the corresponding FA images are in the following row. The total number of colors is 262 144. The resulting template images are located in the left three columns and the images for one subject are located in the right three columns. Template images of FA, trace (TR), radial diffusivity (RD), axial diffusivity (AD) are shown in Fig.5 in the axial, coronal, and sagittal directions. In the upper row, FA images are in the left three columns, while TR images are in the right three columns. In the lower row, RD images are in the left three columns, while AD images are in the right three columns. These are different characteristics of the diffusion tensor, which are useful for the analysis of voxels between different subjects. Here, TR, RD, and AD represent the diffusion rate of the water mole-cule, while FA represents the diffusion anisotropy with values in the range of 0-1. As mentioned in Ref.[29], even if different templates are used, the results of their standardization should be logically consistent.

Fig.3 Flowchart of DTI-TK getting the template

Fig.4 Tensor of template and FA obtained with DTI-TK

Fig.5 FA, TR, RD, AD images of the template obtained with DTI-TK

2 Experiments

The data preprocessing uses the FSL toolkit (a software developed by a lab at Oxford University, which is run on Linux or Mac)[30]to perform eddy correction on the data, primarily for distortions and head movements. The brain tissue extraction, tensor estimation, FA, TR, three eigenvalues and eigenvectors can be obtained by DTIFIT in FSL. The FA, TR of the subjects were obtained by DTI-TK. The implementations of DTI-TK, FSL were based on the Centos 6.3 operating system. ANTS is implemented in a Windows 7, 64-bit operating system. The results of high-dimensional FA-based and tensor-based standardization are shown in Fig.6. Presented on the left are the high dimensional tensor based standardization results of DTI-TK. In the middle are the high dimensional FA-based standardization results of SyN. Lastly, on the right are the elastic high dimensional FA-based standardization results. The images show the standardization results of one subject. The total color number of these color-oriented tensor images is 262 144. The original data in the axial, coronal and sagittal sections are shown in the right three columns in Fig.4. The three standardization algorithms yielded similar results.

Fig.6 Results of high-dimensional FA based standardization and high-dimensional tensor based standardization

3 Evaluation Method

The evaluation of spatial standardization is primarily the performance evaluation of the spatial standardization method, which is to measure the accuracy of white matter alignment, including the speed, precision and robustness of registration. The visual judging criterion of white matter alignmentis mostly based on the expert’s subjective experience. Consequently, a large number of errors are introduced and evaluation results are poor. The diffusion tensor contains a wealth of information. For example, eigenvalues and eigenvectors contain tensor size, shape, and orientation information. In the perspective of the above standardized results, the following evaluation criteria were chosen to evaluate the effect: the diffusion tensor matrix information, orientation, scalar consistency and spatial transforms.

(1)

(2)

② Dyadic coherenceκis used to measure the change in the principal diffusion direction of water molecules in the white matter (the change of the eigenvector), the value is in the range of 0 to 1, 0 is the random direction, and 1 is the direction of the determination.

(3)

where,βj(j=1, 2, 3) is the eigenvalue of the mean dyadic tensor defined as

〈e1eT1〉= e21xe1xe1ye1xe1ze1xe1ye21ye1ye1ze1xe1ze1ye1ze21z() =∑Ni=1ei1eiT1N

(4)

③ Diffusion cross-correlation:

(5)

wherevindexes over all the voxels.X1andX2are two scalar images, i.e. FA or TR. This grey scale value ranges between 0 and 1. The larger the value is, the greater the similarity of the image. Take FA as an example,

(6)

④ Overlap of eigenvalue-eigenvector pairs (OVL)

(7)

⑤ Euclidean distance (ED)

(8)

where D1and D2are two tensors. The smaller the distance is, the more accurate the white matter alignment is.

⑥ Euclidean distance of the deviatoric tensor (DE)

DE=‖D-[trace(D)/3]I‖c

(9)

where D is a tensor. The smaller the distance is, the more accurate the white matter alignment is.

⑦ Deviator of tensors (Ddev)

(10)

The smaller the deviator value is, the more accurate the white matter alignment is.

