Geodesic boundary value problems with symmetry

This paper shows how left and right actions of Lie groups on a manifold may be used to complement one another in a variational reformulation of optimal control problems equivalently as geodesic boundary value problems with symmetry. We prove an equivalence theorem to this effect and illustrate it with several examples. In finite-dimensions, we discuss geodesic flows on the Lie groups SO(3) and SE(3) under the left and right actions of their respective Lie algebras. In an infinite-dimensional example, we discuss optimal large-deformation matching of one closed curve to another embedded in the same plane. In the curve-matching example, the manifold \Emb(S^1, \mathbb{R}^2) comprises the space of closed curves $S^1$ embedded in the plane $\mathbb{R}^2$. The diffeomorphic left action \Diff(\mathbb{R}^2) deforms the curve by a smooth invertible time-dependent transformation of the coordinate system in which it is embedded, while leaving the parameterisation of the curve invariant. The diffeomorphic right action \Diff(S^1) corresponds to a smooth invertible reparameterisation of the $S^1$ domain coordinates of the curve. As we show, this right action unlocks an important degree of freedom for geodesically matching the curve shapes using an equivalent fixed boundary value problem, without being constrained to match corresponding points along the template and target curves at the endpoint in time.Comment: First version -- comments welcome

Visual analytics methods for shape analysis of biomedical images exemplified on rodent skull morphology

In morphometrics and its application fields like medicine and biology experts are interested in causal relations of variation in organismic shape to phylogenetic, ecological, geographical, epidemiological or disease factors - or put more succinctly by Fred L. Bookstein, morphometrics is "the study of covariances of biological form". In order to reveal causes for shape variability, targeted statistical analysis correlating shape features against external and internal factors is necessary but due to the complexity of the problem often not feasible in an automated way. Therefore, a visual analytics approach is proposed in this thesis that couples interactive visualizations with automated statistical analyses in order to stimulate generation and qualitative assessment of hypotheses on relevant shape features and their potentially affecting factors. To this end long established morphometric techniques are combined with recent shape modeling approaches from geometry processing and medical imaging, leading to novel visual analytics methods for shape analysis. When used in concert these methods facilitate targeted analysis of characteristic shape differences between groups, co-variation between different structures on the same anatomy and correlation of shape to extrinsic attributes. Here a special focus is put on accurate modeling and interactive rendering of image deformations at high spatial resolution, because that allows for faithful representation and communication of diminutive shape features, large shape differences and volumetric structures. The utility of the presented methods is demonstrated in case studies conducted together with a collaborating morphometrics expert. As exemplary model structure serves the rodent skull and its mandible that are assessed via computed tomography scans

Accurate, Fast and Controllable Image and Point Cloud Registration

Registration is the process of establishing spatial correspondences between two objects. Many downstream tasks, e.g, in image analysis, shape animation, can make use of these spatial correspondences. A variety of registration approaches have been developed over the last decades, but only recently registration approaches have been developed that make use of and can easily process the large data samples of the big data era. On the one hand, traditional optimization-based approaches are too slow and cannot take advantage of very large data sets. On the other hand, registration users expect more controllable and accurate solutions since most downstream tasks, e.g., facial animation and 3D reconstruction, increasingly rely on highly precise spatial correspondences. In recent years, deep network registration approaches have become popular as learning-based approaches are fast and can benefit from large-scale data during network training. However, how to make such deep-learning-based approached accurate and controllable is still a challenging problem that is far from being completely solved. This thesis explores fast, accurate and controllable solutions for image and point cloud registration. Specifically, for image registration, we first improve the accuracy of deep-learning-based approaches by introducing a general framework that consists of affine and non-parametric registration for both global and local deformation. We then design a more controllable image registration approach that image regions could be regularized differently according to their local attributes. For point cloud registration, existing works either are limited to small-scale problems, hardly handle complicated transformations or are slow to solve. We thus develop fast, accurate and controllable solutions for large-scale real-world registration problems via integrating optimal transport with deep geometric learning.Doctor of Philosoph

Evaluation of Free Form Deformation and Demons Registration with Discontinuities

Medical image registration plays an important part in most today’s clinical procedures. Registration goal is to find transformation which warps one image into the space of another. Registration of moving organs in human body has a significant part in therapy planning. This task is harder in cases when one organ (tissue) slides along another, i.e. in a case of discontinuities in the motion field. Discontinuities introduce unwanted transformations which often lead to poor or unsatisfied registration results. In this paper we evaluate one form of discontinuities for two well-known and used registration algorithms namely Free Form Deformation and Demons

Smooth representation of thin shells and volume structures for isogeometric analysis

