### Abstract

Purpose: We hypothesize that all thoracic CBCT projections, if viewed as points in a high dimensional Euclidian space V, lie on the surface of a 2‐ torus structure. Understanding this structure helps many other projects, e.g. 4DCBCT phase sorting. This work will validate this hypothesis using a manifold learning algorithm. Methods: An NCAT phantom in the thorax region with a smooth respiratory motion pattern is used to generate a series of CBCT projections with realistic scanning and breathing parameters. A manifold learning technique, diffusion map, is employed to extract the topological structure underlying these projections. Specifically, a diffusion matrix is first constructed, whose entries are determined based on the Euclidian distance between those projections in V. The associated transition matrix is then computed and diagonalized. The eigenvectors, when sorted according to the associated eigenvalues in a descending order, represent the inherent topological structure of the manifold in a coarse‐to‐fine manner. The first three eigenvectors are selected as the coordinates to represent the CBCT projections, which reduces the dimensionality from the space V to a 3D space, while preserving the major topological properties. This also allows for a visualization of the hypothesized 2‐torus structure. Results: The points given by the first three eigenvectors form an oscillatory curve on a distorted 2‐torus in the 3D space. The two principle directions of the torus can be interpreted as the directions corresponding to the gantry rotation and the patient breathing, respectively. During CBCT projection acquisition, the gantry angle continuously increases, while the breathing phase oscillates periodically. Hence, those projections, while being represented by points in this 3D space, travel along a spiral curve winding around the torus. Conclusions: The manifold learning method can recover the 2‐torus structure of the thorax CBCT projections, whose two independent intrinsic dimensions correspond to gantry rotation and patient breathing.

Original language | English (US) |
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Pages (from-to) | 3903-3904 |

Number of pages | 2 |

Journal | Medical Physics |

Volume | 39 |

Issue number | 6 |

DOIs | |

State | Published - 2012 |

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### ASJC Scopus subject areas

- Biophysics
- Radiology Nuclear Medicine and imaging

### Cite this

*Medical Physics*,

*39*(6), 3903-3904. https://doi.org/10.1118/1.4735934