Geodesic shape regression with multiple geometries and sparse parameters

International audienceMany problems in medicine are inherently dynamic processes which include the aspect of change over time, such as childhood development, aging, and disease progression. From medical images, numerous geometric structures can be extracted with various representations, such as landmarks, point clouds, curves, and surfaces. Different sources of geometry may characterize different aspects of the anatomy, such as fiber tracts from DTI and subcortical shapes from structural MRI, and therefore require a modeling scheme which can include various shape representations in any combination. In this paper, we present a geodesic regression model in the large deformation (LDDMM) framework applicable to multi-object complexes in a variety of shape representations. Our model decouples the deformation parameters from the specific shape representations, allowing the complexity of the model to reflect the nature of the shape changes, rather than the sampling of the data. As a consequence, the sparse representation of diffeomorphic flow allows for the straightforward embedding of a variety of geometry in different combinations, which all contribute towards the estimation of a single deformation of the ambient space. Additionally, the sparse representation along with the geodesic constraint results in a compact statistical model of shape change by a small number of parameters defined by the user. Experimental validation on multi-object complexes demonstrate robust model estimation across a variety of parameter settings. We further demonstrate the utility of our method to support the analysis of derived shape features, such as volume, and explore shape model extrapolation. Our method is freely available in the software package deformetrica which can be downloaded at www.deformetrica.org

The pore geometry of pharmaceutical coatings: statistical modelling, characterization methods and transport prediction

This thesis contains new methods for bridging the gap between the pore geometry of porous materials and experimentally measured functional properties. The focus has been on diffusive transport in pharmaceutical coatings used in controlled drug delivery systems, but the methods are general and can be applied to other porous materials and functional properties. Relatively large datasets are needed to train realistic models connecting the pore geometry and diffusive transport properties of porous materials. 3-D statistical pore models based on microscopy images of the coating material were in this thesis used to generate large sets of pore structures, in which diffusive transport was computed numerically. Characterization measures capturing important features of the pore geometry were developed and used as predictors of diffusive transport rates in multiplicative regression models. The characterization measures have been implemented in a freely available software, MIST.In Paper I, a Gaussian random field based pore model was developed and fitted to microscopy images of the coating material. Due to the large size of the data, the model was formulated using a Gaussian Markov random field approximation, which allows for efficient inference. A new method for solving linear equations with Kronecker matrices which reduced the complexity of the model fitting algorithm considerably was developed. The pore model was found to fit the microscopy images well. In Paper II, characterization measures that have been shown to provide good regression models for diffusive transport rates were developed further and implemented. Multiplicative regression models were fitted to pore structures from the model from Paper I, and the new methods were shown to give improved results. In Papers III and V characterization measures that capture a type of bottleneck effect which was observed in another set of microscopy images of the coating material (Papers III and IV), but which is not captured by existing methods, were invented. Pore structures with this type of bottleneck were generated using 3-D statistical pore models, and the new type of bottleneck was found to be an important determinant of diffusive transport rates when the regression models were fitted to simple pore structures (Paper V)

Model selection for spatiotemporal modeling of early childhood sub-cortical development

Spatiotemporal shape models capture the dynamics of shape change over time and are an essential tool for monitoring and measuring anatomical growth or degeneration. In this paper we evaluate non-parametric shape regression on the challenging problem of modeling early childhood sub-cortical development starting from birth. Due to the flexibility of the model, it can be challenging to choose parameters which lead to a good model fit yet does not overfit. We systematically test a variety of parameter settings to evaluate model fit as well as the sensitivity of the method to specific parameters, and we explore the impact of missing data on model estimation

MMGP: a Mesh Morphing Gaussian Process-based machine learning method for regression of physical problems under non-parameterized geometrical variability

When learning simulations for modeling physical phenomena in industrial designs, geometrical variabilities are of prime interest. While classical regression techniques prove effective for parameterized geometries, practical scenarios often involve the absence of shape parametrization during the inference stage, leaving us with only mesh discretizations as available data. Learning simulations from such mesh-based representations poses significant challenges, with recent advances relying heavily on deep graph neural networks to overcome the limitations of conventional machine learning approaches. Despite their promising results, graph neural networks exhibit certain drawbacks, including their dependency on extensive datasets and limitations in providing built-in predictive uncertainties or handling large meshes. In this work, we propose a machine learning method that do not rely on graph neural networks. Complex geometrical shapes and variations with fixed topology are dealt with using well-known mesh morphing onto a common support, combined with classical dimensionality reduction techniques and Gaussian processes. The proposed methodology can easily deal with large meshes without the need for explicit shape parameterization and provides crucial predictive uncertainties, which are essential for informed decision-making. In the considered numerical experiments, the proposed method is competitive with respect to existing graph neural networks, regarding training efficiency and accuracy of the predictions

De Casteljau's algorithm in geometric data analysis: Theory and application

For decades, de Casteljau's algorithm has been used as a fundamental building block in curve and surface design and has found a wide range of applications in fields such as scientific computing and discrete geometry, to name but a few. With increasing interest in nonlinear data science, its constructive approach has been shown to provide a principled way to generalize parametric smooth curves to manifolds. These curves have found remarkable new applications in the analysis of parameter-dependent, geometric data. This article provides a survey of the recent theoretical developments in this exciting area as well as its applications in fields such as geometric morphometrics and longitudinal data analysis in medicine, archaeology, and meteorology