## 5 Experiments
### Experiment 3--Reconstruction of Liver Tumor  
In Experiment 3, we reconstructed two human liver tumors from real ultrasound data. In this case, we did not have the original surface model for comparison. We could not perform a quantitative assessment of the reconstructed surface, and thus we inspected the visual differences instead. The visual differences between the reconstructed surfaces and the original surface are illustrated in Figures 8 and 9. As we did not have the original surface, the reconstructions are as follows, from  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Number of Cross Sections** & \begin{tabular}{c} **Surface** \\ **Reconstruction** \\ **Method** \\ \end{tabular}  & \begin{tabular}{c} **Mean** \\ **Distance** \\ \end{tabular}  & **std** & **rms** & \begin{tabular}{c} **Hausdorff** \\ **Distance** \\ \end{tabular}  & \begin{tabular}{c} **Medial** \\ **Distance** \\ \end{tabular}  & \ \begin{tabular}{c} **Area** \\ **Difference** \\ **(\%)** \\ \end{tabular}  & \begin{tabular}{c} **Volume** \\ **Difference** \\ **(\%)** \\ \end{tabular}  \\ \hline \multirow{4}{*}{3} & BPA & 5.40364 & 4.37961 & 6.9542 & 18.9081 & 4.27574 & \(-\)48.10 & \(-\)86.11 \\  & Power Crust & 2.89954 & 2.96081 & 4.14305 & 14.0522 & 1.87489 & \(-\)12.56 & \(-\)26.25 \\  & Poisson & 5.95107 & 5.20793 & 7.90635 & 23.448 & 4.37537 & 110.84 & \(-\)61.14 \\  & VSR & 2.57361 & 2.91418 & 3.88682 & 14.539 & 1.37939 & \(-\)7.49 & \(-\)11 \\ \hline \multirow{4}{*}{4} & BPA & 3.45712 & 3.51744 & 4.93067 & 17.8971 & 2.35204 & \(-\)26.14 & \(-\)54.27 \\  & Power Crust & 2.54163 & 2.37519 & 3.47789 & 9.96287 & 1.76471 & \(-\)9.91 & \(-\)17.72 \\  & Poisson & 6.06799 & 4.58929 & 7.60663 & 24.0824 & 5.02371 & 26.70 & 30.68 \\  & VSR & 1.7113 & 1.8289 & 2.504 & 8.05269 & 0.974841 & \(-\)6.73 & \(-\)6.71 \\ \hline \multirow{4}{*}{6} & BPA & 2.30932 & 2.67167 & 3.53038 & 13.9329 & 1.20211 & \(-\)14.57 & \(-\)34.66 \\  & Power Crust & 10.4462 & 12.8452 & 16.5517 & 42.5017 & 3.72379 & \(-\)39.17 & \(-\)52.38 \\  & Poisson & 2.80168 & 2.66679 & 3.86704 & 12.9421 & 2.04536 & 4.74 & 8.25 \\  & VSR & 1.08773 & 1.32213 & 1.71156 & 7.36376 & 0.530696 & \(-\)3.39 & \(-\)3.05 \\ \hline \multirow{4}{*}{8} & BPA & 1.73672 & 2.15556 & 2.76733 & 11.2383 & 0.88484 & \(-\)7.93 & \(-\)39.12 \\  & Power Crust & 1.32654 & 1.87959 & 2.29979 & 10.6975 & 0.663859 & \(-\)5.69 & \(-\)8.62 \\  & Poisson & 1.5415 & 1.47949 & 2.13873 & 9.09251 & 1.11002 & 0.37 & 3.42 \\  & VSR & 0.862839 & 1.15736 & 1.44313 & 7.16898 & 0.33444 & \(-\)2.70 & \(-\)2.65 \\ \hline \multirow{4}{*}{10} & BPA & 1.49347 & 2.22578 & 2.67947 & 13.3502 & 0.690223 & \(-\)3.80 & \(-\)7.80 \\  & Power Crust & 0.789766 & 0.929686 & 1.2195 & 5.72804 & 0.446035 & \(-\)3.83 & \(-\)4.75 \\  & Poisson & 1.33454 & 1.48037 & 1.99255 & 9.15823 & 0.822045 & 0.78 & 3.51 \\  & VSR & 0.675437 & 0.675437 & 1.0794 & 1.27285 & 7.16558 & 0.25 & \(-\)1.56 \\ \hline \multirow{4}{*}{11} & BPA & 1.29316 & 1.86346 & 2.26743 & 12.2463 & 0.553634 & \(-\)2.97 & \(-\)5.67 \\  & Power Crust & 0.682934 & 0.819252 & 1.06625 & 4.56298 & 0.375812 & \(-\)3.02 & \(-\)5.14 \\  & Poisson & 1.25609 & 1.12206 & 1.67987 & 6.08752 & 0.948939 & \(-\)1.88 & 1.43 \\  & VSR & 0.654974 & 1.08766 & 1.26917 & 7.19773 & 0.213162 & 2 left to right: the contours used, the surface reconstructed by BPA, the surface reconstructed by Power Crust, the surface reconstructed by Poisson, and the surface reconstructed by VSR. Figure 8 also indicated that VSR was the best compared to three other methods, similar to the results of Experiment 1 and Experiment 2. When two contours were used, the VSR method still reconstructed a satisfactory surface while the other three methods failed to create closed surfaces. In Figure 9, only two contours were used to reconstruct the surface of another hepatic tumor. Visually, this figure also indicated that VSR was the best method compared with the other methods, consistent with the results of the previous experiments. Figure 8: Surface reconstruction of one human liver tumor using four methods.