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Impact of Noise and Number of Gradients

The resulting impact of noise for a varying number of gradient directions is quantitatively assessed in Fig. 1. In case of high noise, the DNN achieves a slightly higher NMSE if many gradient directions are acquired. Moreover, the resulting

Resulting NMSE of the MAP approach and the DNN for 10, 20, 40 and 80 gradient directions, SNR={10,1}

Fig. 1 Resulting NMSE of the MAP approach and the DNN for 10, 20, 40 and 80 gradient directions, SNR={10,1}

NMSE increases for the MAP algorithm if only few gradient directions are available, but remains rather stable in case of the DNN.

In addition, it can be seen that both algorithms achieve a low NMSE if no noise is added to the signal. Notably, the DNN achieves a lower NMSE for every gradient set.

Prediction of Another Shell

In order to evaluate the performance of predicting other shells, Tables 2 and 3 contain the resulting prediction NMSEs for every combination of input and target shells that can be generated based on the HCP dataset utilizing 15 and 90 gradient directions. DNN training is performed for each combination individually. Moreover, the resulting shell distance d between two shells is provided.

Table 2 Average NMSE for predicting the signal utilizing the DNN and the MAP approach based on 90 gradient directions on each shell

Input shell

Target shells (90 gradients)

Predicted 1st Shell

Predicted 2nd Shell

Predicted 3rd Shell

DNN

MAP

d

DNN

MAP

d

DNN

MAP

d

1st shell

1.03%

1.04%

0%

3.49%

18.06%

38.13%

5.63%

27.90%

66.21%

2nd shell

2.01%

8.08%

18.50%

2.28%

2.36%

0%

4.39%

17.81%

21.68%

3rd shell

3.58%

24.57%

33.41%

3.43%

25.99%

9.73%

3.45%

4.55%

0%

1st + 2nd shell

-

-

-

-

-

-

4.10%

14.97%

-

1st + 3rd shell

-

-

-

2.69%

16.91%

-

-

-

-

2nd + 3rd shell

1.53%

9.87%

-

-

-

-

-

-

-

In addition, d represents the NMSE between two shells without any prediction, which is comparable to a distance between two shells

Table 3 Average NMSE for predicting the signal utilizing the DNN and the MAP approach based on 15 gradient directions on each shell

Input shell

Target shells (15 gradients)

Predicted 1st shell

Predicted 2nd shell

Predicted 3rd shell

DNN

MAP

d

DNN

MAP

d

DNN

MAP

d

1st shell

1.18%

1.19%

0%

3.67%

18.32%

38.13%

5.98%

28.10%

66.21%

2nd shell

2.15%

8.13%

18.50%

2.58%

2.64%

0%

4.73%

18.09%

21.68%

3rd shell

3.89%

24.66%

33.41%

3.74%

26.11%

9.73%

3.95%

4.98%

0%

1st + 2nd shell

-

-

-

-

-

-

4.27%

20.70%

-

1st + 3rd shell

-

-

-

2.87%

15.46%

-

-

-

-

2nd + 3rd shell

1.72%

9.37%

-

-

-

-

-

-

-

In addition, d represents the NMSE between two shells without any prediction, which is comparable to a distance between two shells

In addition, it should be noted that a prediction of the third shell will always result in a higher NMSE than for the first shell, due to a smaller denominator (see Eq. (3)), which is reflected in the d.

Considering both tables, it can be seen that both algorithms result in similar NMSEs for 90 as well as for 15 gradient directions. Both algorithms achieve their lowest NMSE if the fit is performed from a input to the same target shell.

If a shell is augmented to predict another shell, the DNN generally achieves a more stable fit than the MAP algorithm. The inaccuracy grows with increasing shell distance between input and target shell. Adding a second shell to the input (i.e. predicting a third shell from two input shells) seems to stabilize the MAP algorithm and decreases the NMSE of the resulting augmented shell. Nevertheless, the DNN still achieves a much lower NMSE. In order to evaluate the results in more detail, Figure 2 exemplifies the results using the 3rd shell as target shell for 90 gradient directions. Excluding the 3rd shell as input, the MAP algorithm performs best utilizing two shells as input. In addition, its performance increases as shell

Resulting NMSE of the MAP approach and the DNN utilizing only the 3rd shell as target shell presented as a boxplot for 90 gradient directions distance between input and target shell decreases

Fig. 2 Resulting NMSE of the MAP approach and the DNN utilizing only the 3rd shell as target shell presented as a boxplot for 90 gradient directions distance between input and target shell decreases. The DNN shows similar results, but outperforms the MAP for each input scenario.

 
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