A Liver Atlas Using the Special Euclidean Group
An atlas is a shape model derived using statistics of a population. Standard models treat local deformations as pure translations and apply linear statistics. They are often inadequate for highly variable anatomical shapes. Non-linear methods has been dev
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Queen’s University, Kingston, Ontario, Canada {hefny,ellis}@cs.queensu.ca 2 Tsukuba University, Tsukuba, Japan [email protected] 3 Osaka University, Osaka, Japan [email protected] Nara Institute of Science and Technology, Nara, Japan [email protected]
Abstract. An atlas is a shape model derived using statistics of a population. Standard models treat local deformations as pure translations and apply linear statistics. They are often inadequate for highly variable anatomical shapes. Non-linear methods has been developed but are generally difficult to implement. This paper proposes encoding shapes using the special Euclidean group SE(3) for model construction. SE(3) is a Lie group, so basic linear algebra can be used to analyze data in non-linear higher-dimensional spaces. This group represents non-linear shape variations by decomposing piecewiselocal deformations into rotational and translational components. The method was applied to 49 human liver models that were derived from CT scans. The atlas covered 99% of the population using only three components. Also, the method outperformed the standard method in reconstruction. Encoding shapes as ensembles of elements in the SE(3) group is a simple way of constructing compact shape models. Keywords: Statistical Shape Model · Special Euclidean Group · Lie Groups · Lie Algebras · Anatomical Atlas
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Introduction
Statistical shape models, or atlases, are shape descriptors, derived using statistical analysis on a population, that was introduced by Cootes et al. [2]. Atlases have been widely used in medical applications such as shape analysis, segmentation, and treatment planning. The concept is to reduce the dimensionality of a large dataset by capturing variations and removing redundancies. This is often performed using classical principal component analysis (PCA) [12], which assumes the shape space to be Euclidean so that pure translations will describe point-wise deformations. Although this might be a legitimate assumption in some datasets, it is not useful for highly variable human anatomy such as the liver.
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© Springer International Publishing Switzerland 2015 N. Navab et al. (Eds.): MICCAI 2015, Part II, LNCS 9350, pp. 238–245, 2015. DOI: 10.1007/978-3-319-24571-3_29
A Liver Atlas Using the Special Euclidean Group
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Many attempts have been made to construct compact and accurate shape models of the human liver. Lamecker et al. [13] divided the liver into four patches and encoded local distortions by mapping each patch to a disk. Davatzikos et al. [3] used wavelet transforms to construct hierarchical shape models. Feng et al. [5] used a multi-resolution method for model construction. Okada et al. [14] considered inter-organ relationships to handle large variations of shapes. Foruzan et al. [7] employed classification of samples to construct population-based models. These models adapted the original PCA formulation, with limited success. There has been some previous work on applying non-linear statistical analysis for s
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