# Computing the Surface Area of Three-Dimensional Scanned Human Data

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## Abstract

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## 1. Introduction

- We propose a simple and effective area computation method based on surface reconstruction for the body parts of 3D scanned human models.
- The area computed using the surface reconstruction method has a 95% similarity with that obtained by using the traditional alginate method.
- Our area computation method proves to be a possible substitute for the cumbersome alginate method.

## 2. Related Work

## 3. Computing the Surface Area of 3D Scanned Human Data

#### 3.1. Natural User Interface for Selecting the Region of Interest

#### 3.2. Smooth Surface Reconstruction

**Chart and transition function**: For each vertex of the selected region, we define a chart in the 2D complex plane. The chart shape is determined by the degree of a vertex. Figure 3 shows the charts ${U}_{i}$ and ${U}_{j}$ of two vertices with different degrees 6 and 3, respectively. As shown in Figure 3, adjacent charts share two regions and their correspondence is defined by a transition function ${\theta}_{ij}\left(z\right)$ as follows:

**Local Surface Patches**: For each chart ${U}_{i}$ of a vertex ${\mathbf{v}}_{i}$, we construct a 3D surface patch ${P}_{i}(u,v)$ approximating the 1-ring neighborhood of ${\mathbf{v}}_{i}$. We employ a biquadratic surface patch ${P}_{i}(u,v)$ defined as follows:

**Blending Surface**: We reconstruct a smooth surface by blending the local surface patches. For this, we need a blending function ${w}_{i}(u,v)$ on each chart ${U}_{i}$. To construct a blending function ${w}_{i}(u,v)$, we first construct a piece of blending function $\eta \left(u\right)\eta \left(v\right)$ on the unit square $[0,1]\times [0,1]$, where $\eta \left(t\right)=2{t}^{3}-3{t}^{2}+1$. We then apply conformal mapping to $\eta \left(u\right)\eta \left(v\right)$, followed by rotating and copying. Figure 5 shows the example of a blending function ${w}_{i}(u,v)$ on a chart of degree $k=6$. Note that blending functions ${w}_{i}(u,v)$ satisfy the partition of unit, ${\sum}_{\forall i}{w}_{i}(u,v)=1,$ on overlapping charts.

**Measuring Surface Area**: Now we can measure the surface area on a smooth blending surface rather than a polygon mesh as follows:

## 4. Experimental Results

^{®}Iris Pro Graphics 5200. In this section, we explain our experiment results of area computation and compare the results with those obtained by using alginate. We measure areas using alginate and compute areas using the proposed method from 8 subjects. Figure 10 shows a 3D scanned human model with different rendering options. We select 15 regions of interest to measure area: upper arms, lower arms, upper legs, lower legs, abdomen, back, pelvis, hips, head, face, and neck. Figure 11 shows examples of the selected regions of interest.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Measuring the surface area of a hand by using alginate [1].

**Figure 3.**Charts ${U}_{i}$ and ${U}_{j}$ and their transition function ${\theta}_{ij}\left(z\right)$.

**Figure 6.**(

**a**) Polygon approximations to a sphere of radius = 5 cm; (

**b**) blending surfaces reconstructed from (

**a**).

**Figure 7.**(

**a**) Polygon approximations to a hyperboloid ${x}^{2}+{y}^{2}-{z}^{2}=1$; (

**b**) blending surfaces reconstructed from (

**a**).

**Figure 8.**(

**a**) Polygon approximations to a torus of radii $r=1$ cm and $R=2$ cm; (

**b**) smooth blending surfaces reconstructed from (

**a**).

**Figure 9.**Comparison of errors of (

**a**) a sphere; (

**b**) a hyperboloid and (

**c**) a torus; (

**d**) ratios of polygon error to surface error.

**Figure 10.**A 3D scanned human model with different rendering options: (

**a**) skin texture; (

**b**) front view; (

**c**) back view; (

**d**) side view; (

**e**) wireframe.

**Figure 11.**Selected regions of interest: (

**a**) left upper arm; (

**b**) left lower arm; (

**c**) left upper leg; (

**d**) left lower leg; (

**e**) abdomen; (

**f**) back; (

**g**) pelvis; (

**h**) hips; (

**i**) head; (

**j**) face; (

**k**) neck.

**Figure 12.**Areas of body parts of eight people. The red broken line shows eight values of area obtained by using alginate and the blue line shows those obtained by surface reconstruction; (

