# Quantitative Assessment of Variational Surface Reconstruction from Sparse Point Clouds in Freehand 3D Ultrasound Imaging during Image-Guided Tumor Ablation

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## Abstract

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## 1. Introduction

#### 1.1. Freehand 3D Ultrasound Imaging

#### 1.2. Volume Reconstruction for Freehand 3D Ultrasound Imaging

#### 1.3. Surface Reconstruction for Freehand 3D Ultrasound Imaging

#### 1.4. Contribution and Structure of this Paper

## 2. Framework of Surface Reconstruction for Freehand 3D US Using VSR

^{1}-continuous, i.e., smooth.

## 3. Four Methods of Surface Reconstruction from Point Clouds

#### 3.1. VSR

#### 3.1.1. Constraint Definition

- Step 1:
- Begin with the first contour. Mark it as the current contour.
- Step 2:
- For each point ${c}_{i}$ of the current contour, assign a scalar value 0 to ${c}_{i}$. After the assignment, ${c}_{i}$ becomes an on-surface constraint ${c}_{i}{}^{\text{on}}$.
- Step 3:
- For each point ${c}_{i}$ of the current contour, calculate its corresponding off-surface constraint ${c}_{i}{}^{\text{off}}$ according to Equations (1) and (2).$${\mathit{c}}_{i}^{\text{off}}={\mathit{c}}_{i}^{N}={\mathit{c}}_{i}^{\text{on}}+{\mathit{n}}_{i}$$$${\mathit{n}}_{i}=\frac{{\mathit{n}}^{C}\times \left({\mathit{c}}_{i}^{\text{on}}-{\mathit{c}}_{i-1}^{\text{on}}\right)+{\mathit{n}}^{C}\times \left({\mathit{c}}_{i+1}^{\text{on}}-{\mathit{c}}_{i}^{\text{on}}\right)}{\Vert {\mathit{n}}^{C}\times \left({\mathit{c}}_{i}^{\text{on}}-{\mathit{c}}_{i-1}^{\text{on}}\right)+{\mathit{n}}^{C}\times \left({\mathit{c}}_{i+1}^{\text{on}}-{\mathit{c}}_{i}^{\text{on}}\right)\Vert}$$
- Step 4:
- When all the points of the current contour are finished, go to the next contour. Mark it as the current contour.
- Step 5:
- Loop from step 2 to step 4 until all contours are finished.

#### 3.1.2. Variational Interpolation

- Step 1:
- For any pair of constraints ${c}_{i}$ and ${c}_{j}$, calculate $\varphi ({c}_{i}-{c}_{j})={\Vert {c}_{i}-{c}_{j}\Vert}^{3}.$
- Step 2:
- After all pairs of ${c}_{i}$ and ${c}_{j}$ $(i=1\dots n,j=1\dots n)$ are calculated, form the coefficient matrix A in Equation (3):$$A=\left[\begin{array}{cccccccc}{\varphi}_{11}& {\varphi}_{12}& \dots & {\varphi}_{1n}& 1& {c}_{1}^{x}& {c}_{1}^{y}& {c}_{1}^{z}\\ {\varphi}_{21}& {\varphi}_{22}& \dots & {\varphi}_{2n}& 1& {c}_{2}^{x}& {c}_{2}^{y}& {c}_{2}^{z}\\ \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ {\varphi}_{n1}& {\varphi}_{n2}& \dots & {\varphi}_{nn}& 1& {c}_{n}^{x}& {c}_{n}^{y}& {c}_{n}^{z}\\ 1& 1& \dots & 1& 0& 0& 0& 0\\ {c}_{1}^{x}& {c}_{2}^{x}& \dots & {c}_{n}^{x}& 0& 0& 0& 0\\ {c}_{1}^{y}& {c}_{2}^{y}& \dots & {c}_{n}^{y}& 0& 0& 0& 0\\ {c}_{1}^{z}& {c}_{2}^{z}& \dots & {c}_{n}^{z}& 0& 0& 0& 0\end{array}\right]$$
- Step 3:
- Calculate vector B using Equation (4). Each ${h}_{i}$ denotes the corresponding value assigned to the constraint point ${c}_{i}$.$$B={\left[\begin{array}{cccccccc}{h}_{1}& {h}_{2}& \dots & {h}_{n}& 0& 0& 0& 0\end{array}\right]}^{T}$$
- Step 4:
- Solve Equation (5). The solution vector
**v**contains the weights d_{j}of $f(\mathit{x})$ and the coefficients of $P(\mathit{x})$.$$Av=B$$$$v={\left[\begin{array}{cccccccc}{d}_{1}& {d}_{2}& \dots & {d}_{n}& {p}_{0}& {p}_{1}& {p}_{2}& {p}_{3}\end{array}\right]}^{T}$$$$f(\mathit{x})={\displaystyle \sum _{j=1}^{n}{d}_{j}\varphi (\mathit{x}-{\mathit{c}}_{j})+P(\mathit{x})}$$$$P(\mathit{x})={p}_{0}+{p}_{1}x+{p}_{2}y+{p}_{3}z$$

