Ensembles of contours arise in various applications like simulation, computer-Aided design, and semantic segmentation. Uncovering ensemble patterns and analyzing individual members is a challenging task that suffers from clutter. Ensemble statistical summarization can alleviate this issue by permitting analyzing ensembles' distributional components like the mean and median, confidence intervals, and outliers. Contour boxplots, powered by Contour Band Depth (CBD), are a popular non-parametric ensemble summarization method that benefits from CBD's generality, robustness, and theoretical properties. In this work, we introduce Inclusion Depth (ID), a new notion of contour depth with three defining characteristics. First, ID is a generalization of functional Half-Region Depth, which offers several theoretical guarantees. Second, ID relies on a simple principle: The inside/outside relationships between contours. This facilitates implementing ID and understanding its results. Third, the computational complexity of ID scales quadratically in the number of members of the ensemble, improving CBD's cubic complexity. This also in practice speeds up the computation enabling the use of ID for exploring large contour ensembles or in contexts requiring multiple depth evaluations like clustering. In a series of experiments on synthetic data and case studies with meteorological and segmentation data, we evaluate ID's performance and demonstrate its capabilities for the visual analysis of contour ensembles.

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The contour depth methodology enables non-parametric summarization of contour ensembles by extracting their representatives, confidence bands, and outliers for visualization (via contour boxplots) and robust downstream procedures. We address two shortcomings of these methods. Firstly, we significantly expedite the computation and recomputation of Inclusion Depth (ID), introducing a linear-time algorithm for epsilon ID, a variant used for handling ensembles with contours with multiple intersections. We also present the inclusion matrix, which contains the pairwise inclusion relationships between contours, and leverage it to accelerate the recomputation of ID. Secondly, extending beyond the single distribution assumption, we present the Relative Depth (ReD), a generalization of contour depth for ensembles with multiple modes. Building upon the linear-time eID, we introduce CDclust, a clustering algorithm that untangles ensemble modes of variation by optimizing ReD. Synthetic and real datasets from medical image segmentation and meteorological forecasting showcase the speed advantages, illustrate the use case of progressive depth computation and enable non-parametric multimodal analysis. To promote research and adoption, we offer the contour-depth Python package.

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Background and purpose: Retrospective dose evaluation for organ-at-risk auto-contours has previously used small cohorts due to additional manual effort required for treatment planning on auto-contours. We aimed to do this at large scale, by a) proposing and assessing an automated plan optimization workflow that used existing clinical plan parameters and b) using it for head-and-neck auto-contour dose evaluation. Materials and methods: Our automated workflow emulated our clinic's treatment planning protocol and reused existing clinical plan optimization parameters. This workflow recreated the original clinical plan (P_OG) with manual contours (P_MC) and evaluated the dose effect (P_OG-P_MC) on 70 photon and 30 proton plans of head-and-neck patients. As a use-case, the same workflow (and parameters) created a plan using auto-contours (P_AC) of eight head-and-neck organs-at-risk from a commercial tool and evaluated their dose effect (P_MC-P_AC). Results: For plan recreation (P_OG-P_MC), our workflow had a median impact of 1.0% and 1.5% across dose metrics of auto-contours, for photon and proton respectively. Computer time of automated planning was 25% (photon) and 42% (proton) of manual planning time. For auto-contour evaluation (P_MC-P_AC), we noticed an impact of 2.0% and 2.6% for photon and proton radiotherapy. All evaluations had a median ΔNTCP (Normal Tissue Complication Probability) less than 0.3%. Conclusions: The plan replication capability of our automated program provides a blueprint for other clinics to perform auto-contour dose evaluation with large patient cohorts. Finally, despite geometric differences, auto-contours had a minimal median dose impact, hence inspiring confidence in their utility and facilitating their clinical adoption.

