The design of an optimal gradient encoding scheme (GES) is a fundamental problem in diffusion MRI. It is
well studied for the case of secondorder tensor imaging (Gaussian diffusion). However, it has not been investigated
for the wide range of nonGaussian diffusion models. The optimal GES is the one that minimizes the variance of
the estimated parameters. Such a GES can be realized by minimizing the condition number of the design matrix
(
Diffusionweighted MRI is a noninvasive imaging technique to probe microstructures in living tissues, for example, the human brain. It involves acquiring a series of diffusionweighted images, each corresponding to diffusion sensitization along a particular gradient direction. NonGaussian diffusion models have gained wide attention among researchers because of their potential ability to resolve complex multifiber microstructures. Özarslan and Mareci [
The need for robust estimation of diffusion parameters in a limited acquisition time has given rise to many studies on optimal gradient encoding scheme (GES) design. In the case of the classical secondorder model they include [
This section briefly reviews the basics of HOTbased ADC profile estimation both for the sake of completeness and to define notation. The reader is referred to [
In the framework described above, the precision of the estimation problem is dependent on the experiment designs
For isotropic diffusion, it has been shown that (
In Section
The condition number measures the sensitivity of the solution to changes in measurements [
It is possible to relax the constraints
The
If
The minimum condition number is independent of
The
And
The task of extracting the design points (
Optimal gradient encoding scheme (










0.1514  −0.9883  −0.0161  0.3527  −0.8791  0.3207  −0.9125  0.2478  −0.3253 
0.4840  0.1736  −0.8576  −0.0048  1.0000  −0.0068  −0.0163  0.0245  0.9996 
−0.5357  0.0645  −0.8419  0.9960  −0.0292  0.0842  0.0160  0.0349  0.9993 
−0.0633  −0.1941  0.9789  0.8959  −0.1044  −0.4317  0.3204  −0.3626  −0.8751 
−0.4457  −0.8893  −0.1024  −0.0111  0.0185  0.9998  −0.1819  −0.8503  0.4938 
−0.8564  −0.4798  0.1908  −0.1289  0.4227  −0.8970  0.0248  0.9996  −0.0146 
0.9998  0.0123  0.0169  0.9988  −0.0129  0.0481  0.1318  0.9903  −0.0441 
0.8391  −0.5377  −0.0829  0.0341  0.9994  −0.0089  −0.0149  −0.0427  −0.9990 
−0.2315  −0.3334  −0.9139  0.0851  0.8468  0.5251  0.8780  0.3205  0.3556 
0.3072  −0.9185  −0.2490  0.9867  0.0077  −0.1623  0.9973  0.0662  −0.0304 




0.9096  0.0000  0.4155 
0.0000  0.4155  0.9096 
0.4155  0.9096  0.0000 
0.0000  0.4155  −0.9096 
0.4155  −0.9096  0.0000 
−0.9096  0.0000  0.4155 
The set of secondorder tensors can be seen as a subset of fourthorder tensors. As an example, the equality
In this section we evaluate the proposed
Table
Comparison of the proposed
GES 

DISCOBALL  Jones  MCN  Wong 


1.9141  3.6392  3.8039  4.9473  4.9849 
Reference  [Proposed]  [ 
[ 
[ 
[ 
Signal deviation is defined in [
Tenfourthorder tensors used for evaluation of the proposed method. These tensors correspond to different fiber architectures as illustrated in Figure











2  0.60  0.54  0.73  0.64  0.45  0.69  0.56  0.70  8.50 
0  0  0  0  0  0  0  0  0  0 
1  0.38  0  0  0.03  0.79  0.29  0.84  0.34  0 
0  0  0  0  0  0  0  0  0  0 
2  13.31  23.77  12.51  8.66  5.47  10.86  7.48  7.25  8.50 
0  0  0  0  0  0  0  0  0  0 
0  −1.23  0  0  0  0  0.13  0.38  −0.05  0 
0  0  0  0  0  0  0  0  0  0 
0  30.02  0  0  0  0  10.37  6.80  15.23  0 
3  0.99  2.16  0  0  0  0.02  0.04  0.01  0 
0  0  0  0  0  0  0  0  0  0 
3  27.03  0  0  0  0  3.93  2.32  12.49  0 
0  0  0  0  0  0  0  0  0  0 
0  9.70  0  0  0  0  0.69  0.35  4.35  0 
17  1.83  0.29  12.27  16.20  19.33  12.32  16.18  13.01  8.50 
Comparison of the proposed
Tensor 

DISCOBALL  Jones  MCN  Wong 

Mean 

0.0553  0.0552  0.0525  0.0527 

0.0010  0.0009  0.0009  0.0012  0.0011 
Mean 

0.2083  0.2155  0.1524  0.1769 

0.0457  0.0311  0.0310  0.0341  0.0281 
Mean 

0.0648  0.0648  0.0598  0.0607 

0.0033  0.0015  0.0017  0.0029  0.0026 
Mean 

0.0541  0.0538  0.0512  0.0515 

0.0016  0.0009  0.0009  0.0015  0.0010 
Mean 

0.0555  0.0553  0.0525  0.0527 

0.0016  0.0009  0.0010  0.0017  0.0012 
Mean 

0.0579  0.0577  0.0544  0.0546 

0.0018  0.0011  0.0011  0.0020  0.0015 
Mean 

0.0723  0.0725  0.0651  0.0663 

0.0032  0.0020  0.0022  0.0031  0.0029 
Mean 

0.0642  0.0639  0.0594  0.0597 

0.0016  0.0011  0.0012  0.0021  0.0015 
Mean 

0.1096  0.1081  0.0879  0.0930 

0.0122  0.0102  0.0091  0.0109  0.0083 
Mean 

0.0476  0.0474  0.0461  0.0466 

0.0008  0.0007  0.0007  0.0008  0.0007 
Shape of 10 fourthorder tensors used for the evaluation of the proposed method: (a) singlefiber with orientation
Results of rotational variance test for
The distribution of gradient encoding directions over the unit sphere for the evaluated GESs is plotted in Figure
Distribution of directions over the unit sphere for different GESs (
The proposed approach can be applied in experiment design for other tensors, although the current work focuses on its application and results on fourthorder tensor estimation. In Section
In Section
As we mentioned above the
We showed that the
See Algorithm
The authors declare that there is no conflict of interests regarding the publication of this paper.