Multidirectional and Topography-based Dynamic-scale Varifold Representations with Application to Matching Developing Cortical Surfaces

The human cerebral cortex is marked by great complexity as well as substantial dynamic changes during early postnatal development. To obtain a fairly comprehensive picture of its age-induced and/or disorder-related cortical changes, one needs to match cortical surfaces to one another, while maximizing their anatomical alignment. Methods that geodesically shoot surfaces into one another as currents (a distribution of oriented normals) and varifolds (a distribution of non-oriented normals) provide an elegant Riemannian framework for generic surface matching and reliable statistical analysis. However, both conventional current and varifold matching methods have two key limitations. First, they only use the normals of the surface to measure its geometry and guide the warping process, which overlooks the importance of the orientations of the inherently convoluted cortical sulcal and gyral folds. Second, the ‘conversion’ of a surface into a current or a varifold operates at a fixed scale under which geometric surface details will be neglected, which ignores the dynamic scales of cortical foldings. To overcome these limitations and improve varifold-based cortical surface registration, we propose two different strategies. The first strategy decomposes each cortical surface into its normal and tangent varifold representations, by integrating principal curvature direction field into the varifold matching framework, thus providing rich information of the orientation of cortical folding and better characterization of the complex cortical geometry. The second strategy explores the informative cortical geometric features to perform a dynamic-scale measurement of the cortical surface that depends on the local surface topography (e.g., principal curvature), thereby we introduce the concept of a topography-based dynamic-scale varifold. We tested the proposed varifold variants for registering 12 pairs of dynamically developing cortical surfaces from 0 to 6 months of age. Both variants improved the matching accuracy in terms of closeness to the target surface and the goodness of alignment with regional anatomical boundaries, when compared with three state-of-the-art methods: (1) diffeomorphic spectral matching, (2) conventional current-based surface matching, and (3) conventional varifold-based surface matching

LEARNING TO RIG CHARACTERS

With the emergence of 3D virtual worlds, 3D social media, and massive online games, the need for diverse, high-quality, animation-ready characters and avatars is greater than ever. To animate characters, artists hand-craft articulation structures, such as animation skeletons and part deformers, which require significant amount of manual and laborious interaction with 2D/3D modeling interfaces. This thesis presents deep learning methods that are able to significantly automate the process of character rigging. First, the thesis introduces RigNet, a method capable of predicting an animation skeleton for an input static 3D shape in the form of a polygon mesh. The predicted skeletons match the animator expectations in joint placement and topology. RigNet also estimates surface skin weights which determine how the mesh is animated given the different skeletal poses. In contrast to prior work that fits pre-defined skeletal templates with hand-tuned objectives, RigNet is able to automatically rig diverse characters, such as humanoids, quadrupeds, toys, birds, with varying articulation structure and geometry. RigNet is based on a deep neural architecture that directly operates on the mesh representation. The architecture is trained on a diverse dataset of rigged models that we mined online and curated. The dataset includes 2.7K polygon meshes, along with their associated skeletons and corresponding skin weights. Second, the thesis introduces Morig, a method that automatically rigs character meshes driven by single-view point cloud streams capturing the motion of performing characters. Compared to RigNet, MoRig\u27s rigging is \emph{motion-aware}: its neural network encodes motion cues from the point clouds into compact feature representations that are informative about the articulated parts of the performing character. These motion-aware features guide the inference of an appropriate skeletal rig for the input mesh. Furthermore, Morig is able to animate the rig according to the captured point cloud motion. Morig can handle diverse characters with different morphologies (e.g., humanoids, quadrupeds, toy characters). It also accounts for occluded regions in the point clouds and mismatches in the part proportions between the input mesh and captured character. Third, the thesis introduces APES, a method that takes as input 2D raster images depicting a small set of poses of a character shown in a sprite sheet, and identifies articulated parts useful for rigging the character. APES uses a combination of neural network inference and integer linear programming to identify a compact set of articulated body parts, e.g. head, torso and limbs, that best reconstruct the input poses. Compared to Morig and RigNet that require a large collection of training models with associated skeletons and skinning weights, APES\u27 neural architecture relies on less effortful supervision from (i) pixel correspondences readily available in existing large cartoon image datasets (e.g., Creative Flow), (ii) a relatively small dataset of 57 cartoon characters segmented into moving parts. Finally, the thesis discusses future research directions related to combining neural rigging with 3D and 4D reconstruction of characters from point cloud data and 2D video as well as automating the process of motion synthesis for 3D characters

