Shape Analysis Using Spectral Geometry

Shape analysis is a fundamental research topic in computer graphics and computer vision. To date, more and more 3D data is produced by those advanced acquisition capture devices, e.g., laser scanners, depth cameras, and CT/MRI scanners. The increasing data demands advanced analysis tools including shape matching, retrieval, deformation, etc. Nevertheless, 3D Shapes are represented with Euclidean transformations such as translation, scaling, and rotation and digital mesh representations are irregularly sampled. The shape can also deform non-linearly and the sampling may vary. In order to address these challenging problems, we investigate Laplace-Beltrami shape spectra from the differential geometry perspective, focusing more on the intrinsic properties. In this dissertation, the shapes are represented with 2 manifolds, which are differentiable. First, we discuss in detail about the salient geometric feature points in the Laplace-Beltrami spectral domain instead of traditional spatial domains. Simultaneously, the local shape descriptor of a feature point is the Laplace-Beltrami spectrum of the spatial region associated to the point, which are stable and distinctive. The salient spectral geometric features are invariant to spatial Euclidean transforms, isometric deformations and mesh triangulations. Both global and partial matching can be achieved with these salient feature points. Next, we introduce a novel method to analyze a set of poses, i.e., near-isometric deformations, of 3D models that are unregistered. Different shapes of poses are transformed from the 3D spatial domain to a geometry spectral one where all near isometric deformations, mesh triangulations and Euclidean transformations are filtered away. Semantic parts of that model are then determined based on the computed geometric properties of all the mapped vertices in the geometry spectral domain while semantic skeleton can be automatically built with joints detected. Finally we prove the shape spectrum is a continuous function to a scale function on the conformal factor of the manifold. The derivatives of the eigenvalues are analytically expressed with those of the scale function. The property applies to both continuous domain and discrete triangle meshes. On the triangle meshes, a spectrum alignment algorithm is developed. Given two closed triangle meshes, the eigenvalues can be aligned from one to the other and the eigenfunction distributions are aligned as well. This extends the shape spectra across non-isometric deformations, supporting a registration-free analysis of general motion data

Spectral/hp element methods: recent developments, applications, and perspectives

The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on coarse finite element-type meshes. The spatial approximation is based upon orthogonal polynomials, such as Legendre or Chebychev polynomials, modified to accommodate C0-continuous expansions. Computationally and theoretically, by increasing the polynomial order p, high-precision solutions and fast convergence can be obtained and, in particular, under certain regularity assumptions an exponential reduction in approximation error between numerical and exact solutions can be achieved. This method has now been applied in many simulation studies of both fundamental and practical engineering flows. This paper briefly describes the formulation of the spectral/hp element method and provides an overview of its application to computational fluid dynamics. In particular, it focuses on the use the spectral/hp element method in transitional flows and ocean engineering. Finally, some of the major challenges to be overcome in order to use the spectral/hp element method in more complex science and engineering applications are discussed

SAGA: Spectral Adversarial Geometric Attack on 3D Meshes

A triangular mesh is one of the most popular 3D data representations. As such, the deployment of deep neural networks for mesh processing is widely spread and is increasingly attracting more attention. However, neural networks are prone to adversarial attacks, where carefully crafted inputs impair the model's functionality. The need to explore these vulnerabilities is a fundamental factor in the future development of 3D-based applications. Recently, mesh attacks were studied on the semantic level, where classifiers are misled to produce wrong predictions. Nevertheless, mesh surfaces possess complex geometric attributes beyond their semantic meaning, and their analysis often includes the need to encode and reconstruct the geometry of the shape. We propose a novel framework for a geometric adversarial attack on a 3D mesh autoencoder. In this setting, an adversarial input mesh deceives the autoencoder by forcing it to reconstruct a different geometric shape at its output. The malicious input is produced by perturbing a clean shape in the spectral domain. Our method leverages the spectral decomposition of the mesh along with additional mesh-related properties to obtain visually credible results that consider the delicacy of surface distortions. Our code is publicly available at https://github.com/StolikTomer/SAGA

Multimodal Characterization of the Atrial Substrate - Risks and Rewards of Electrogram and Impedance Mapping