⑧ The angular separationαiof the three sets of eigenvectors of two tensors

(11)

The smaller the angle separation is, the better matched the eigenvector directions will be, and the more accurate the fiber bundle alignment will be.

4 Evaluation Results

The high dimensional FA-based and tensor-based standardization method are both evaluated for the template obtained by DTI-TK. The standardized results are shown in Fig.6. A good spatial standardization method can be a good way to maintain the consistency between anatomical structures. For the evaluation methods listed above, the images and figures of the cumulative distribution functions (CDFs) are illustrated separately. The red solid line represents the high dimensional tensor-based standardization algorithm. The blue dotted line represents the high dimensional FA-based standardization algorithm by SyN. Lastly, the green double-crossed line represents the high dimensional FA-based standardization algorithm by elastic.

Fig.9 Standardized FA, TR images with DTI-TK, SyN, elastic algorithms

4.1 Normalized standard deviation

The statistical results of the CDF of the FA and TR-normalized standard deviation in the three algorithms are shown in Figs.7-8. The normalized standard deviations of DTI-TK on FA and TR were significantly smaller. The smaller normalized standard deviation of FA and TR indicate that the higher consistency of the eigenvectors, that is the spatial standardization, is able to better maintain the consistency of the anatomical structures. The standardized FA, TR images of the three algorithms are shown in Fig.9. The left three columns are FA images with DTI-TK, SyN, and elastic in turn, while the right three columns are TR images with DTI-TK, SyN, and elastic in turn. From top to bottom are axial, coronal and sagittal sections. It is clear that the standardized FA and TR diagrams with DTI-TK are smoother. The high dimensional tensor-based standardization algorithm has good consistency, in which the normalized standard deviations of FA and TR are significantly smaller than those of the other two algorithms. High dimensional FA-based standardization algorithms by SyN and elastic performed similarly.

Fig.7 CDFs of normalized standard deviation on FA

Fig.8 CDFs of normalized standard deviation on TR

4.2 Dyadic coherence κ

The statistical results from the CDFs of the dyadic coherenceκin three algorithms are shown in Fig.10. The greater the dyadic coherenceκvalue is, the better the alignment degree of the eigenvectors is. Furthermore, the higher the accuracy of the fiber bundle alignment, the better the consistency of anatomical structures. In Fig.10, the statistical results of dyadic coherenceκvalues are very close, indicating that the consistencies of the three algorithms are similar. The statistical results of dyadic tensorβ1are shown in Fig.11. The fiber alignment degree of the high dimensional tensor-based standardization algorithm is better than that of high dimensional FA-based standardization algorithm by SyN and elastic. The dyadic coherenceκcurves after three standardization algorithms are shown in Fig.12. The three columns are images of dyadic coherenceκwith DTI-TK, SyN and elastic in turn. From top to bottom are axial, coronal and sagittal sections.

Fig.10 CDFs of dyadic coherence κ

Fig.11 CDFs of dyadic tensor β1

4.3 Diffusion cross-correlation

The statistical results of the diffusion cross-correlation CDFs of the three algorithms are shown in Fig.13 and Fig.14. The larger the value, the more similar the images are. It can be seen from Fig.13 and Fig.14 that the similarity drawn from the high dimensional tensor-based standardization algorithm is higher than that of high dimensional FA-based standardization algorithm. The correlations of the two FA standardized algorithms are similar.

Fig.12 Images of dyadic coherence κ after three standardization algorithms

Fig.13 CDFs of diffusion cross-correlation on FA

Fig.14 CDFs of diffusion cross-correlation on TR

4.4 Overlap of eigenvalue-eigenvector pairs

The CDF statistical results of OVLs in the three algorithms are shown in Fig.15. The higher the OVL value is, the higher the consistency after standardization is. It can be seen from Fig.15 that the OVL of the high dimensional tensor-based standardization algorithm is higher than that of FA-based algorithm, that is, the consistency of the high dimensional tensor-based standardization algorithm is better than that of FA-based algorithm. Therefore, high dimensional tensor-based standardization algorithm can better maintain the consistency of the anatomical structures. Moreover, the OVL values in the two high dimensional FA based standardization algorithms are close.