#### 3.2. Ball Pivoting

#### 3.3. Power Crust

#### 3.4. Poisson Reconstruction

## 4. Quantitative Metrics for Assessment of Surface Reconstruction

- mean distance (mean)
- standard deviation from the mean distance (std)
- root mean square distance (rms)
- maximum distance (Hausdorff)
- medial distance (median).

## 5. Experiments

#### 5.1. Experiment 1—Reconstruction of Acorn

- In all cases, the surface reconstructed by the VSR method best visually resembles the original surface compared with the other methods. Most metrics show that the VSR-reconstructed surface shows fewer differences from the original surface compared with the other methods. The only exception is observed in volume difference for four contours, where the Poisson method produces a volume that is more similar to the original surface than that of the VSR method.
- For the other three methods, the quality of the reconstructed surfaces drops dramatically as the number of contours decreases (as seen for two or three contours in Figure 4 and Figure 5). However, as the number of contours decreases, both the quality and the Hausdorff distance for the VSR method show little variation. Even with only two contours, the VSR surface reconstruction closely approximates the original surface.
- Visually and quantitatively, the Poisson method performs the worst, except when reconstructed with two contours. Although Poisson and VSR are implicit-function-based methods, the Poisson method fails to reconstruct a satisfactory surface in all cases.
- Although the Power Crust method claims to be capable of producing watertight surfaces, it fails to do so when it is performed on two contours. This makes it the worst method for surface reconstruction with only two contours.
- As an intuitive and efficient method, BPA succeeds in constructing a triangle mesh for all cases. When the input data are dense (i.e., seven contours), it can produce a satisfactory surface of the acorn. However, its performance decreases dramatically as the number of contours decreases.

#### 5.2. Experiment 2—Reconstruction of Human Kidney

#### 5.3. Experiment 3—Reconstruction of Liver Tumor

#### 5.4. Experiment 4—Assessing the Reproducibility of Four Methods

- The contours used greatly affect the quality of the reconstructed surfaces by all four methods. Visually, the contours drawn by User 3 produce the best approximation to the original surface, and the contours drawn by User 4 produce the poorest. Figure 11 confirms this conclusion graphically. This result indicates that the sparse input contours should cover the key contours of the original surface to produce a close approximation to it.
- In all cases, the surface reconstructed by the VSR method best visually resembles the original surface compared with the other methods. All metrics show that the VSR-reconstructed surface shows fewer differences from the original surface compared with the other methods, which indicates that the reproducibility of the VSR method is the best of the four methods.

- The contours used greatly affect the quality of the reconstructed surfaces by all four methods. Visually, the contours drawn by User 3 produce the best approximation to the reconstructed surface from the contours drawn by User 1 using the BPA, Power Crust, and VSR methods. The contours drawn by User 4 produce the poorest approximation, especially when the Power Crust method is used; it fails to produce a surface. Figure 13 confirms this conclusion graphically. It should be noted that the Hausdorff distance of the Power Crust method is missing in Figure 13 since its value is much greater than that of the other methods.
- In all cases, the surface reconstructed by the VSR method best visually resembles the original surface compared with the other methods. All metrics show that the VSR-reconstructed surfaces show fewer differences from the surface reconstructed from the contours drawn by User 1 compared with the other methods, which indicates that the reproducibility of the VSR method is the best of four methods.

## 6. Discussion

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Freehand 3D US system. This is also the setup for the experiments. The spatial localizer is an electromagnetic tracking device called AURORA from Northern Digital Inc. (Waterloo, ON, Canada). IGS (image-guided surgery) software serves as the 3D surface reconstruction and visualization program.

**Figure 4.**Surface reconstruction of an acorn by four methods. For each row, from left to right: the contours used, the original surface, the surface reconstructed by BPA, the surface reconstructed by Power Crust, the surface reconstructed by Poisson, and the surface reconstructed by VSR.

**Figure 6.**Surface reconstruction of a human kidney by four methods. For each row, from left to right: the contours used, the original surface, the surface reconstructed by BPA, the surface reconstructed by Power Crust, the surface reconstructed by Poisson, and the surface reconstructed by VSR.