An Unified Multiscale Framework for Planar, Surface, and Curve Skeletonization

Computing skeletons of 2D shapes, and medial surface and curve skeletons of 3D shapes, is a challenging task. In particular, there is no unified framework that detects all types of skeletons using a single model, and also produces a multiscale representation which allows to progressively simplify, or regularize, all skeleton types. In this paper, we present such a framework. We model skeleton detection and regularization by a conservative mass transport process from a shape's boundary to its surface skeleton, next to its curve skeleton, and finally to the shape center. The resulting density field can be thresholded to obtain a multiscale representation of progressively simplified surface, or curve, skeletons. We detail a numerical implementation of our framework which is demonstrably stable and has high computational efficiency. We demonstrate our framework on several complex 2D and 3D shapes

Learning distributions of shape trajectories from longitudinal datasets: a hierarchical model on a manifold of diffeomorphisms

We propose a method to learn a distribution of shape trajectories from longitudinal data, i.e. the collection of individual objects repeatedly observed at multiple time-points. The method allows to compute an average spatiotemporal trajectory of shape changes at the group level, and the individual variations of this trajectory both in terms of geometry and time dynamics. First, we formulate a non-linear mixed-effects statistical model as the combination of a generic statistical model for manifold-valued longitudinal data, a deformation model defining shape trajectories via the action of a finite-dimensional set of diffeomorphisms with a manifold structure, and an efficient numerical scheme to compute parallel transport on this manifold. Second, we introduce a MCMC-SAEM algorithm with a specific approach to shape sampling, an adaptive scheme for proposal variances, and a log-likelihood tempering strategy to estimate our model. Third, we validate our algorithm on 2D simulated data, and then estimate a scenario of alteration of the shape of the hippocampus 3D brain structure during the course of Alzheimer's disease. The method shows for instance that hippocampal atrophy progresses more quickly in female subjects, and occurs earlier in APOE4 mutation carriers. We finally illustrate the potential of our method for classifying pathological trajectories versus normal ageing

Multi-scale active shape description in medical imaging

Shape description in medical imaging has become an increasingly important research field in recent years. Fast and high-resolution image acquisition methods like Magnetic Resonance (MR) imaging produce very detailed cross-sectional images of the human body - shape description is then a post-processing operation which abstracts quantitative descriptions of anatomically relevant object shapes. This task is usually performed by clinicians and other experts by first segmenting the shapes of interest, and then making volumetric and other quantitative measurements. High demand on expert time and inter- and intra-observer variability impose a clinical need of automating this process. Furthermore, recent studies in clinical neurology on the correspondence between disease status and degree of shape deformations necessitate the use of more sophisticated, higher-level shape description techniques. In this work a new hierarchical tool for shape description has been developed, combining two recently developed and powerful techniques in image processing: differential invariants in scale-space, and active contour models. This tool enables quantitative and qualitative shape studies at multiple levels of image detail, exploring the extra image scale degree of freedom. Using scale-space continuity, the global object shape can be detected at a coarse level of image detail, and finer shape characteristics can be found at higher levels of detail or scales. New methods for active shape evolution and focusing have been developed for the extraction of shapes at a large set of scales using an active contour model whose energy function is regularized with respect to scale and geometric differential image invariants. The resulting set of shapes is formulated as a multiscale shape stack which is analysed and described for each scale level with a large set of shape descriptors to obtain and analyse shape changes across scales. This shape stack leads naturally to several questions in regard to variable sampling and appropriate levels of detail to investigate an image. The relationship between active contour sampling precision and scale-space is addressed. After a thorough review of modem shape description, multi-scale image processing and active contour model techniques, the novel framework for multi-scale active shape description is presented and tested on synthetic images and medical images. An interesting result is the recovery of the fractal dimension of a known fractal boundary using this framework. Medical applications addressed are grey-matter deformations occurring for patients with epilepsy, spinal cord atrophy for patients with Multiple Sclerosis, and cortical impairment for neonates. Extensions to non-linear scale-spaces, comparisons to binary curve and curvature evolution schemes as well as other hierarchical shape descriptors are discussed

