Learning shape correspondence with anisotropic convolutional neural networks

Establishing correspondence between shapes is a fundamental problem in geometry processing, arising in a wide variety of applications. The problem is especially difficult in the setting of non-isometric deformations, as well as in the presence of topological noise and missing parts, mainly due to the limited capability to model such deformations axiomatically. Several recent works showed that invariance to complex shape transformations can be learned from examples. In this paper, we introduce an intrinsic convolutional neural network architecture based on anisotropic diffusion kernels, which we term Anisotropic Convolutional Neural Network (ACNN). In our construction, we generalize convolutions to non-Euclidean domains by constructing a set of oriented anisotropic diffusion kernels, creating in this way a local intrinsic polar representation of the data (`patch'), which is then correlated with a filter. Several cascades of such filters, linear, and non-linear operators are stacked to form a deep neural network whose parameters are learned by minimizing a task-specific cost. We use ACNNs to effectively learn intrinsic dense correspondences between deformable shapes in very challenging settings, achieving state-of-the-art results on some of the most difficult recent correspondence benchmarks

Three Dimensional Nonlinear Statistical Modeling Framework for Morphological Analysis

This dissertation describes a novel three-dimensional (3D) morphometric analysis framework for building statistical shape models and identifying shape differences between populations. This research generalizes the use of anatomical atlases on more complex anatomy as in case of irregular, flat bones, and bones with deformity and irregular bone growth. The foundations for this framework are: 1) Anatomical atlases which allow the creation of homologues anatomical models across populations; 2) Statistical representation for output models in a compact form to capture both local and global shape variation across populations; 3) Shape Analysis using automated 3D landmarking and surface matching. The proposed framework has various applications in clinical, forensic and physical anthropology fields. Extensive research has been published in peer-reviewed image processing, forensic anthropology, physical anthropology, biomedical engineering, and clinical orthopedics conferences and journals. The forthcoming discussion of existing methods for morphometric analysis, including manual and semi-automatic methods, addresses the need for automation of morphometric analysis and statistical atlases. Explanations of these existing methods for the construction of statistical shape models, including benefits and limitations of each method, provide evidence of the necessity for such a novel algorithm. A novel approach was taken to achieve accurate point correspondence in case of irregular and deformed anatomy. This was achieved using a scale space approach to detect prominent scale invariant features. These features were then matched and registered using a novel multi-scale method, utilizing both coordinate data as well as shape descriptors, followed by an overall surface deformation using a new constrained free-form deformation. Applications of output statistical atlases are discussed, including forensic applications for the skull sexing, as well as physical anthropology applications, such as asymmetry in clavicles. Clinical applications in pelvis reconstruction and studying of lumbar kinematics and studying thickness of bone and soft tissue are also discussed

Non-isometric 3D shape registration.

3D shape registration is an important task in computer graphics and computer vision. It has been widely used in the area of film industry, 3D animation, video games and AR/VR assets creation. Manually creating the 3D model of a character from scratch is tedious and time consuming, and it can only be completed by professional trained artists. With the development of 3D geometry acquisition technology, it becomes easier and cheaper to capture high-resolution and highly detailed 3D geometries. However, the scanned data are often incomplete or noisy and therefore cannot be employed directly. To deal with the above two problems, one typical and efficient solution is to deform an existing high-quality model (template) to fit the scanned data (target). Shape registration as an essential technique to do so has been arousing intensive attention. In last decades, various shape registration approaches have been proposed for accurate template fitting. However, there are still some remaining challenges. It is well known that the template can be largely different with the target in respect of size and pose. With the large (usually non-isometric) deformation between them, the shear distortion can easily occur, which may lead to poor results, such as degenerated triangles, fold-overs. Before deforming the template towards the target, reliable correspondences between them should be found first. Incorrect correspondences give the wrong deformation guidance, which can also easily produce fold-overs. As mentioned before, the target always comes with noise. This is the part we want to filter out and try not to fit the template on it. Hence, non-isometric shape registration robust to noise is highly desirable in the scene of geometry modelling from the scanned data. In this PhD research, we address existing challenges in shape registration, including how to prevent the deformation distortion, how to reduce the foldover occurrence and how to deal with the noise in the target. Novel methods including consistent as-similar as-possible surface deformation and robust Huber-L1 surface registration are proposed, which are validated through experimental comparison with state-of-the-arts. The deformation technique plays an important role in shape registration. In this research, a consistent as similar-as-possible (CASAP) surface deformation approach is proposed. Starting from investigating the continuous deformation energy, we analyse the existing term to make the discrete energy converge to the continuous one, whose property we called as energy consistency. Based on the deformation method, a novel CASAP non-isometric surface registration method is proposed. The proposed registration method well preserves the angles of triangles in the template surface so that least distortion is introduced during the surface deformation and thus reduce the risk of fold-over and self-intersection. To reduce the noise influence, a Huber-L1 based non-isometric surface registration is proposed, where a Huber-L1 regularized model constrained on the transformation variation and position difference. The proposed method is robust to noise and produces piecewise smooth results while still preserving fine details on the target. We evaluate and validate our methods through extensive experiments, whose results have demonstrated that the proposed methods in this thesis are more accurate and robust to noise in comparison of the state-of-the arts and enable us to produce high quality models with little efforts

Surface Networks

We study data-driven representations for three-dimensional triangle meshes, which are one of the prevalent objects used to represent 3D geometry. Recent works have developed models that exploit the intrinsic geometry of manifolds and graphs, namely the Graph Neural Networks (GNNs) and its spectral variants, which learn from the local metric tensor via the Laplacian operator. Despite offering excellent sample complexity and built-in invariances, intrinsic geometry alone is invariant to isometric deformations, making it unsuitable for many applications. To overcome this limitation, we propose several upgrades to GNNs to leverage extrinsic differential geometry properties of three-dimensional surfaces, increasing its modeling power. In particular, we propose to exploit the Dirac operator, whose spectrum detects principal curvature directions --- this is in stark contrast with the classical Laplace operator, which directly measures mean curvature. We coin the resulting models \emph{Surface Networks (SN)}. We prove that these models define shape representations that are stable to deformation and to discretization, and we demonstrate the efficiency and versatility of SNs on two challenging tasks: temporal prediction of mesh deformations under non-linear dynamics and generative models using a variational autoencoder framework with encoders/decoders given by SNs

Automated Fragmentary Bone Matching

Identification, reconstruction and matching of fragmentary bones are basic tasks required to accomplish quantification and analysis of fragmentary human remains derived from forensic contexts. Appropriate techniques for three-dimensional surface matching have received great attention in computer vision literature, and various methods have been proposed for matching fragmentary meshes; however, many of these methods lack automation, speed and/or suffer from high sensitivity to noise. In addition, reconstruction of fragementary bones along with identification in the presence of reference model to compare with in an automatic scheme have not been addressed. In order to address these issues, we used a multi-stage technique for fragment identification, matching and registration. The study introduces an automated technique for matching of fragmentary human skeletal remains for improving forensic anthropology practice and policy. The proposed technique involves creation of surfaces models for the fragmentary elements which can be done using computerized tomographic scans followed by segmentation. Upon creation of the fragmentary elements models, the models go through feature extraction technique where the surface roughness map of each model is measured using local shape analysis measures. Adaptive thesholding is then used to extract model features. A multi-stage technique is then used to identify, match and register bone fragments to their corresponding template bone model. First, extracted features are used for matching with different template bone models using iterative closest point algorithm with different positions and orientations. The best match score, in terms of minimum root-mean-square error, is used along with the position and orientation and the resulting transformation to register the fragment bone model with the corresponding template bone model using iterative closest point algorithm