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The need to understand the structure of hierarchical or high-dimensional data is present in a variety of fields. Hyperbolic spaces have proven to be an important tool for embedding computations and analysis tasks as their non-linear nature lends itself well to tree or graph data. Subsequently, they have also been used in the visualization of high-dimensional data, where they exhibit increased embedding performance. However, none of the existing dimensionality reduction methods for embedding into hyperbolic spaces scale well with the size of the input data. That is because the embeddings are computed via iterative optimization schemes and the computation cost of every iteration is quadratic in the size of the input. Furthermore, due to the non-linear nature of hyperbolic spaces, euclidean acceleration structures cannot directly be translated to the hyperbolic setting. This article introduces the first acceleration structure for hyperbolic embeddings, building upon a polar quadtree. We compare our approach with existing methods and demonstrate that it computes embeddings of similar quality in significantly less time. Implementation and scripts for the experiments can be found at https://graphics.tudelft.nl/accelerating-hyperbolic-tsne .@en

Artificial Intelligence (AI)-based auto-delineation technologies rapidly delineate multiple structures of interest like organs-at-risk and tumors in 3D medical images, reducing personnel load and facilitating time-critical therapies. Despite its accuracy, the AI may produce flawed delineations, requiring clinician attention. Quality assessment (QA) of these delineations is laborious and demanding. Delineation error detection systems aim to aid QA, yet questions linger about clinician adoption, challenges, and time-saving potential. In this study, we address these queries in two stages. First, we investigate the error detection workflow of a radiotherapy technologist and a radiation oncologist from Holland Proton Therapy Center, a Dutch cancer treatment center. The user study revealed which information sources clinicians prefer to use for the error prioritization task and elucidated clinicians' slice-based navigation workflows with and without system assistance. Based on the findings from the user study, we developed a simulation model of the QA process, which we used to assess different error detection workflows on a retrospective cohort of 42 head and neck cancer patients. The simulation study results indicate potential time savings through error and dose information, contingent on per-slice analysis time remaining near the current baseline. Our findings encourage the development of user-centric delineation error detection systems and provide a new way to model and evaluate these systems' potential clinical value.@en

We introduce a geometric multigrid method for solving linear systems arising from variational problems on surfaces in geometry processing, Gravo MG. Our scheme uses point clouds as a reduced representation of the levels of the multigrid hierarchy to achieve a fast hierarchy construction and to extend the applicability of the method from triangle meshes to other surface representations like point clouds, nonmanifold meshes, and polygonal meshes. To build the prolongation operators, we associate each point of the hierarchy to a triangle constructed from points in the next coarser level. We obtain well-shaped candidate triangles by computing graph Voronoi diagrams centered around the coarse points and determining neighboring Voronoi cells. Our selection of triangles ensures that the connections of each point to points at adjacent coarser and finer levels are balanced in the tangential directions. As a result, we obtain sparse prolongation matrices with three entries per row and fast convergence of the solver. Code is available at https://graphics.tudelft.nl/gravo_mg.@en

Post-translational modifications (PTMs) affecting a protein's residues (amino acids) can disturb its function, leading to illness. Whether or not a PTM is pathogenic depends on its type and the status of neighboring residues. In this paper, we present the ProtoFold Neighborhood Inspector (PFNI), a visualization system for analyzing residues neighborhoods. The main contribution is a visualization idiom, the Residue Constellation (RC), for identifying and comparing three-dimensional neighborhoods based on per-residue features and spatial characteristics. The RC leverages two-dimensional representations of the protein's three-dimensional structure to overcome problems like occlusion, easing the analysis of neighborhoods that often have complicated spatial arrangements. Using the PFNI, we explored proteins' structural PTM data, which allowed us to identify patterns in the distribution and quantity of per-neighborhood PTMs that might be related to their pathogenic status. In the following, we define the tasks that guided the development of the PFNI and describe the data sources we derived and used. Then, we introduce the PFNI and illustrate its usage through an example of an analysis workflow. We conclude by reflecting on preliminary findings obtained while using the tool on the provided data and future directions concerning the development of the PFNI.@en

