# Application of a new discreet form of Gauss’ theorem for measuring volume

Hughes, Stephen W., Arcy, Tom J. D., Maxwell, Daryl J., Saunders, J. E., Chiu, Wilson S. C., & Sheppard , Rod J.
(1996)
Application of a new discreet form of Gauss’ theorem for measuring volume.
*Physics in Medicine and Biology*, *41*, pp. 1809-1821.

## Abstract

Volume measurements are useful in many branches of science and medicine. They are usually accomplished by acquiring a sequence of cross sectional images through the object using an appropriate scanning modality, for example x-ray computed tomography (CT), magnetic resonance (MR) or ultrasound (US). In the cases of CT and MR, a dividing cubes algorithm can be used to describe the surface as a triangle mesh. However, such algorithms are not suitable for US data, especially when the image sequence is multiplanar (as it usually is). This problem may be overcome by manually tracing regions of interest (ROIs) on the registered multiplanar images and connecting the points into a triangular mesh. In this paper we describe and evaluate a new discreet form of Gauss’ theorem which enables the calculation of the volume of any enclosed surface described by a triangular mesh. The volume is calculated by summing the vector product of the centroid, area and normal of each surface triangle. The algorithm was tested on computer-generated objects, US-scanned balloons, livers and kidneys and CT-scanned clay rocks. The results, expressed as the mean percentage difference ± one standard deviation were 1.2 ± 2.3, 5.5 ± 4.7, 3.0 ± 3.2 and −1.2 ± 3.2% for balloons, livers, kidneys and rocks respectively. The results compare favourably with other volume estimation methods such as planimetry and tetrahedral decomposition.

Impact and interest:

**Citation counts** are sourced monthly from **Scopus** and **Web of Science®** citation databases.

These databases contain citations from different subsets of available publications and different time periods and thus the citation count from each is usually different. Some works are not in either database and no count is displayed. Scopus includes citations from articles published in 1996 onwards, and Web of Science® generally from 1980 onwards.

Citations counts from the **Google Scholar™** indexing service can be viewed at the linked Google Scholar™ search.

Full-text downloads:

**146**since deposited on 23 Apr 2012

**35**in the past twelve months

**Full-text downloads** displays the total number of times this work’s files (e.g., a PDF) have been downloaded from QUT ePrints as well as the number of downloads in the previous 365 days. The count includes downloads for all files if a work has more than one.

ID Code: | 49845 |
---|---|

Item Type: | Journal Article |

Additional URLs: | |

Keywords: | gauss theorem, volume, 3D, surface rendering, CT scanning, ultrasound, magnetic resonance, archimedes |

DOI: | 10.1088/0031-9155/41/9/016 |

ISSN: | 0031-9155 |

Subjects: | Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > NUMERICAL AND COMPUTATIONAL MATHEMATICS (010300) > Numerical and Computational Mathematics not elsewhere classified (010399) |

Divisions: | Current > Schools > School of Chemistry, Physics & Mechanical Engineering Current > QUT Faculties and Divisions > Science & Engineering Faculty |

Copyright Owner: | Copyright 1996 IOP Publishing Ltd |

Deposited On: | 23 Apr 2012 22:40 |

Last Modified: | 27 Apr 2012 04:23 |

Export: EndNote | Dublin Core | BibTeX

Repository Staff Only: item control page