## 4 Discussion and Summary  
In this paper, the motion tracking accuracy and computational efficiency of the three methods in the CPU serial computing environment and GPU parallel computing environment are compared and systematically analyzed. Under the CPU environment, the Lou-Konofagou method can substantially improve computational efficiency (by 95% or more). In contrast, using Luo-Konofagou method, the 3-D tracking can still be accelerated under the GPU parallel environment. However, the rate of improvements only ranged between 17% and 46% (see Figure 6). The rate of improvements obtained by the [PERSON] method was considerably less (7-23%; see Figure 6). In order to further improve the GPU-based 3-D motion tracking, one strategy is to enhance the memory access efficiency. That requires us to make full use of on-chip memory. Recall that, according to Nvidia's GPU specifications, the latency of on-chip memory is significantly better than that of off-chip memory. However, on-chip memory capacity is limited on GPUs. With the development of GPU hardware technology, the on-chip memory capacity may continue to increase. At the same time, the current GPU implementation can also be further optimized. In multi-instruction and multi-data (MIMD) mode, when one instruction is waiting for loading data, another segment of data can be used to perform some computing tasks at the same time. Therefore, the MIMD mode may further improve the computational efficiency of GPU implementation. We expect that with the improvements of GPU hardware, real-time 3-D ultrasound motion tracking may become a reality in the near future. As the tissue deformation increases, the required search range becomes inevitably large. Consequently, the creation of those sum-tables requires a longer time (see Appendix B). It is also found that the Lou-Konofagou method demands a high usage of memory (see Appendix C). Also, when the search range increases, the number of required sum-tables becomes bigger accordingly. In this sense, in order to deal with large tissue deformation, the Luo-Konofagou method can be used in conjunction with a multiple-compression tracking strategy [39; 40]. This is because the multi-compression method can effectively reduce the required search range at the expense of performing motion tracking multiple times. Alternatively, the Luo-Konofagou method can be used together with a 3-D region-growing motion tracking method [20]. The advantage of using a region-growing motion tracking method is that the search region would be fairly small. Both directions will be explored in our future work. Our preliminary results demonstrated that the Lewis method accelerated the 3-D motion tracking at a slow rate (see Figures 6 and 7) in both CPU and GPU computing environments. This is largely because the FFT requires a large number of calculations in order to estimate the numerator in Equation (1) regardless of the computing environment. \begin{table}
\begin{tabular}{c c} \hline \hline
**Implementation Method** &
\begin{tabular}{c} **Computing Time** \\ **(milliseconds)** \\ \end{tabular} \\ \hline Standard-NCC-CPU & 5760.0 \(\pm\) 9.3 \\ Luo-Konofagou-CPU & 168.5 \(\pm\) 2.2 \\ Lewis-CPU & 2120.0 \(\pm\) 5.5 \\ Standard-NCC-GPU & 278.1 \(\pm\) 0.9 \\ Luo-Konofagou-GPU & 159.9 \(\pm\) 0.4 \\ Lewis-GPU & 225.2 \(\pm\) 3.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: A summary of computing time (mean \(\pm\) one standard deviation) using 6 different implementations. 3D Displacement fields were first calculated for 30 slices. Then, the mean values (\(\pm\) one standard deviation) were derived and displayed below. The tracking kernel size and search region were \(69\times 9\times 3\) and \(18\times 5\times 3\), respectively. **Author Contributions:** All authors discussed the original idea.