**Figure 7.**Hausdorff distance between the reconstructed surface and the original surface (HUMAN KIDNEY).

**Figure 8.**Surface reconstruction of one human liver tumor using four methods. For each row, from 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 9.**Surface reconstruction of another human liver tumor by four methods. For each row, from 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 10.**Surface reconstruction of a human kidney by four methods. The number of contours are the same in different rows, but the contours were drawn by different users: (

**a**) User 1; (

**b**) User 2; (

**c**) User 3; (

**d**) User 4. For each row, from left to right: the contours used, the original surface, the surface reconstructed by BPA, the surface reconstructed by Power Crust, the surface reconstructed by Poisson, and the surface reconstructed by VSR.

**Figure 11.**Hausdorff distances between the reconstructed surfaces and the original surface (HUMAN KIDNEY). The contours were drawn by four different users.

**Figure 12.**Surface reconstruction of a hepatic tumor by four methods. The numbers of contours are the same in different rows, but the contours were drawn by different users: (

**a**) User 1; (

**b**) User 2; (

**c**) User 3; (

**d**) User 4. For each row, from 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 13.**Hausdorff distances between the reconstructed surfaces from the contours drawn by Users 2, 3, and 4 and the reconstructed surface from the contours drawn by User 1 (LIVER TUMOR). The contours were drawn by four different users.

**Figure 14.**Surface reconstruction of contradictory contours of the kidney. (

**a**) Contradictory contour: the original horizontal contour (yellow) should intersect with the vertical contour (blue), but the edited version of the horizontal contour (red) did not intersect with the vertical contour (blue); (

**b**) surface reconstruction from the original contours; (

**c**) surface reconstruction from the contradictory contours. For row b and c, from left to right: the original surface of the kidney, the surface reconstructed by BPA, the surface reconstructed by Power Crust, the surface reconstructed by Poisson, and the surface reconstructed by VSR. (

**d**) Detail of the surface reconstruction near the intersection of the contradictory contours. From left to right: the original surface of the kidney and the contradictory contours, the surface reconstructed by BPA, the surface reconstructed by Power Crust, the surface reconstructed by Poisson, and the surface reconstructed by VSR.

Number of Cross Sections | Surface Reconstruction Method | Mean Distance | std | rms | Hausdorff Distance | Medial Distance | Area Difference (%) | Volume Difference (%) |
---|---|---|---|---|---|---|---|---|