Bayesian Neural Nets (BNN) are increasingly used for robust organ auto-contouring. Uncertainty heatmaps extracted from BNNs have been shown to correspond to inaccurate regions. To help speed up the mandatory quality assessment (QA) of contours in radiotherapy, these heatmaps could be used as stimuli to direct visual attention of clinicians to potential inaccuracies. In practice, this is non-trivial to achieve since many accurate regions also exhibit uncertainty. To influence the output uncertainty of a BNN, we propose a modified accuracy-versus-uncertainty (AvU) metric as an additional objective during model training that penalizes both accurate regions exhibiting uncertainty as well as inaccurate regions exhibiting certainty. For evaluation, we use an uncertainty-ROC curve that can help differentiate between Bayesian models by comparing the probability of uncertainty in inaccurate versus accurate regions. We train and evaluate a FlipOut BNN model on the MICCAI2015 Head and Neck Segmentation challenge dataset and on the DeepMind-TCIA dataset, and observed an increase in the AUC of uncertainty-ROC curves by 5.6% and 5.9%, respectively, when using the AvU objective. The AvU objective primarily reduced false positives regions (uncertain and accurate), drawing less visual attention to these regions, thereby potentially improving the speed of error detection.

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We introduce Dynamic Constrained Grid (DCGrid), a hierarchical and adaptive grid structure for fluid simulation combined with a scheme for effectively managing the grid adaptations. DCGrid is designed to be implemented on the GPU and used in high-performance simulations. Specifically, it allows us to efficiently vary and adjust the grid resolution across the spatial domain and to rapidly evaluate local stencils and individual cells in a GPU implementation. A special feature of DCGrid is that the control of the grid adaption is modeled as an optimization under a constraint on the maximum available memory, which addresses the memory limitations in GPU-based simulation. To further advance the use of DCGrid in high-performance simulations, we complement DCGrid with an efficient scheme for approximating collisions between fluids and static solids on cells with different resolutions. We demonstrate the effectiveness of DCGrid for smoke flows and complex cloud simulations in which terrain-atmosphere interaction requires working with cells of varying resolution and rapidly changing conditions. Finally, we compare the performance of DCGrid to that of alternative adaptive grid structures.

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Sparse eigenproblems are important for various applications in computer graphics. The spectrum and eigenfunctions of the Laplace–Beltrami operator, for example, are fundamental for methods in shape analysis and mesh processing. The Subspace Iteration Method is a robust solver for these problems. In practice, however, Lanczos schemes are often faster. In this article, we introduce the Hierarchical Subspace Iteration Method (HSIM), a novel solver for sparse eigenproblems that operates on a hierarchy of nested vector spaces. The hierarchy is constructed such that on the coarsest space all eigenpairs can be computed with a dense eigensolver. HSIM uses these eigenpairs as initialization and iterates from coarse to fine over the hierarchy. On each level, subspace iterations, initialized with the solution from the previous level, are used to approximate the eigenpairs. This approach substantially reduces the number of iterations needed on the finest grid compared to the non-hierarchical Subspace Iteration Method. Our experiments show that HSIM can solve Laplace–Beltrami eigenproblems on meshes faster than state-of-the-art methods based on Lanczos iterations, preconditioned conjugate gradients, and subspace iterations.@en

