Diffeomorphic image registration with applications to deformation modelling between multiple data sets

Over last years, the diffeomorphic image registration algorithms have been successfully introduced into the field of the medical image analysis. At the same time, the particular usability of these techniques, in majority derived from the solid mathematical background, has been only quantitatively explored for the limited applications such as longitudinal studies on treatment quality, or diseases progression. The thesis considers the deformable image registration algorithms, seeking out those that maintain the medical correctness of the estimated dense deformation fields in terms of the preservation of the object and its neighbourhood topology, offer the reasonable computational complexity to satisfy time restrictions coming from the potential applications, and are able to cope with low quality data typically encountered in Adaptive Radiotherapy (ART). The research has led to the main emphasis being laid on the diffeomorphic image registration to achieve one-to-one mapping between images. This involves introduction of the log-domain parameterisation of the deformation field by its approximation via a stationary velocity field. A quantitative and qualitative examination of existing and newly proposed algorithms for pairwise deformable image registration presented in this thesis, shows that the log-Euclidean parameterisation can be successfully utilised in the biomedical applications. Although algorithms utilising the log-domain parameterisation have theoretical justification for maintaining diffeomorphism, in general, the deformation fields produced by them have similar properties as these estimated by classical methods. Having this in mind, the best compromise in terms of the quality of the deformation fields has been found for the consistent image registration framework. The experimental results suggest also that the image registration with the symmetrical warping of the input images outperforms the classical approaches, and simultaneously can be easily introduced to most known algorithms. Furthermore, the log-domain implicit group-wise image registration is proposed. By linking the various sets of images related to the different subjects, the proposed image registration approach establishes a common subject space and between-subject correspondences therein. Although the correspondences between groups of images can be found by performing the classic image registration, the reference image selection (not required in the proposed implementation), may lead to a biased mean image being estimated and the corresponding common subject space not adequate to represent the general properties of the data sets. The approaches to diffeomorphic image registration have been also utilised as the principal elements for estimating the movements of the organs in the pelvic area based on the dense deformation field prediction system driven by the partial information coming from the specific type of the measurements parameterised using the implicit surface representation, and recognising facial expressions where the stationary velocity fields are used as the facial expression descriptors. Both applications have been extensively evaluated based on the real representative data sets of three-dimensional volumes and two-dimensional images, and the obtained results indicate the practical usability of the proposed techniques

Multiple Shape Registration using Constrained Optimal Control

Lagrangian particle formulations of the large deformation diffeomorphic metric mapping algorithm (LDDMM) only allow for the study of a single shape. In this paper, we introduce and discuss both a theoretical and practical setting for the simultaneous study of multiple shapes that are either stitched to one another or slide along a submanifold. The method is described within the optimal control formalism, and optimality conditions are given, together with the equations that are needed to implement augmented Lagrangian methods. Experimental results are provided for stitched and sliding surfaces

Gaussian Process Morphable Models

Statistical shape models (SSMs) represent a class of shapes as a normal distribution of point variations, whose parameters are estimated from example shapes. Principal component analysis (PCA) is applied to obtain a low-dimensional representation of the shape variation in terms of the leading principal components. In this paper, we propose a generalization of SSMs, called Gaussian Process Morphable Models (GPMMs). We model the shape variations with a Gaussian process, which we represent using the leading components of its Karhunen-Loeve expansion. To compute the expansion, we make use of an approximation scheme based on the Nystrom method. The resulting model can be seen as a continuous analogon of an SSM. However, while for SSMs the shape variation is restricted to the span of the example data, with GPMMs we can define the shape variation using any Gaussian process. For example, we can build shape models that correspond to classical spline models, and thus do not require any example data. Furthermore, Gaussian processes make it possible to combine different models. For example, an SSM can be extended with a spline model, to obtain a model that incorporates learned shape characteristics, but is flexible enough to explain shapes that cannot be represented by the SSM. We introduce a simple algorithm for fitting a GPMM to a surface or image. This results in a non-rigid registration approach, whose regularization properties are defined by a GPMM. We show how we can obtain different registration schemes,including methods for multi-scale, spatially-varying or hybrid registration, by constructing an appropriate GPMM. As our approach strictly separates modelling from the fitting process, this is all achieved without changes to the fitting algorithm. We show the applicability and versatility of GPMMs on a clinical use case, where the goal is the model-based segmentation of 3D forearm images

Reduction by Lie Group symmetries in diffeomorphic image registration and deformation modelling

We survey the role of reduction by symmetry in the large deformation diffeomorphic metric mapping framework for registration of a variety of data types (landmarks, curves, surfaces, images and higher-order derivative data). Particle relabelling symmetry allows the equations of motion to be reduced to the Lie algebra allowing the equations to be written purely in terms of the Eulerian velocity field. As a second use of symmetry, the infinite dimensional problem of finding correspondences between objects can be reduced for a range of concrete data types, resulting in compact representations of shape and spatial structure. Using reduction by symmetry, we describe these models in a common theoretical framework that draws on links between the registration problem and geometric mechanics. We outline these constructions and further cases where reduction by symmetry promises new approaches to the registration of complex data types

Information-Theoretic Registration with Explicit Reorientation of Diffusion-Weighted Images

We present an information-theoretic approach to the registration of images with directional information, and especially for diffusion-Weighted Images (DWI), with explicit optimization over the directional scale. We call it Locally Orderless Registration with Directions (LORD). We focus on normalized mutual information as a robust information-theoretic similarity measure for DWI. The framework is an extension of the LOR-DWI density-based hierarchical scale-space model that varies and optimizes the integration, spatial, directional, and intensity scales. As affine transformations are insufficient for inter-subject registration, we extend the model to non-rigid deformations. We illustrate that the proposed model deforms orientation distribution functions (ODFs) correctly and is capable of handling the classic complex challenges in DWI-registrations, such as the registration of fiber-crossings along with kissing, fanning, and interleaving fibers. Our experimental results clearly illustrate a novel promising regularizing effect, that comes from the nonlinear orientation-based cost function. We show the properties of the different image scales and, we show that including orientational information in our model makes the model better at retrieving deformations in contrast to standard scalar-based registration.Comment: 16 pages, 19 figure