2 | BPA | 2.6412 | 2.19942 | 3.43703 | 11.6783 | 2.16232 | −48 | −60 |

PowerCrust | 9.50837 | 6.57579 | 11.5607 | 23.4187 | 9.52517 | −55 | −81 | |

Poisson | 2.74878 | 2.24832 | 3.55114 | 11.1559 | 2.15936 | 99 | −35 | |

VSR | 0.710704 | 0.709221 | 1.00403 | 3.40886 | 0.477692 | −12 | −4 | |

3 | BPA | 1.65938 | 1.34633 | 2.13684 | 6.20571 | 1.36133 | −27 | −35 |

PowerCrust | 1.68578 | 1.57828 | 2.30927 | 6.71023 | 1.23402 | −12 | −32 | |

Poisson | 2.13187 | 1.64198 | 2.69089 | 7.98914 | 1.73299 | 73 | −77 | |

VSR | 0.926695 | 1.01773 | 1.37641 | 4.35121 | 0.537287 | −19 | −17 | |

4 | BPA | 1.41748 | 1.34832 | 1.95631 | 6.20571 | 1.02828 | −22 | −24 |

PowerCrust | 1.51101 | 1.66481 | 2.24826 | 6.71748 | 0.778636 | −17 | −26 | |

Poisson | 2.59704 | 2.0811 | 3.32798 | 9.51066 | 2.05565 | 45 | −6 | |

VSR | 0.876429 | 1.12105 | 1.42297 | 4.59235 | 0.350886 | −18 | −15 | |

5 | BPA | 0.888351 | 0.79243 | 1.19042 | 4.05483 | 0.658741 | −11 | −4 |

PowerCrust | 0.639664 | 0.675445 | 0.930259 | 3.68787 | 0.394742 | 2 | −9 | |

Poisson | 1.96806 | 1.64165 | 2.56285 | 7.70361 | 1.4991 | 67 | 30 | |

VSR | 0.411081 | 0.476312 | 0.629169 | 3.02812 | 0.227498 | −8 | −1 | |

6 | BPA | 0.865362 | 0.801298 | 1.17937 | 4.05483 | 0.608282 | −10 | −5 |

PowerCrust | 0.656822 | 0.805945 | 1.03968 | 4.19336 | 0.321317 | 1 | −13 | |

Poisson | 1.91001 | 1.51941 | 2.44062 | 7.74737 | 1.53498 | 79 | 43 | |

VSR | 0.456283 | 0.632145 | 0.779608 | 3.41585 | 0.191891 | −9 | −6 | |

7 | BPA | 0.876748 | 0.854095 | 1.22399 | 4.54538 | 0.561955 | −11 | −2 |

PowerCrust | 0.498742 | 0.591533 | 0.773721 | 4.13959 | 0.285986 | 5 | −11 | |

Poisson | 1.83339 | 1.63492 | 2.45646 | 7.77791 | 1.34775 | 98 | 26 | |

VSR | 0.274174 | 0.315076 | 0.417661 | 2.12882 | 0.154248 | −4 | −2 |

Number of Cross Sections | Surface Reconstruction Method | Mean Distance | std | rms | Hausdorff Distance | Medial Distance | Area Difference (%) | Volume Difference (%) |
---|---|---|---|---|---|---|---|---|

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 | |

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 | |

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 | |

8 | BPA | 1.73677 | 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 | 1.54515 | 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 | |

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 | |

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 | 1.25069 | 1.12206 | 1.67987 | 6.08752 | 0.949839 | −1.88 | 1.43 | |

VSR | 0.654974 | 1.08766 | 1.26917 | 7.19773 | 0.213162 | 2.03 | −3.20 | |

12 | BPA | 1.25462 | 1.86828 | 2.24967 | 12.2463 | 0.503546 | −2.84 | −5.09 |

Power Crust | 0.639903 | 0.820045 | 1.03984 | 4.96253 | 0.343724 | −2.80 | −4.90 | |

Poisson | 1.03552 | 1.01728 | 1.45125 | 6.10591 | 0.725931 | 0.12 | 3.43 | |

VSR | 0.538622 | 0.905817 | 1.05347 | 6.72458 | 0.181576 | 1.79 | −2.23 | |

14 | BPA | 1.29556 | 2.19755 | 2.55007 | 13.8442 | 0.457283 | −2.85 | −3.76 |

Power Crust | 0.5029 | 0.591473 | 0.776141 | 3.61966 | 0.289134 | −2.52 | −3.58 | |

Poisson | 0.883901 | 1.0258 | 1.35369 | 6.70781 | 0.554642 | −1.83 | 1.42 | |

VSR | 0.473663 | 0.758462 | 0.893891 | 6.0587 | 0.16908 | 1.91 | −1.43 |

**Table 3.**Difference between the reconstructed surface and the original surface (HUMAN KIDNEY; the contours were drawn by different users).

User | Surface Reconstruction Method | Mean Distance | std | rms | Hausdorff Distance | Medial Distance | Area Difference (%) | Volume Difference (%) |
---|---|---|---|---|---|---|---|---|

User 1 | 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 | |

User 2 | BPA | 4.48905 | 5.10272 | 6.79433 | 22.0837 | 2.09925 | −39.04 | −95.96 |

Power Crust | 3.2693 | 3.37085 | 4.69462 | 14.7158 | 1.9469 | −13.94 | −29.77 | |

Poisson | 3.61952 | 3.42344 | 4.98086 | 15.594 | 2.48503 | 37.66 | 26.67 | |

VSR | 1.99204 | 2.78775 | 3.42519 | 14.6415 | 0.946489 | −7.72 | −7.85 | |

User 3 | BPA | 4.40708 | 4.73455 | 6.4665 | 21.115 | 2.68107 | −50.46 | −71.71 |

Power Crust | 2.72485 | 2.63747 | 3.79131 | 11.5233 | 1.96596 | −10.82 | −23.71 | |

Poisson | 3.3612 | 2.8271 | 4.39114 | 13.3416 | 2.59396 | 47.83 | 22.53 | |

VSR | 1.34403 | 1.71604 | 2.17905 | 9.10527 | 0.616535 | −5.71 | −5.89 | |

User 4 | BPA | 4.91483 | 4.93414 | 6.96252 | 23.0959 | 3.22832 | −49.22 | −46.76 |

Power Crust | 2.8246 | 2.99256 | 4.11397 | 12.8419 | 1.85294 | −13.05 | −20.85 | |

Poisson | 6.73954 | 5.39803 | 8.63311 | 24.1895 | 5.40374 | 80.34 | −26.53 | |

VSR | 2.06118 | 2.59825 | 3.3155 | 12.6043 | 0.963157 | −10.40 | −11 |

**Table 4.**Difference between the reconstructed surfaces from the contours drawn by Users 2, 3, and 4 and the reconstructed surface from the contours drawn by User 1 (LIVER TUMOR).