**a**) areas of left upper arms; (

**b**) areas of right upper arms; (

**c**) areas of left lower arms; (

**d**) areas of right lower arms; (

**e**) areas of left upper legs; (

**f**) areas of right upper legs; (

**g**) areas of left lower legs; (

**h**) areas of right lower legs; (

**i**) areas of abdomens; (

**j**) areas of backs; (

**k**) areas of pelvises; (

**l**) areas of hips; (

**m**) areas of heads; (

**n**) areas of faces; (

**o**) areas of necks.

Cases | # of Triangles | Area (time) (a) | Area (time) (b) | Error (1) | Error (2) | (1)/(2) |
---|---|---|---|---|---|---|

1 | 60 | 272.46179 (0.03) | 293.46164 (3) | 41.69747 | 20.69763 | 2.01460 |

2 | 180 | 299.35513 (0.05) | 308.94577 (9) | 14.80413 | 5.21349 | 2.83958 |

3 | 420 | 307.64926 (0.06) | 312.20694 (22) | 6.510004 | 1.95233 | 3.33449 |

4 | 760 | 310.52105 (0.11) | 313.14139 (40) | 3.638208 | 1.01788 | 3.57431 |

5 | 1740 | 312.55352 (0.18) | 313.73544 (92) | 1.605738 | 0.42382 | 3.78871 |

Cases | # of Triangles | Area (time) (a) | Area (time) (b) | Error (1) | Error (2) | (1)/(2) |
---|---|---|---|---|---|---|

1 | 32 | 17.19809 (0.03) | 18.06601 (2) | 2.81723 | 1.94931 | 1.44524 |

2 | 162 | 19.39459 (0.05) | 19.68971 (8) | 0.62073 | 0.32561 | 1.90636 |

3 | 722 | 19.87309 (0.09) | 19.95923 (37) | 0.14223 | 0.05609 | 2.53575 |

4 | 1682 | 19.95392 (0.17) | 19.99632 (89) | 0.0614 | 0.019 | 3.23158 |

Cases | # of Triangles | Area (time) (a) | Area (time) (b) | Error (1) | Error (2) | (1)/(2) |
---|---|---|---|---|---|---|

1 | 50 | 62.64104 (0.04) | 71.86401 (3) | 16.31580 | 7.09283 | 2.30032 |

2 | 200 | 74.53550 (0.05) | 78.27505 (12) | 4.42134 | 0.68179 | 6.48495 |

3 | 800 | 77.82805 (0.1) | 78.87682 (45) | 1.12879 | 0.08002 | 14.10562 |

4 | 1800 | 78.45343 (0.18) | 78.92898 (98) | 0.50341 | 0.02786 | 18.06795 |

**Table 4.**Similarity and correlation between the results of alginate and the proposed surface reconstruction methods.

Region | Similarity | Correlation |
---|---|---|

left upper arm | 0.99232920 | 0.98651408 |

right upper arm | 0.99050425 | 0.97836923 |

left lower arm | 0.97442492 | 0.94239832 |

right lower arm | 0.97565553 | 0.88847152 |

left upper leg | 0.96904881 | 0.82351873 |

right upper leg | 0.97294687 | 0.91311208 |

left lower leg | 0.98809628 | 0.97643038 |

right lower leg | 0.99031974 | 0.98423465 |

abdomen | 0.98108957 | 0.97599424 |

back | 0.97219378 | 0.89756368 |

pelvis | 0.94844035 | 0.50870081 |

hips | 0.95367837 | 0.64129904 |

head | 0.96274736 | 0.63287971 |

neck | 0.97341437 | 0.88813431 |

face | 0.97505372 | 0.87872788 |

average | 0.94925100 | 0.75430084 |

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**MDPI and ACS Style**

Yoon, S.-H.; Lee, J.
Computing the Surface Area of Three-Dimensional Scanned Human Data. *Symmetry* **2016**, *8*, 67.
https://doi.org/10.3390/sym8070067

**AMA Style**

Yoon S-H, Lee J.
Computing the Surface Area of Three-Dimensional Scanned Human Data. *Symmetry*. 2016; 8(7):67.
https://doi.org/10.3390/sym8070067

**Chicago/Turabian Style**

Yoon, Seung-Hyun, and Jieun Lee.
2016. "Computing the Surface Area of Three-Dimensional Scanned Human Data" *Symmetry* 8, no. 7: 67.
https://doi.org/10.3390/sym8070067