The purpose of this study is to develop self-contained methods for obtaining smooth meshes which are compatible with isogeometric analysis (IGA). The study contains three main parts. We start by developing a better understanding of shapes and splines through the study of an image-related problem. Then we proceed towards obtaining smooth volumetric meshes of the given voxel-based images. Finally, we treat the smoothness issue on the multi-patch domains with C1 coupling. Following are the highlights of each part. First, we present a B-spline convolution method for boundary representation of voxel-based images. We adopt the filtering technique to compute the B-spline coefficients and gradients of the images effectively. We then implement the B-spline convolution for developing a non-rigid images registration method. The proposed method is in some sense of “isoparametric”, for which all the computation is done within the B-splines framework. Particularly, updating the images by using B-spline composition promote smooth transformation map between the images. We show the possible medical applications of our method by applying it for registration of brain images. Secondly, we develop a self-contained volumetric parametrization method based on the B-splines boundary representation. We aim to convert a given voxel-based data to a matching C1 representation with hierarchical cubic splines. The concept of the osculating circle is employed to enhance the geometric approximation, where it is done by a single template and linear transformations (scaling, translations, and rotations) without the need for solving an optimization problem. Moreover, we use the Laplacian smoothing and refinement techniques to avoid irregular meshes and to improve mesh quality. We show with several examples that the method is capable of handling complex 2D and 3D configurations. In particular, we parametrize the 3D Stanford bunny which contains irregular shapes and voids. Finally, we propose the B´ezier ordinates approach and splines approach for C1 coupling. In the first approach, the new basis functions are defined in terms of the B´ezier Bernstein polynomials. For the second approach, the new basis is defined as a linear combination of C0 basis functions. The methods are not limited to planar or bilinear mappings. They allow the modeling of solutions to fourth order partial differential equations (PDEs) on complex geometric domains, provided that the given patches are G1 continuous. Both methods have their advantages. In particular, the B´ezier approach offer more degree of freedoms, while the spline approach is more computationally efficient. In addition, we proposed partial degree elevation to overcome the C1-locking issue caused by the over constraining of the solution space. We demonstrate the potential of the resulting C1 basis functions for application in IGA which involve fourth order PDEs such as those appearing in Kirchhoff-Love shell models, Cahn-Hilliard phase field application, and biharmonic problems

Proceedings of the First International Workshop on Mathematical Foundations of Computational Anatomy (MFCA'06) - Geometrical and Statistical Methods for Modelling Biological Shape Variability

International audienceNon-linear registration and shape analysis are well developed research topic in the medical image analysis community. There is nowadays a growing number of methods that can faithfully deal with the underlying biomechanical behaviour of intra-subject shape deformations. However, it is more difficult to relate the anatomical shape of different subjects. The goal of computational anatomy is to analyse and to statistically model this specific type of geometrical information. In the absence of any justified physical model, a natural attitude is to explore very general mathematical methods, for instance diffeomorphisms. However, working with such infinite dimensional space raises some deep computational and mathematical problems. In particular, one of the key problem is to do statistics. Likewise, modelling the variability of surfaces leads to rely on shape spaces that are much more complex than for curves. To cope with these, different methodological and computational frameworks have been proposed. The goal of the workshop was to foster interactions between researchers investigating the combination of geometry and statistics for modelling biological shape variability from image and surfaces. A special emphasis was put on theoretical developments, applications and results being welcomed as illustrations. Contributions were solicited in the following areas: * Riemannian and group theoretical methods on non-linear transformation spaces * Advanced statistics on deformations and shapes * Metrics for computational anatomy * Geometry and statistics of surfaces 26 submissions of very high quality were recieved and were reviewed by two members of the programm committee. 12 papers were finally selected for oral presentations and 8 for poster presentations. 16 of these papers are published in these proceedings, and 4 papers are published in the proceedings of MICCAI'06 (for copyright reasons, only extended abstracts are provided here)

Geodesic shooting for anatomical curve registration on the plane

The aim of the work presented in this thesis is to develop a method of characterising the shape of curves in the plane that is independent of the parameterisation of the curve. It is important to remove the effect of a specific parameterisation of a curve because it is possible for two curves to have the same shape while having different parameterisations. The characterisation is accomplished by matching curves via deformations, and using the deformations to characterise the difference between them. We specifically aim for a method that is able to characterise the kind of complex curves found in cross sections of the human nasal cavity. In order to match one curve to another, we derive the equations of motion for a geodesic flow, and seeking the flow that deforms an embedded reference curve into the target curve we wish to characterise. The geodesic flow is itself characterised by a conjugate momentum on, and normal to, the reference curve, giving a one dimensional descriptive signal of the deformation. This descriptive signal contains all of the information required to generate the target curve from the reference curve. We therefore say that this descriptive signal characterises the target curve with respect to the reference curve. The descriptive signal is found using a shooting approach, requiring a functional to measure how closely overlaid are two curves. Formulating the problem as an optimisation problem, we first present a parameterisation-independent functional based on geometric currents, but show that we encounter problems in this matching functional due to local minima. We then present a second approach in which we formulate the problem as a landmark matching problem. Since we seek a characterisation that is independent of the choice of landmarks, and the landmark matching functional is parameterisation dependent, we minimise the functional over all reparameterisations of the reference curve. These two approaches solve equivalent problems. We present the results of the reparameterisation-based matching, and show that they overcome the problems observed in the currents-based method. In particular we demonstrate that the method is able to match complex nasal geometries, and show how the descriptive signal can be used to interpolate between two dimensional slices of three dimensional objects to reconstruct three dimensional surfaces representing the objects. Though here we implement the geodesic flow in two dimensions, we note that the flow could be extended to three dimensional space. Since the reparameterisation based matching functional is trivial to implement in three dimensions, this would allow for the characterisation of both curves and surfaces in three dimensional space