Deep learning models for organ contouring in radiotherapy are poised for clinical usage, but currently, there exist few tools for automated quality assessment (QA) of the predicted contours. Bayesian models and their associated uncertainty, can potentially automate the process of detecting inaccurate predictions. We investigate two Bayesian models for auto-contouring, DropOut and FlipOut, using a quantitative measure – expected calibration error (ECE) and a qualitative measure – region-based accuracy-vs-uncertainty (R-AvU) graphs. It is well understood that a model should have low ECE to be considered trustworthy. However, in a QA context, a model should also have high uncertainty in inaccurate regions and low uncertainty in accurate regions. Such behaviour could direct visual attention of expert users to potentially inaccurate regions, leading to a speed-up in the QA process. Using R-AvU graphs, we qualitatively compare the behaviour of different models in accurate and inaccurate regions. Experiments are conducted on the MICCAI2015 Head and Neck Segmentation Challenge and on the DeepMindTCIA CT dataset using three models: DropOut-DICE, Dropout-CE (Cross Entropy) and FlipOut-CE. Quantitative results show that DropOut-DICE has the highest ECE, while Dropout-CE and FlipOut-CE have the lowest ECE. To better understand the difference between DropOut-CE and FlipOut-CE, we use the R-AvU graph which shows that FlipOut-CE has better uncertainty coverage in inaccurate regions than DropOut-CE. Such a combination of quantitative and qualitative metrics explores a new approach that helps to select which model can be deployed as a QA tool in clinical settings.
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Deep learning has improved vanishing point detection in images. Yet, deep networks require expensive annotated datasets trained on costly hardware and do not generalize to even slightly different domains, and minor problem variants. Here, we address these issues by injecting deep vanishing point detection networks with prior knowledge. This prior knowledge no longer needs to be learned from data, saving valuable annotation efforts and compute, unlocking realistic few-sample scenarios, and reducing the impact of domain changes. Moreover, the interpretability of the priors allows to adapt deep networks to minor problem variations such as switching between Manhattan and non-Manhattan worlds. We seamlessly incorporate two geometric priors: (i) Hough Transform -- mapping image pixels to straight lines, and (ii) Gaussian sphere -- mapping lines to great circles whose intersections denote vanishing points. Experimentally, we ablate our choices and show comparable accuracy to existing models in the large-data setting. We validate our model's improved data efficiency, robustness to domain changes, adaptability to non-Manhattan settings.@en

Delineation of tumors and organs-at-risk permits detecting and correcting changes in the patients' anatomy throughout the treatment, making it a core step of adaptive proton therapy (APT). Although AI-based auto-contouring technologies have sped up this process, the time needed to perform the quality assessment (QA) of the generated contours remains a bottleneck, taking clinicians between several minutes up to an hour to complete. This paper introduces a fast contouring workflow suitable for time-critical APT, enabling detection of anatomical changes in shorter time frames and with a lower demand of clinical resources. The proposed AI-infused workflow follows two principles uncovered after reviewing the APT literature and conducting several interviews and an observational study in two radiotherapy centers in the Netherlands. First, enable targeted inspection of the generated contours by leveraging AI uncertainty and clinically-relevant features such as the proximity of the organs-at-risk to the tumor. Second, minimize the number of interactions needed to edit faulty delineations with redundancy-aware editing tools that provide the user a sense of predictability and control. We use a proof of concept that we validated with clinicians to demonstrate how current and upcoming AI capabilities support the workflow and how it would fit into clinical practice.@en

Delineation of tumours and organs-at-risk permits detecting and correcting changes in the patients' anatomy throughout the treatment, making it a core step of adaptive external beam radiotherapy. Although auto-contouring technologies have sped up this process, the time needed to perform the quality assessment of the generated contours remains a bottleneck, taking clinicians between several minutes and an hour to complete. The authors of this article conducted several interviews and an observational study at two treatment centres in the Netherlands to identify challenges and opportunities for speeding up the delineation process in adaptive therapies. The study revealed three contextual variables that influence contouring performance: usable additional information, applicable domain-specific knowledge, and available editing capabilities in contouring software. In practice, clinicians leverage these variables to accelerate contouring in two ways. First, they use domain-specific knowledge and relevant clinical features such as the proximity of the organs-at-risk to the tumour to enable targeted inspection of the delineation. Second, clinicians modulate editing precision depending on the effect they anticipate the edit will have on the patient outcome. By implementing these acceleration strategies in guidelines and contouring tools, developers and workflow builders could increase contouring efficiency and consistency without affecting the patient outcome.@en

Learning from 3D point-cloud data has rapidly gained momentum, motivated by the success of deep learning on images and the increased availability of 3D~data. In this paper, we aim to construct anisotropic convolution layers that work directly on the surface derived from a point cloud. This is challenging because of the lack of a global coordinate system for tangential directions on surfaces. We introduce DeltaConv, a convolution layer that combines geometric operators from vector calculus to enable the construction of anisotropic filters on point clouds. Because these operators are defined on scalar- and vector-fields, we separate the network into a scalar- and a vector-stream, which are connected by the operators. The vector stream enables the network to explicitly represent, evaluate, and process directional information. Our convolutions are robust and simple to implement and match or improve on state-of-the-art approaches on several benchmarks, while also speeding up training and inference.@en