The treatment of atrial rhythm disorders such as atrial fibrillation has remained a major challenge predominantly for patients with severely remodeled substrate. Individualized ablation strategies beyond pulmonary vein isolation in combination with real-time assess- ment of ablation lesion formation have been striven for insistently. Current approaches for identifying arrhythmogenic regions predominantly rely on electrogram-based features such as activation time and voltage or electrogram fractionation as a surrogate for tissue pathology. Despite bending every effort, large-scale clinical trials have yielded ambiguous results on the efficacy of various substrate mapping approaches without significant improvement of patient outcomes. This work focuses on enhancing the understanding of electrogram features and local impedance measurements in the atria towards the extraction of clinically relevant and predic- tive substrate characteristics. Features were extracted from intra-atrial electrograms with particular reference to the un- derlying excitation patterns to address morphological alterations caused by structural and functional changes. The noise level of unipolar electrograms was estimated and reduced by tailored filtering to enhance unipolar signal quality. Electrogram features exhibited nar- row distributions for healthy substrate across patients while a wide range was observed for pathologically altered excitation. Additionally, local impedance was investigated as a novel parameter and mapping modality. Having been introduced to the medical device market recently for monitoring ablative lesion formation, initial clinical experiences with local impedance-enabled catheters lack comple- mentary systematic investigations. Confounding factors and the potential for application as a tool for substrate mapping need elucidation. This work pursued a trimodal approach combining in human, in vitro, and in silico experiments to quantitatively understand the effect of distinct ambient conditions on the measured local impedance. Forward simulations of the spread of the electrical field with a finite element approach as well as the application of inverse solution methods to reconstruct tissue conductivity were implemented in silico. Adequate preprocessing steps were developed for measurements in human to eliminate artefacts automatically. Two clinical studies on local impedance as an indicator for ablation lesion formation and on local impedance based substrate mapping were conducted. Local impedance recordings identified both previously ablated and native scar areas irrespective of local excitation. A highly detailed in silico environment for local impedance measurements was validated with in vitro recordings and provided quantitative insights into the influence of changes in clinically relevant scenarios. Inverse reconstruction of relative tissue conductivity yielded promising results in silico. This work demonstrates that local impedance mapping shows great potential to comple- ment electrogram-based substrate mapping. A validated in silico environment for local impedance measurements can facilitate and optimize the development of next generation local impedance-enabled catheters. Conduction velocity, electrogram features, and recon- structed tissue conductivity suggest to be promising candidates for enhancing future clinical mapping systems

Shape analysis of the human brain.

Autism is a complex developmental disability that has dramatically increased in prevalence, having a decisive impact on the health and behavior of children. Methods used to detect and recommend therapies have been much debated in the medical community because of the subjective nature of diagnosing autism. In order to provide an alternative method for understanding autism, the current work has developed a 3-dimensional state-of-the-art shape based analysis of the human brain to aid in creating more accurate diagnostic assessments and guided risk analyses for individuals with neurological conditions, such as autism. Methods: The aim of this work was to assess whether the shape of the human brain can be used as a reliable source of information for determining whether an individual will be diagnosed with autism. The study was conducted using multi-center databases of magnetic resonance images of the human brain. The subjects in the databases were analyzed using a series of algorithms consisting of bias correction, skull stripping, multi-label brain segmentation, 3-dimensional mesh construction, spherical harmonic decomposition, registration, and classification. The software algorithms were developed as an original contribution of this dissertation in collaboration with the BioImaging Laboratory at the University of Louisville Speed School of Engineering. The classification of each subject was used to construct diagnoses and therapeutic risk assessments for each patient. Results: A reliable metric for making neurological diagnoses and constructing therapeutic risk assessment for individuals has been identified. The metric was explored in populations of individuals having autism spectrum disorders, dyslexia, Alzheimers disease, and lung cancer. Conclusion: Currently, the clinical applicability and benefits of the proposed software approach are being discussed by the broader community of doctors, therapists, and parents for use in improving current methods by which autism spectrum disorders are diagnosed and understood

Schnelle Löser für Partielle Differentialgleichungen

This workshop was well attended by 52 participants with broad geographic representation from 11 countries and 3 continents. It was a nice blend of researchers with various backgrounds