Bridge Simulation and Metric Estimation on Landmark Manifolds

We present an inference algorithm and connected Monte Carlo based estimation procedures for metric estimation from landmark configurations distributed according to the transition distribution of a Riemannian Brownian motion arising from the Large Deformation Diffeomorphic Metric Mapping (LDDMM) metric. The distribution possesses properties similar to the regular Euclidean normal distribution but its transition density is governed by a high-dimensional PDE with no closed-form solution in the nonlinear case. We show how the density can be numerically approximated by Monte Carlo sampling of conditioned Brownian bridges, and we use this to estimate parameters of the LDDMM kernel and thus the metric structure by maximum likelihood

Optical Flow on Moving Manifolds

Optical flow is a powerful tool for the study and analysis of motion in a sequence of images. In this article we study a Horn-Schunck type spatio-temporal regularization functional for image sequences that have a non-Euclidean, time varying image domain. To that end we construct a Riemannian metric that describes the deformation and structure of this evolving surface. The resulting functional can be seen as natural geometric generalization of previous work by Weickert and Schn\"orr (2001) and Lef\`evre and Baillet (2008) for static image domains. In this work we show the existence and wellposedness of the corresponding optical flow problem and derive necessary and sufficient optimality conditions. We demonstrate the functionality of our approach in a series of experiments using both synthetic and real data.Comment: 26 pages, 6 figure

Statistical M-Estimation and Consistency in Large Deformable Models for Image Warping

The problem of defining appropriate distances between shapes or images and modeling the variability of natural images by group transformations is at the heart of modern image analysis. A current trend is the study of probabilistic and statistical aspects of deformation models, and the development of consistent statistical procedure for the estimation of template images. In this paper, we consider a set of images randomly warped from a mean template which has to be recovered. For this, we define an appropriate statistical parametric model to generate random diffeomorphic deformations in two-dimensions. Then, we focus on the problem of estimating the mean pattern when the images are observed with noise. This problem is challenging both from a theoretical and a practical point of view. M-estimation theory enables us to build an estimator defined as a minimizer of a well-tailored empirical criterion. We prove the convergence of this estimator and propose a gradient descent algorithm to compute this M-estimator in practice. Simulations of template extraction and an application to image clustering and classification are also provided

Measuring Deformations and Illumination Changes in Images with Applications to Face Recognition

This thesis explores object deformation and lighting change in images, proposing methods that account for both variabilities within a single framework. We construct a deformation- and lighting-insensitive metric that assigns a cost to a pair of images based on their similarity. The primary applications discussed will be in the domain of face recognition, because faces provide a good and important example of highly structured yet deformable objects with readily available datasets. However, our methods can be applied to any domain with deformations and lighting change. In order to model variations in expression, establishing point correspondences between faces is essential, and a primary goal of this thesis is to determine dense correspondences between pairs of face images, assigning a cost to each point pairing based on a novel image metric. We show that an image manifold can be defined to model deformations and illumination changes. Images are considered as points on a high-dimensional manifold given local structure by our new metric, where costs are based on changes in shape and intensity. Curves on this manifold describe transformations such as deformations and lighting changes to connect nearby images, or larger identity changes connecting images far apart. This allows deformations to be introduced gradually over the course of several images, where correspondences are well-defined between every pair of adjacent images along a path. The similarity between two images on the manifold can be defined as the length of the geodesic that connects them. The new local metric is validated in an optical flow-like framework where it is used to determine a dense correspondence vector field between pairs of images. We then demonstrate how to find geodesics between pairs of images on a Riemannian image manifold. The new lighting-insensitive metric is described in the wavelet domain where it is able to handle moderate amounts of deformation, and allows us to derive an algorithm where the analytic geodesics between images can be computed extremely efficiently. To handle larger deformations in addition to changes in illumination, we consider an algorithmic framework where deformations are modeled with diffeomorphisms. We present preliminary implementations of the diffeomorphic framework, and suggest how this work can be extended for further applications

Image computing tools for the investigation of the neurological effects of preterm birth and corticosteroid administration

In this thesis we present a range of computational tools for medical imaging purposes within two main research projects. The first one is a methodological project oriented towards the improvement of the performance of a numerical computation utilised in diffeomorphic image registration. The second research project is a pre-clinical study aimed at the investigation of the effects of antenatal corticosteroids in a preterm rabbit animal model. In the first part we addressed the problem of integrating stationary velocity fields. This mathematical challenge had originated with early studies in fluid dynamics and had been subsequently mathematically formalised in the Lie group theory. Given a tangent velocity field defined in the tridimensional space as in input, the goal is to compute the position of the particles to which the velocity field is applied. This computation, also called numerical Lie exponential, is a fundamental component of several medical image registration algorithm based on diffeomorphisms, i.e. bijective differentiable maps with differentiable inverse. It is as well a widely utilised tool in computational anatomy to quantify the differences between two anatomical shapes measuring the parameters of the transformation that belongs to a metric vector space. The resulting new class of algorithms introduced in this thesis was created combining the known scaling and squaring algorithm with a class of numerical integrators aimed to solve systems of ordinary differential equations called exponential integrators. The introduced scaling and squaring based approximated exponential integrator algorithm have improved the computational time and accuracy respect to the state- of-the-art methods. The second part of the research is a pre-clinical trial carried forward in collab- oration with the Department of Development and Regeneration, Woman and Child Cluster at the KU Leuven University. The clinical research question is related to the understanding of the possible negative effects of administering antenatal cor- ticosteroids for preterm birth. To tackle this problem we designed and started a pre-clinical study using a New Zealand perinatal rabbit model. In this part of the research I was involved in the research team to provide the tools to automatise the data analysis and to eliminate the time consuming and non reproducible manual segmentation step. The main result of this collaboration is the creation of the first multi-modal multi-atlas for the newborn rabbit brain. This is embedded in a segmentation propagation and label fusion algorithm at the core of the proposed open-sourced automatic pipeline, having as input the native scanner format and as output the main MRI readouts, such as volume, fractional anisotropy and mean diffusivity