This paper is concerned with a fundamental problem in geometric deep learning that arises in the construction of convolutional neural networks on surfaces. Due to curvature, the transport of filter kernels on surfaces results in a rotational ambiguity, which prevents a uniform alignment of these kernels on the surface. We propose a network architecture for surfaces that consists of vector-valued, rotation-equivariant features. The equivariance property makes it possible to locally align features, which were computed in arbitrary coordinate systems, when aggregating features in a convolution layer. The resulting network is agnostic to the choices of coordinate systems for the tangent spaces on the surface. We implement our approach for triangle meshes. Based on circular harmonic functions, we introduce convolution filters for meshes that are rotation-equivariant at the discrete level. We evaluate the resulting networks on shape correspondence and shape classifications tasks and compare their performance to other approaches.@en

Geometric optimization problems are at the core of many applications in geometry processing. The choice of a representation fitting an optimization problem can considerably simplify solving the problem. We consider the Nonlinear Rotation-Invariant Coordinates (NRIC) that represent the nodal positions of a discrete triangular surface with fixed combinatorics as a vector that stacks all edge lengths and dihedral angles of the mesh. It is known that this representation associates a unique vector to an equivalence class of nodal positions that differ by a rigid body motion. Moreover, integrability conditions that ensure the existence of nodal positions that match a given vector of edge lengths and dihedral angles have been established. The goal of this paper is to develop the machinery needed to use the NRIC for solving geometric optimization problems. First, we use the integrability conditions to derive an implicit description of the space of discrete surfaces as a submanifold of an Euclidean space and a corresponding description of its tangent spaces. Secondly, we reformulate the integrability conditions using quaternions and provide explicit formulas for their first and second derivatives facilitating the use of Hessians in NRIC-based optimization problems. Lastly, we introduce a fast and robust algorithm that reconstructs nodal positions from almost integrable NRIC. We demonstrate the benefits of this approach on a collection of geometric optimization problems. Comparisons to alternative approaches indicate that NRIC-based optimization is particularly effective for problems involving near-isometric deformations.

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We introduce a construction of subspaces of the spaces of tangential vector, n-vector, and tensor fields on surfaces. The resulting subspaces can be used as the basis of fast approximation algorithms for design and processing problems that involve tangential fields. Important features of our construction are that it is based on a general principle, from which constructions for different types of tangential fields can be derived, and that it is scalable, making it possible to efficiently compute and store large subspace bases for large meshes. Moreover, the construction is adaptive, which allows for controlling the distribution of the degrees of freedom of the subspaces over the surface. We evaluate our construction in several experiments addressing approximation quality, scalability, adaptivity, computation times and memory requirements. Our design choices are justified by comparing our construction to possible alternatives. Finally, we discuss examples of how subspace methods can be used to build interactive tools for tangential field design and processing tasks.

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This paper introduces the construction of a low-dimensional nonlinear space capturing the variability of a non-rigid shape from a data set of example poses. The core of the approach is a Sparse Principal Geodesic Analysis (SPGA) on the Riemannian manifold of discrete shells, in which a pose of a non-rigid shape is a point. The SPGA is invariant to rigid body motions of the poses and supports large deformation. Since the Riemannian metric measures the membrane and bending distortions of the shells, the sparsity term forces the modes to describe largely decoupled and localized deformations. This property facilitates the analysis of articulated shapes. The modes often represent characteristic articulations of the shape and usually come with a decomposing of the spanned subspace into low-dimensional widely decoupled subspaces. For example, for human models, one expects distinct, localized modes for the bending of elbow or knee whereas some more modes are required to represent shoulder articulation. The decoupling property can be used to construct useful starting points for the computation of the nonlinear deformations via a superposition of shape submanifolds resulting from the decoupling. In a preprocessing stage, samples of the individual subspaces are computed, and, in an online phase, these are interpolated multilinearly. This accelerates the construction of nonlinear deformations and makes the method applicable for interactive applications. The method is compared to alternative approaches and the benefits are demonstrated on different kinds of input data.

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