Group-wise similarity registration of point sets using Student’s t-mixture model for statistical shape models

Abstract

A probabilistic group-wise similarity registration technique based on Student's t-mixture model (TMM) and a multi-resolution extension of the same (mr-TMM) are proposed in this study, to robustly align shapes and establish valid correspondences, for the purpose of training statistical shape models (SSMs). Shape analysis across large cohorts requires automatic generation of the requisite training sets. Automated segmentation and landmarking of medical images often result in shapes with varying proportions of outliers and consequently require a robust method of alignment and correspondence estimation. Both TMM and mrTMM are validated by comparison with state-of-the-art registration algorithms based on Gaussian mixture models (GMMs), using both synthetic and clinical data. Four clinical data sets are used for validation: (a) 2D femoral heads (K= 1000 samples generated from DXA images of healthy subjects); (b) control-hippocampi (K= 50 samples generated from T1-weighted magnetic resonance (MR) images of healthy subjects); (c) MCI-hippocampi (K= 28 samples generated from MR images of patients diagnosed with mild cognitive impairment); and (d) heart shapes comprising left and right ventricular endocardium and epicardium (K= 30 samples generated from short-axis MR images of: 10 healthy subjects, 10 patients diagnosed with pulmonary hypertension and 10 diagnosed with hypertrophic cardiomyopathy). The proposed methods significantly outperformed the state-of-the-art in terms of registration accuracy in the experiments involving synthetic data, with mrTMM offering significant improvement over TMM. With the clinical data, both methods performed comparably to the state-of-the-art for the hippocampi and heart data sets, which contained few outliers. They outperformed the state-of-the-art for the femur data set, containing large proportions of outliers, in terms of alignment accuracy, and the quality of SSMs trained, quantified in terms of generalization, compactness and specificity.

Generalised coherent point drift for group-wise multi-dimensional analysis of diffusion brain MRI data

“…In equation 5a, T k represents the similarity transformation (comprising rotation R k , scaling s k and translation b k ), to align the positions m p j defining the mean template, to the k th sample in the group. In our recent work (Ravikumar et al, 2016), (Ravikumar et al, 2018), we showed that the form of Q to be maximised, to estimate the desired similarity transformations T k ∈ T and mixture component parameters Θ Q with respect to the weights, similarly to (Myronenko and Song, 2010). In…”

“…Previously, we proposed a group-wise rigid point set registration framework based on Student's t-mixture model (Ravikumar et al, 2016), (Ravikumar et al, 2018), which exploits the inherent robustness of Student's tdistribution for robust registration of shapes in the presence of missing data and significant proportions of outliers. Additionally, in a more recent study (Ravikumar et al, 2017) we proposed a variant of the hybrid mixture modelbased registration framework formulated in this study.…”

“…Group-wise point set registration using mixture models assumes that the point sets to be aligned are transformed observations of a central mixture model (which we refer to as the mean template) (Gooya et al, 2015). Consequently, the optimal transformations that align the template to the group of shapes are those that maximise the likelihood of the data (or equivalently, in the E-step and used to compute a set of corrected posterior probabilities P ⋆ kij , estimated identically to (Ravikumar et al, 2016), (Ravikumar et al, 2018) (refer to the Appendix); and (2) Student's t-distributions are defined by three parameters (as opposed to two for Gaussians). The additional parameter is referred to as the degrees of freedom/shape parameter ν, which is responsible for controlling the heaviness of the tails of the distribution (and consequently, the degree of robustness to outliers).…”

“…Closed form expressions are derived for the Mstep update equations of all TMM and transformation parameters, which are presented in the Appendix. Fibre orientations and FA are invariant to translation b k and scaling s k , consequently, these transformation parameters are estimated identically as in(Ravikumar et al, 2016),(Ravikumar et al, 2018). Although the former are rotationally dependent, the contribution of fibre orientations to the estimation of R k is ignored as the direction of the observed diffusion eigenvectors tend to be random.…”

“…Although the former are rotationally dependent, the contribution of fibre orientations to the estimation of R k is ignored as the direction of the observed diffusion eigenvectors tend to be random. Consequently, rotations R k are derived based on the spatial positions of hybrid point sets alone, by optimising the form of Q shown in equation 5b, similar to(Ravikumar et al, 2016),(Ravikumar et al, 2018). However, following estimation of the desired rotations R k at each EM-iteration, the current estimate of themean template is transformed by rotating both spatial positions m p j and their associated fibre orientations m d j , to align it with the k th sample in the group.…”