Joint Rigid Registration of Multiple Generalized Point Sets With Anisotropic Positional Uncertainties in Image-Guided Surgery

In medical image analysis (MIA) and computer-assisted surgery (CAS), aligning two multiple point sets (PSs) together is an essential but also a challenging problem. For example, rigidly aligning multiple point sets into one common coordinate frame is a prerequisite for statistical shape modelling (SSM). Accurately aligning the pre-operative space with the intra-operative space in CAS is very crucial to successful interventions. In this article, we formally formulate the multiple generalized point set registration problem (MGPSR) in a probabilistic manner, where both the positional and the normal vectors are used. The six-dimensional vectors consisting of both positional and normal vectors are called as generalized points. In the formulated model, all the generalized PSs to be registered are considered to be the realizations of underlying unknown hybrid mixture models (HMMs). By assuming the independence of the positional and orientational vectors (i.e., the normal vectors), the probability density function (PDF) of an observed generalized point is computed as the product of Gaussian and Fisher distributions. Furthermore, to consider the anisotropic noise in surgical navigation, the positional error is assumed to obey a multi-variate Gaussian distribution. Finally, registering PSs is formulated as a maximum likelihood (ML) problem, and solved under the expectation maximization (EM) technique. By using more enriched information (i.e., the normal vectors), our algorithm is more robust to outliers. By treating all PSs equally, our algorithm does not bias towards any PS. To validate the proposed approach, extensive experiments have been conducted on surface points extracted from CT images of (i) a human femur bone model; (ii) a human pelvis bone model. Results demonstrate our algorithm's high accuracy, robustness to noise and outliers

Spatial-temporal data mining procedure: LASR

This paper is concerned with the statistical development of our spatial-temporal data mining procedure, LASR (pronounced ``laser''). LASR is the abbreviation for Longitudinal Analysis with Self-Registration of large-$p$-small-$n$ data. It was motivated by a study of ``Neuromuscular Electrical Stimulation'' experiments, where the data are noisy and heterogeneous, might not align from one session to another, and involve a large number of multiple comparisons. The three main components of LASR are: (1) data segmentation for separating heterogeneous data and for distinguishing outliers, (2) automatic approaches for spatial and temporal data registration, and (3) statistical smoothing mapping for identifying ``activated'' regions based on false-discovery-rate controlled $p$-maps and movies. Each of the components is of interest in its own right. As a statistical ensemble, the idea of LASR is applicable to other types of spatial-temporal data sets beyond those from the NMES experiments.Comment: Published at http://dx.doi.org/10.1214/074921706000000707 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

GOGMA: Globally-Optimal Gaussian Mixture Alignment

Gaussian mixture alignment is a family of approaches that are frequently used for robustly solving the point-set registration problem. However, since they use local optimisation, they are susceptible to local minima and can only guarantee local optimality. Consequently, their accuracy is strongly dependent on the quality of the initialisation. This paper presents the first globally-optimal solution to the 3D rigid Gaussian mixture alignment problem under the L2 distance between mixtures. The algorithm, named GOGMA, employs a branch-and-bound approach to search the space of 3D rigid motions SE(3), guaranteeing global optimality regardless of the initialisation. The geometry of SE(3) was used to find novel upper and lower bounds for the objective function and local optimisation was integrated into the scheme to accelerate convergence without voiding the optimality guarantee. The evaluation empirically supported the optimality proof and showed that the method performed much more robustly on two challenging datasets than an existing globally-optimal registration solution.Comment: Manuscript in press 2016 IEEE Conference on Computer Vision and Pattern Recognitio