3.1.2. The Image Patch Radius *RLP*

Figure 6 shows the TRE values and computational time of the FMIND method while using the different *RLP* values. Likewise, one-way ANOVA with α = 0.05 is used to evaluate the influence of *RLP* on the registration results. The obtained significance value *P* is 0.001 for registration accuracy and *P* is 0.007 for registration efficiency. Therefore, *RLP* has the significant impact on both registration accuracy and efficiency. It can be seen from Figure 6a that the TRE significantly declines when *RLP* varies from 2 to 7, and it tends to be stable when *RLP* varies from 7 to 10. Besides, Figure 6b indicates that the registration time gradually increases for the increasing *RLP*. The reason is that, for a larger *RLP*, the more pixels need to be processed in the vertex partition algorithm. Therefore, we have set *RLP* as 7 to achieve the trade-off between registration accuracy and efficiency.

**Figure 6.** The TRE and computation time with different *RLP*. (**a**) TRE (voxels); (**b**) Time (minutes).

### 3.1.3. The Similarity Threshold δ

Figure 7 shows the effect of δ. For one-way ANOVA with α = 0.05, the obtained *P* values are 0.001 for both registration accuracy and efficiency, which means the significant impact of δ on the registration performance of the FMIND method. From Figure 7a, we can see that the TRE significantly declines when δ varies from 0.3 to 0.8. The reason is that for a larger δ in this range, fewer control vertices are

divided into the static ones and the number of dynamic control vertices increases, thereby resulting in a smaller TRE. Meanwhile, the TRE tends to be stable when δ varies from 0.8 to 1.0. The reason is that, for the threshold δ in this range, the number of dynamic control vertices will increase to a certain value, so that the variation of δ will have little effect on the TRE. Besides, Figure 7b indicates that the registration time significantly increases with the increasing δ. It is easy to understand that, for a larger δ, the increasing dynamic control vertices will lead to more processing time. Therefore, we set δ as 0.8 to balance the registration accuracy and efficiency.

**Figure 7.** The TRE and computation time with different δ. (**a**) TRE (voxels); and, (**b**) Time (minutes).

### 3.1.4. The Static Factor ε

Figure 8 shows the effect of the static factor ε on registration accuracy and efficiency. As ε is also used to divide the control vertices, it has a similar effect on registration performance to the similarity threshold δ. According to Figure 8a,b, ε has the opposite effect on TRE and computational time. We have chosen ε = 0.9 for the FMIND method based on the comprehensive consideration of registration performance.

**Figure 8.** The TRE and computation time with different ε. (**a**) TRE (voxels); and, (**b**) Time (minutes).

### *3.2. Comparison of Registration Performance*

### 3.2.1. Registration Results of Simulated T1, T2 and PD Images

In order to quantitatively and qualitatively compare the registration performance of the FMIND method and other methods on 3D T1, T2 and PD weighted MR images, we will test them on three pairs of simulated T2-T1, PD-T2, and PD-T1 images of size 256 × 256 × 32. For all evaluated methods, the mean and the standard deviation (std) of TRE values as well as the *P* values for the *t*-test with the significance level α = 0.05 are computed and are shown in Table 1. In Table 1, "/" means that no registration is implemented. It is shown that all of the *P* values are less than 0.002, which indicates that there exists significant difference between the FMIND method and any other compared method in terms of TRE. Specifically, as regards the registration of T2-T1 images, the mean and the standard deviation of TRE values for the MIND method are 2.2 voxel and 0.5 voxel, respectively. By comparison, the FMIND method has the lower mean (1.8 voxel) and standard deviation (0.2 voxel) of TRE values than the MIND method. This is mainly due to the advantage of the proposed FMIND in describing the structural information of multi-modal MR images over the MIND method.


**Table 1.** The TRE for all evaluated methods and the *P* values for the *t*-test between the FMIND method and other compared methods operating on the T2-T1, PD-T2, and PD-T1 image pairs.

Figure 9 visually shows the registration results of 3D PD-T1 images for all the evaluated methods. Here, it should be noted that the background regions in these images are removed and the same operation will be implemented for other experiments in the rest of this paper. The comparison among Figures 9f and 9c–e shows that the registration result of the FMIND method is more similar to the reference image that is shown in Figure 9a than those of the ESSD, MIND, and HLCSO methods. Especially for the tissue indicated by the red boxes in Figure 9, the FMIND can recover its deformation better than other evaluated methods.

**Figure 9.** The registration results of all evaluated methods operating on 3D PD-T1 images. (**a**) PD image (reference image); (**b**) T1 image (float image); (**c**) ESSD; (**d**) MIND; (**e**) hybrid L-BFGS-B and cat swarm optimization (HLCSO); and, (**f**) FMIND.

Table 2 lists the implementation time of all evaluated methods on 3D T1, T2, and PD weighted MR images. It can be observed that the FMIND method, on average, takes approximately 44 min to produce the registration results, and it has the highest computational efficiency among all of the methods. The reason lies in that the FMIND method can generally reduce the number of deformation field variables by utilizing the FMIND based spatial constraint for MRF optimization.

**Table 2.** Computation time for all evaluated methods operating on the T2-T1, PD-T2, and PD-T1 image pairs.