ADVANCED MOTION MODELS FOR RIGID AND DEFORMABLE REGISTRATION IN IMAGE-GUIDED INTERVENTIONS

Image-guided surgery (IGS) has been a major area of interest in recent decades that continues to transform surgical interventions and enable safer, less invasive procedures. In the preoperative contexts, diagnostic imaging, including computed tomography (CT) and magnetic resonance (MR) imaging, offers a basis for surgical planning (e.g., definition of target, adjacent anatomy, and the surgical path or trajectory to the target). At the intraoperative stage, such preoperative images and the associated planning information are registered to intraoperative coordinates via a navigation system to enable visualization of (tracked) instrumentation relative to preoperative images. A major limitation to such an approach is that motions during surgery, either rigid motions of bones manipulated during orthopaedic surgery or brain soft-tissue deformation in neurosurgery, are not captured, diminishing the accuracy of navigation systems. This dissertation seeks to use intraoperative images (e.g., x-ray fluoroscopy and cone-beam CT) to provide more up-to-date anatomical context that properly reflects the state of the patient during interventions to improve the performance of IGS. Advanced motion models for inter-modality image registration are developed to improve the accuracy of both preoperative planning and intraoperative guidance for applications in orthopaedic pelvic trauma surgery and minimally invasive intracranial neurosurgery. Image registration algorithms are developed with increasing complexity of motion that can be accommodated (single-body rigid, multi-body rigid, and deformable) and increasing complexity of registration models (statistical models, physics-based models, and deep learning-based models). For orthopaedic pelvic trauma surgery, the dissertation includes work encompassing: (i) a series of statistical models to model shape and pose variations of one or more pelvic bones and an atlas of trajectory annotations; (ii) frameworks for automatic segmentation via registration of the statistical models to preoperative CT and planning of fixation trajectories and dislocation / fracture reduction; and (iii) 3D-2D guidance using intraoperative fluoroscopy. For intracranial neurosurgery, the dissertation includes three inter-modality deformable registrations using physic-based Demons and deep learning models for CT-guided and CBCT-guided procedures

3-D lung deformation and function from respiratory-gated 4-D x-ray CT images : application to radiation treatment planning.

Many lung diseases or injuries can cause biomechanical or material property changes that can alter lung function. While the mechanical changes associated with the change of the material properties originate at a regional level, they remain largely asymptomatic and are invisible to global measures of lung function until they have advanced significantly and have aggregated. In the realm of external beam radiation therapy of patients suffering from lung cancer, determination of patterns of pre- and post-treatment motion, and measures of regional and global lung elasticity and function are clinically relevant. In this dissertation, we demonstrate that 4-D CT derived ventilation images, including mechanical strain, provide an accurate and physiologically relevant assessment of regional pulmonary function which may be incorporated into the treatment planning process. Our contributions are as follows: (i) A new volumetric deformable image registration technique based on 3-D optical flow (MOFID) has been designed and implemented which permits the possibility of enforcing physical constraints on the numerical solutions for computing motion field from respiratory-gated 4-D CT thoracic images. The proposed optical flow framework is an accurate motion model for the thoracic CT registration problem. (ii) A large displacement landmark-base elastic registration method has been devised for thoracic CT volumetric image sets containing large deformations or changes, as encountered for example in registration of pre-treatment and post-treatment images or multi-modality registration. (iii) Based on deformation maps from MOFIO, a novel framework for regional quantification of mechanical strain as an index of lung functionality has been formulated for measurement of regional pulmonary function. (iv) In a cohort consisting of seven patients with non-small cell lung cancer, validation of physiologic accuracy of the 4-0 CT derived quantitative images including Jacobian metric of ventilation, Vjac, and principal strains, (V?1, V?2, V?3, has been performed through correlation of the derived measures with SPECT ventilation and perfusion scans. The statistical correlations with SPECT have shown that the maximum principal strain pulmonary function map derived from MOFIO, outperforms all previously established ventilation metrics from 40-CT. It is hypothesized that use of CT -derived ventilation images in the treatment planning process will help predict and prevent pulmonary toxicity due to radiation treatment. It is also hypothesized that measures of regional and global lung elasticity and function obtained during the course of treatment may be used to adapt radiation treatment. Having objective methods with which to assess pre-treatment global and regional lung function and biomechanical properties, the radiation treatment dose can potentially be escalated to improve tumor response and local control