User | Surface Reconstruction Method | Mean Distance | std | rms | Hausdorff Distance | Medial Distance | Area Difference (%) | Volume Difference (%) |
---|---|---|---|---|---|---|---|---|

User 2 | BPA | 0.606759 | 0.709767 | 0.932945 | 2.85121 | 0.304864 | −2.72 | 119.87 |

Power Crust | 0.565943 | 0.586972 | 0.814723 | 2.37748 | 0.383221 | 11.60 | −25.79 | |

Poisson | 1.04422 | 0.845751 | 1.34295 | 3.33231 | 0.737621 | 44.81 | 13.57 | |

VSR | 0.199237 | 0.253187 | 0.321874 | 1.31926 | 0.0916052 | 19.80 | -9 | |

User 3 | BPA | 0.179038 | 0.311993 | 0.359301 | 1.56732 | 0 | 6.94 | −94.72 |

Power Crust | 0.196016 | 0.31971 | 0.374599 | 1.2938 | 0.0102139 | 16.88 | −25.98 | |

Poisson | 0.66439 | 0.61721 | 0.906199 | 3.37821 | 0.492152 | 90.58 | 23.22 | |

VSR | 0.0703013 | 0.123795 | 0.142199 | 0.718006 | 0.019699 | 23.09 | −6 | |

User 4 | BPA | 0.542069 | 1.06084 | 1.18987 | 5.79188 | 0.153002 | 1.63 | 345.95 |

Power Crust | 1.68e + 009 | 3.75e + 009 | 4.10e + 009 | 1e + 010 | 13.5631 | −100 | −100 | |

Poisson | 1.85742 | 1.97787 | 2.71109 | 8.48791 | 0.996965 | 91.69 | 1140.05 | |

VSR | 0.166469 | 0.265851 | 0.313325 | 1.31816 | 0.0547988 | 18.85 | −11 |

**Table 5.**Difference between the reconstructed surfaces from contradictory contours and the original surface (HUMAN KIDNEY).

Case | Surface Reconstruction Method | Mean Distance | std | rms | Hausdorff Distance | Medial Distance | Area Difference (%) | Volume Difference (%) |
---|---|---|---|---|---|---|---|---|

Non-contradictory contours | BPA | 5.4036 | 4.3796 | 6.9542 | 18.9081 | 4.2757 | −48.10 | −86.11 |

Power Crust | 2.8995 | 2.9608 | 4.1430 | 14.0522 | 1.8748 | −12.56 | −26.25 | |

Poisson | 5.9510 | 5.2079 | 7.9063 | 23.448 | 4.3753 | 110.84 | −61.14 | |

VSR | 2.5736 | 2.9142 | 3.8868 | 14.539 | 1.3793 | −7.49 | −11 | |

Contradictory contours | BPA | 6.1428 | 6.0208 | 8.5992 | 29.3689 | 3.9117 | −54.21 | −89.94 |

Power Crust | 3.7181 | 3.3860 | 5.0277 | 15.5735 | 2.7963 | −17.79 | −35.19 | |

Poisson | 7.8823 | 6.5397 | 10.239 | 26.9386 | 5.9497 | 40.99 | −62.38 | |

VSR | 2.6500 | 2.8585 | 3.8968 | 14.6123 | 1.6135 | −8.74 | −13 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Deng, S.; Li, Y.; Jiang, L.; Liang, P. Quantitative Assessment of Variational Surface Reconstruction from Sparse Point Clouds in Freehand 3D Ultrasound Imaging during Image-Guided Tumor Ablation. *Appl. Sci.* **2016**, *6*, 114.
https://doi.org/10.3390/app6040114

**AMA Style**

Deng S, Li Y, Jiang L, Liang P. Quantitative Assessment of Variational Surface Reconstruction from Sparse Point Clouds in Freehand 3D Ultrasound Imaging during Image-Guided Tumor Ablation. *Applied Sciences*. 2016; 6(4):114.
https://doi.org/10.3390/app6040114

**Chicago/Turabian Style**

Deng, Shuangcheng, Yunhua Li, Lipei Jiang, and Ping Liang. 2016. "Quantitative Assessment of Variational Surface Reconstruction from Sparse Point Clouds in Freehand 3D Ultrasound Imaging during Image-Guided Tumor Ablation" *Applied Sciences* 6, no. 4: 114.
https://doi.org/10.3390/app6040114