Fig.15 CDFs of the OVL values by three algorithms

4.5 Euclidean distance of diffusion tensor

The CDF statistical results of the ED values of diffusion tensors produced by the three algorithms are shown in Fig.16. The smaller the ED value is, the higher the consistency is between different subjects after tensor-based standardization. It is evident from Fig.16 that the ED value of the high-dimensional tensor-based standardization algorithm is smaller than that of FA-based algorithm, that is, the consistency of the high-dimensional tensor-based standardization algorithm is better than that of FA-based algorithm. Thus, the high-dimensional tensor-based standardization algorithm is better at maintaining the consistency of the anatomical structures. Moreover, the performances of the two high-dimensional FA based standardization algorithms are close.

Fig.16 CDFs of the ED values by three algorithms

4.6 Euclidean distance of the deviatoric tensor

The statistical results from the CDF of the DE values in the three algorithms are shown in Fig.17. The smaller the DE value is, the higher the consistency is between the different subjects after tensor-based standardization. It is evident in Fig.14 that the DE values of the high-dimensional tensor-based standardization algorithm are significantly smaller than those of FA-based algorithm, that is, the consistency of the high-dimensional tensor-based standardization algorithm is better than FA-based algorithm. Therefore, the high-dimensional tensor-based standardization algorithm better maintains the consistency of the anatomical structures. Moreover, the performances of the two high-dimensional FA based standardization algorithms are close.

Fig.17 CDFs of DE values by three algorithms

Fig.18 ED and DE after three standardization algorithms

The ED and DE images after three standardization algorithms are shown in Fig.18. The left three columns are ED images produced from DTI-TK, SyN and elastic, while the right three columns are the DE images from DTI-TK, SyN and Elastic. From top to bottom are axial, coronal and sagittal sections. It is obvious that a standardized ED and DE with DTI-TK produces no artifacts, while the other two methods produce artifacts.

4.7 Deviatoric of tensors (Ddev)

Fig.19 CDFs of Ddev values by three algorithms

The statistical results from the CDFs ofDdevvalues in the three algorithms are shown in Fig.19. The smaller theDdevvalue is, the higher the consistency is between different subjects after tensor-based standardization. It is evident from Fig.19 that theDdevvalue of the high-dimensional tensor-based standardization algorithm is significantly smaller than that of FA-based algorithm, that is, the consistency of the high-dimensional tensor-based standardization algorithm is better than FA-based algorithm. It follows that the high-dimensional tensor-based standardization algorithm better maintains the consistency of the anatomical structures. Moreover, the performances of the two high-dimensional FA-based standardization algorithms are close. TheDdevtensor images after three standardization algorithms are shown in Fig.20. From top to bottom are results from DTI-TK, SyN and elastic in turn, and from left to right are axial, coronal and sagittal sections.

Fig.20 Ddev tensor images after three standardization algorithms

4.8 Angular separation αi of the three sets of eigenvectors of two tensors

The statistical results from the CDFs of the mean angle separation values in the three algorithms are shown in Fig.21. The smaller the angle separationαi, the better the eigenvectors match, and the more accurate the fiber bundle aligns. As seen from Fig.21, the angle separationαivalues after applying the high-dimensional tensor-based standardization algorithm are smaller than those of FA-based algorithm, that is, the high-dimensional tensor based standardization algorithm makes the fiber bundle alignment degree better than FA-based algorithm. Moreover, the two high-dimensional FA-based standardization algorithms perform similarly.

Fig.21 CDFs of the angle separation mean of the three sets of eigenvectors by three algorithms

5 Conclusion

In this paper, high-dimensional standardization algorithms of diffusion tensor images were used, including the high-dimensional FA-based (e.g. SyN and elastic) and tensor-based standardization algorithm (e.g. DTI-TK). Seven evaluation methods were used to evaluate the three algorithms using the characteristics of the diffusion tensor. The statistical results show that the high-dimensional tensor-based standardization algorithm has good performance and can maintain the consistency of anatomical structures, which is a good way to improve the direction of white matter fiber bundles.