Bayesian alignment of similarity shapes

Abstract

We develop a Bayesian model for the alignment of two point configurations under the full similarity transformations of rotation, translation and scaling. Other work in this area has concentrated on rigid body transformations, where scale information is preserved, motivated by problems involving molecular data; this is known as form analysis. We concentrate on a Bayesian formulation for statistical shape analysis. We generalize the model introduced by Green and Mardia for the pairwise alignment of two unlabeled configurations to full similarity transformations by introducing a scaling factor to the model. The generalization is not straight-forward, since the model needs to be reformulated to give good performance when scaling is included. We illustrate our method on the alignment of rat growth profiles and a novel application to the alignment of protein domains. Here, scaling is applied to secondary structure elements when comparing protein folds; additionally, we find that one global scaling factor is not in general sufficient to model these data and, hence, we develop a model in which multiple scale factors can be included to handle different scalings of shape components.

Size and Shape Analysis of Error-Prone Shape Data

“…This raises concerns especially when the size of objects is the focal point of a study, such as in the study of rat skulls' growth described in Section 1, which was recently revisited by Mardia et al (2013) who carried out a Bayesian analysis with size being a key concept in their investigation. In the upcoming section, we derive unbiased score functions for the measurement error models formulated in Section 2, followed by estimating equations based on these scores.…”

“…A detailed description of the data of rat skulls brought up in Section 1 is given in Bookstein (1991). Besides Kenobi, Dryden, and Le (2010) and Mardia et al (2013), many researchers have analyzed the data assuming the recorded skull configurations are precise (Goodall and Lange 1989;Monteiro 1999;Le and Kume 2000;. For illustration purposes, we assume that the skull configurations, each consisting of eight landmarks, are contaminated with measurement error.…”

“…A natural question is whether or not this naive matching can reveal the same information regarding size and shape comparisons as when one matches the error-free configurations. For example, in the study of rat skulls (Bookstein 1991;Kenobi, Dryden, and Le 2010;Mardia et al 2013), X-rays of rat skulls were recorded from age 7 days to 150 days, and it is of interest to estimate the change in size and shape of the skull as a rat grows. Assuming configurations of skulls are measured precisely, if one implements OPA to match the skull of a rat recorded at an earlier time onto the other skull of the same rat recorded later, then the amount of scaling entailed in OPA reflects the amount of growth of the skull during this time window.…”

Bayesian Protein Sequence and Structure Alignment

“…For instance, Green and Mardia () considered rigid body transformations in their applications, but Mardia et al . () demonstrated applications using full similarity transformations. Forbes et al .…”

“…The underlying formulation is very flexible. For instance, Green and Mardia (2006) consid-ered rigid body transformations in their applications, but Mardia et al (2013) demonstrated applications using full similarity transformations. Forbes et al (2014) also used this approach with similarity transformations in the context of fingerprint matching.…”