Learning-based Single-step Quantitative Susceptibility Mapping Reconstruction Without Brain Extraction

Quantitative susceptibility mapping (QSM) estimates the underlying tissue magnetic susceptibility from MRI gradient-echo phase signal and typically requires several processing steps. These steps involve phase unwrapping, brain volume extraction, background phase removal and solving an ill-posed inverse problem. The resulting susceptibility map is known to suffer from inaccuracy near the edges of the brain tissues, in part due to imperfect brain extraction, edge erosion of the brain tissue and the lack of phase measurement outside the brain. This inaccuracy has thus hindered the application of QSM for measuring the susceptibility of tissues near the brain edges, e.g., quantifying cortical layers and generating superficial venography. To address these challenges, we propose a learning-based QSM reconstruction method that directly estimates the magnetic susceptibility from total phase images without the need for brain extraction and background phase removal, referred to as autoQSM. The neural network has a modified U-net structure and is trained using QSM maps computed by a two-step QSM method. 209 healthy subjects with ages ranging from 11 to 82 years were employed for patch-wise network training. The network was validated on data dissimilar to the training data, e.g. in vivo mouse brain data and brains with lesions, which suggests that the network has generalized and learned the underlying mathematical relationship between magnetic field perturbation and magnetic susceptibility. AutoQSM was able to recover magnetic susceptibility of anatomical structures near the edges of the brain including the veins covering the cortical surface, spinal cord and nerve tracts near the mouse brain boundaries. The advantages of high-quality maps, no need for brain volume extraction and high reconstruction speed demonstrate its potential for future applications.Comment: 26 page

Efficient dense non-rigid registration using the free-form deformation framework

Medical image registration consists of finding spatial correspondences between two images or more. It is a powerful tool which is commonly used in various medical image processing tasks. Even though medical image registration has been an active topic of research for the last two decades, significant challenges in the field remain to be solved. This thesis addresses some of these challenges through extensions to the Free-Form Deformation (FFD) registration framework, which is one of the most widely used and well-established non-rigid registration algorithm. Medical image registration is a computationally expensive task because of the high degrees of freedom of the non-rigid transformations. In this work, the FFD algorithm has been re-factored to enable fast processing, while maintaining the accuracy of the results. In addition, parallel computing paradigms have been employed to provide near real-time image registration capabilities. Further modifications have been performed to improve the registration robustness to artifacts such as tissues non-uniformity. The plausibility of the generated deformation field has been improved through the use of bio-mechanical models based regularization. Additionally, diffeomorphic extensions to the algorithm were also developed. The work presented in this thesis has been extensively validated using brain magnetic resonance imaging of patients diagnosed with dementia or patients undergoing brain resection. It has also been applied to lung X-ray computed tomography and imaging of small animals. Alongside with this thesis an open-source package, NiftyReg, has been developed to release the presented work to the medical imaging community

Non-rigid multi-frame registration of cell nuclei in live cell microscopy image data

To gain a better understanding of cellular and molecular processes it is important to quantitatively analyze the motion of subcellular particles in live cell microscopy image sequences. For accurate quantification of the subcellular particle motion, compensation of the motion and deformation of the cell nucleus is required. This thesis deals with non-rigid registration of cell nuclei in 2D and 3D live cell fluorescence microscopy images. We developed two multi-frame non-rigid registration approaches which simultaneously exploit information from multiple consecutive frames of an image sequence to improve the registration accuracy. The multi-frame registration approaches are based on local optic flow estimation, use information from multiple consecutive images, and take into account computed transformations from previous time steps. The first approach comprises three intensity-based variants and two different temporal weighting schemes. The second approach determines diffeomorphic transformations in the log-domain which allows efficient computation of the inverse transformations. We use a temporally weighted mean image which is constructed based on inverse transformations and multiple consecutive frames. In addition, we employ a flow boundary preserving method for regularization of computed deformation vector fields. Both multi-frame registration approaches have been successfully applied to 2D and 3D synthetic as well as real live cell microscopy image sequences. We have performed an extensive quantitative evaluation of our approaches and compared their performance with previous non-rigid pairwise, multi-frame, and temporal groupwise registration approaches

On Motion Parameterizations in Image Sequences from Fixed Viewpoints

This dissertation addresses the problem of parameterizing object motion within a set of images taken with a stationary camera. We develop data-driven methods across all image scales: characterizing motion observed at the scale of individual pixels, along extended structures such as roads, and whole image deformations such as lungs deforming over time. The primary contributions include: a) fundamental studies of the relationship between spatio-temporal image derivatives accumulated at a pixel, and the object motions at that pixel,: b) data driven approaches to parameterize breath motion and reconstruct lung CT data volumes, and: c) defining and offering initial results for a new class of Partially Unsupervised Manifold Learning: PUML) problems, which often arise in medical imagery. Specifically, we create energy functions for measuring how consistent a given velocity vector is with observed spatio-temporal image derivatives. These energy functions are used to fit parametric snake models to roads using velocity constraints. We create an automatic data-driven technique for finding the breath phase of lung CT scans which is able to replace external belt measurements currently in use clinically. This approach is extended to automatically create a full deformation model of a CT lung volume during breathing or heart MRI during breathing and heartbeat. Additionally, motivated by real use cases, we address a scenario in which a dataset is collected along with meta-data which describes some, but not all, aspects of the dataset. We create an embedding which displays the remaining variability in a dataset after accounting for variability related to the meta-data