Geometry, Imaging and Computing

Volume 1 (2014)

Number 4

Detection of $n$-dimensional shape deformities using $n$-dimensional quasi-conformal maps

Pages: 395 – 415

DOI: http://dx.doi.org/10.4310/GIC.2014.v1.n4.a1

Authors

Hei Long Chan (Department of Mathematics, Chinese University of Hong Kong)

Lok Ming Lui (Department of Mathematics, Chinese University of Hong Kong)

Abstract

Detecting deformities on objects is a typical topic in shape analysis, and has much applications such as abnormalities detection in medical imaging (e.g. growth of tumor, spread of cancer). While many algorithms are already well-established in a 2-dimensional case when the object is indeed a surface, a model that still performs well in the general n-dimensional case is still missing. It is our goal in this paper to complete this missing piece, by introducing an indicator in order to effectively distinguish between normal and abnormal deformities. The proposed framework is closely related to the classic 2-dimensional conformal geometry and quasiconformal geometry. In this work, we model abnormal deformations by anisotropic deformations. Given any two objects of the same dimension (with landmark constraints in between), we define the Anisotropic Indicator, a locally defined real-valued function on the original object, which demonstrates the abnormalities in the deformation between them. Both global and local features about the abnormalities between the two objects can be tracked by analyzing the indicator. We tested the algorithm by detecting deformations on synthetic data and real data, and results show that our algorithm can detect deformations of different types and degrees.

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