### Abstract

In order to compare and integrate brain data more effectively, data from multiple subjects are typically mapped into a canonical space. One method to do this is to conformally map cortical surfaces to the sphere. It is well known that any genus zero Riemann surface can be mapped conformally to a sphere. Cortical surface is a genus zero surface. Therefore, conformal mapping offers a convenient method to parameterize cortical surfaces without angular distortion, generating an orthogonal grid on the cortex that locally preserves the metric. Although conformal mapping preserves the local geometry well, the important anatomical features, such as the sulci landmarks, are usually not aligned consistently. To compare cortical surfaces more effectively, it is advantageous to adjust the conformal parameterizations to match consistent anatomical features across subjects. This matching of cortical patterns improves the alignment of data across subjects, although it is more challenging to create a consistent conformal (orthogonal) parameterization of anatomy across subjects when landmarks are constrained to lie at specific locations in the spherical parameter space. Here we describe two methods to accomplish the task. The first approach is based on pursuing an optimal Möbius transformation to minimize the landmark mismatch error. The second approach is based on a new energy functional, to optimize the conformal parameterization of cortical surfaces by using landmarks. Experimental results on a dataset of 40 brain hemispheres showed that the landmark mismatch energy can be significantly reduced while effectively preserving conformality. The key advantage of these conformal parameterization approaches is that any local adjustments of the mapping to match landmarks do not affect the conformality of the mapping significantly. A detailed comparison between the two approaches will be discussed. The first approach can generate a map which is exactly conformal, although the landmark mismatch error is not reduced as effective as the second approach. The second approach can generate a map which significantly reduces the landmark mismatch error, but some conformality will be lost.

Original language | English (US) |
---|---|

Pages (from-to) | 847-858 |

Number of pages | 12 |

Journal | Applied Numerical Mathematics |

Volume | 57 |

Issue number | 5-7 SPEC. ISS. |

DOIs | |

State | Published - May 2007 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Applied Mathematics
- Computational Mathematics
- Modeling and Simulation

### Cite this

*Applied Numerical Mathematics*,

*57*(5-7 SPEC. ISS.), 847-858. https://doi.org/10.1016/j.apnum.2006.07.031

**Landmark constrained genus zero surface conformal mapping and its application to brain mapping research.** / Lui, Lok Ming; Wang, Yalin; Chan, Tony F.; Thompson, Paul.

Research output: Contribution to journal › Article

*Applied Numerical Mathematics*, vol. 57, no. 5-7 SPEC. ISS., pp. 847-858. https://doi.org/10.1016/j.apnum.2006.07.031

}

TY - JOUR

T1 - Landmark constrained genus zero surface conformal mapping and its application to brain mapping research

AU - Lui, Lok Ming

AU - Wang, Yalin

AU - Chan, Tony F.

AU - Thompson, Paul

PY - 2007/5

Y1 - 2007/5

N2 - In order to compare and integrate brain data more effectively, data from multiple subjects are typically mapped into a canonical space. One method to do this is to conformally map cortical surfaces to the sphere. It is well known that any genus zero Riemann surface can be mapped conformally to a sphere. Cortical surface is a genus zero surface. Therefore, conformal mapping offers a convenient method to parameterize cortical surfaces without angular distortion, generating an orthogonal grid on the cortex that locally preserves the metric. Although conformal mapping preserves the local geometry well, the important anatomical features, such as the sulci landmarks, are usually not aligned consistently. To compare cortical surfaces more effectively, it is advantageous to adjust the conformal parameterizations to match consistent anatomical features across subjects. This matching of cortical patterns improves the alignment of data across subjects, although it is more challenging to create a consistent conformal (orthogonal) parameterization of anatomy across subjects when landmarks are constrained to lie at specific locations in the spherical parameter space. Here we describe two methods to accomplish the task. The first approach is based on pursuing an optimal Möbius transformation to minimize the landmark mismatch error. The second approach is based on a new energy functional, to optimize the conformal parameterization of cortical surfaces by using landmarks. Experimental results on a dataset of 40 brain hemispheres showed that the landmark mismatch energy can be significantly reduced while effectively preserving conformality. The key advantage of these conformal parameterization approaches is that any local adjustments of the mapping to match landmarks do not affect the conformality of the mapping significantly. A detailed comparison between the two approaches will be discussed. The first approach can generate a map which is exactly conformal, although the landmark mismatch error is not reduced as effective as the second approach. The second approach can generate a map which significantly reduces the landmark mismatch error, but some conformality will be lost.

AB - In order to compare and integrate brain data more effectively, data from multiple subjects are typically mapped into a canonical space. One method to do this is to conformally map cortical surfaces to the sphere. It is well known that any genus zero Riemann surface can be mapped conformally to a sphere. Cortical surface is a genus zero surface. Therefore, conformal mapping offers a convenient method to parameterize cortical surfaces without angular distortion, generating an orthogonal grid on the cortex that locally preserves the metric. Although conformal mapping preserves the local geometry well, the important anatomical features, such as the sulci landmarks, are usually not aligned consistently. To compare cortical surfaces more effectively, it is advantageous to adjust the conformal parameterizations to match consistent anatomical features across subjects. This matching of cortical patterns improves the alignment of data across subjects, although it is more challenging to create a consistent conformal (orthogonal) parameterization of anatomy across subjects when landmarks are constrained to lie at specific locations in the spherical parameter space. Here we describe two methods to accomplish the task. The first approach is based on pursuing an optimal Möbius transformation to minimize the landmark mismatch error. The second approach is based on a new energy functional, to optimize the conformal parameterization of cortical surfaces by using landmarks. Experimental results on a dataset of 40 brain hemispheres showed that the landmark mismatch energy can be significantly reduced while effectively preserving conformality. The key advantage of these conformal parameterization approaches is that any local adjustments of the mapping to match landmarks do not affect the conformality of the mapping significantly. A detailed comparison between the two approaches will be discussed. The first approach can generate a map which is exactly conformal, although the landmark mismatch error is not reduced as effective as the second approach. The second approach can generate a map which significantly reduces the landmark mismatch error, but some conformality will be lost.

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UR - http://www.scopus.com/inward/citedby.url?scp=34147112466&partnerID=8YFLogxK

U2 - 10.1016/j.apnum.2006.07.031

DO - 10.1016/j.apnum.2006.07.031

M3 - Article

VL - 57

SP - 847

EP - 858

JO - Applied Numerical Mathematics

JF - Applied Numerical Mathematics

SN - 0168-9274

IS - 5-7 SPEC. ISS.

ER -