Extracting 3D parametric curves from 2D images of Helical objects

Helical objects occur in medicine, biology, cosmetics, nanotechnology, and engineering. Extracting a 3D parametric curve from a 2D image of a helical object has many practical applications, in particular being able to extract metrics such as tortuosity, frequency, and pitch. We present a method that is able to straighten the image object and derive a robust 3D helical curve from peaks in the object boundary. The algorithm has a small number of stable parameters that require little tuning, and the curve is validated against both synthetic and real-world data. The results show that the extracted 3D curve comes within close Hausdorff distance to the ground truth, and has near identical tortuosity for helical objects with a circular profile. Parameter insensitivity and robustness against high levels of image noise are demonstrated thoroughly and quantitatively

Atlas Diffeomorphisms Via Object Models

To tackle the problem of segmenting several closely-spaced objects from 3D medical images, I propose a hybrid of two segmentation approaches: one image-based and one model-based. A major contribution takes the image-based approach by diffeomorphically mapping a fully segmented atlas image to a partially segmented target patient image preserving any `correspondence' inferred from the partial segmentation of the target. The mapping is produced by solving the steady-state heat flow equation where the temperature is a coordinate vector and corresponding points have the same temperature. Objects carried over from the atlas into the target serve as reasonable initial segmentations and can be further refined by a model-based segmentation method. Good quality segmentations are added to the list of the initial partial segmentations, and the process is repeated. Another contribution takes the model-based approach in developing shape models of quasi-tubular objects and statistics on those models. Whereas medial models were previously only developed for slab-shaped objects, this contribution provides an approximately medial method to stably represent nearly tubular objects. I test my method on segmenting objects from 3D Computed Tomography (CT) scans of the head and neck obtained for radiotherapy treatment planning

Doctor of Philosophy

dissertationAn important aspect of medical research is the understanding of anatomy and its relation to function in the human body. For instance, identifying changes in the brain associated with cognitive decline helps in understanding the process of aging and age-related neurological disorders. The field of computational anatomy provides a rich mathematical setting for statistical analysis of complex geometrical structures seen in 3D medical images. At its core, computational anatomy is based on the representation of anatomical shape and its variability as elements of nonflat manifold of diffeomorphisms with an associated Riemannian structure. Although such manifolds effectively represent natural biological variability, intrinsic methods of statistical analysis within these spaces remain deficient at large. This dissertation contributes two critical missing pieces for statistics in diffeomorphisms: (1) multivariate regression models for cross-sectional study of shapes, and (2) generalization of classical Euclidean, mixed-effects models to manifolds for longitudinal studies. These models are based on the principle that statistics on manifold-valued information must respect the intrinsic geometry of that space. The multivariate regression methods provide statistical descriptors of the relationships of anatomy with clinical indicators. The novel theory of hierarchical geodesic models (HGMs) is developed as a natural generalization of hierarchical linear models (HLMs) to describe longitudinal data on curved manifolds. Using a hierarchy of geodesics, the HGMs address the challenge of modeling the shape-data with unbalanced designs typically arising as a result of follow-up medical studies. More generally, this research establishes a mathematical foundation to study dynamics of changes in anatomy and the associated clinical progression with time. This dissertation also provides efficient algorithms that utilize state-of-the-art high performance computing architectures to solve models on large-scale, longitudinal imaging data. These manifold-based methods are applied to predictive modeling of neurological disorders such as Alzheimer's disease. Overall, this dissertation enables clinicians and researchers to better utilize the structural information available in medical images