Vector graphics provide continuous and often even smooth geometric representations of images. While recent approaches to automatically vectorize images lead to relatively good results, they typically leave ample room for improvement: the geometry and color of the vector graphics primitives can be further (automatically) optimized. We propose a novel method that generates high-quality vectorizations based on optimizing input curved triangle meshes (optionally with mesh colors). To overcome the key challenge of establishing a differentiable mapping between the input parameters, i.e., geometry and (mesh) colors of the gradient mesh, and the difference between the vectorized and input image, we treat the input image as a continuous bilinear interpolatory spline and employ Monte Carlo integration. We test our algorithm on various images and show that it can effectively and efficiently improve the quality of an initial vectorization. • We introduce a differentiable error function to drive optimization. • We derive an explicit gradient from the error function using the Monte Carlo method. • We illustrate the effectiveness of the method on various examples.

Gradient meshes are powerful vector graphic primitives known for producing smooth and detailed color transitions. However, their fixed rectangular topology complicates editing, as adding detail in one region introduces control points across the entire mesh. To improve editability and better support artist workflows, we propose a simplification method for gradient meshes based on local refinement. Our method transforms a traditional, globally-refined mesh into a locally-refined one by iteratively merging adjacent faces and eliminating redundant data while preserving visual quality. We achieve this by rasterizing the mesh and applying established visual quality metrics to ensure consistency with the original. Additionally, we offer artists control over the simplification process by introducing an error threshold, allowing them to balance the level of simplification with visual fidelity. Finally, we conduct a thorough comparison of various mesh simplification strategies to analyze the trade-offs between simplification quality and speed so as to inform users to the optimal one that they can use to obtain the desired trade-off.

Fluid simulation is a central topic in computer graphics, encompassing a wide range of methodologies for modeling Newtonian, non-Newtonian, and viscoelastic behaviors across both single-phase and multiphase settings. Existing single-phase frameworks have achieved high visual fidelity, yet multiphase simulations remain limited in accurately capturing complex phase interactions, particularly under high-viscosity-ratio or viscoelastic conditions. To address these challenges, we develop a unified multiphase viscoelastic formulation capable of handling diverse fluid types-including Newtonian, shear-dependent non-Newtonian, and viscoelastic flows-within a single consistent framework. The formulation extends mixture-model approaches through a multi-mode conformation tensor representation, which enhances numerical stability via phase-level stress corrections and efficiently captures a broad spectrum of rheological behaviors. Compared with existing techniques, our framework achieves improved momentum-mass consistency and numerical stability, maintaining physically plausible results across wide viscosity ranges, advancing the state of the art in multiphase viscoelastic fluid simulation.

Simulating the interactions between fluids and porous media has attracted significant attention in computer graphics. A key challenge in this domain is modeling the Poro-Elasto-Capillary (PEC) coupling effect which describes the intricate interplay of three physical phenomena in soft porous materials: pore-structure evolution, elastic deformation, and wetting driven by capillary pressure. These phenomena collectively govern dynamic behavior such as the softening and fracturing of biscuits upon water absorption or the swelling of cellulose sponges due to liquid infiltration. Most existing simulation methods model porous media either as static grids or as solid particles with augmented water content attributes, failing to capture the full spectrum of PEC-driven effects due to the lack of physical modeling for elasticity, dynamic porosity changes, and capillary interactions. We propose a multiphase particle-based framework to holistically simulate PEC coupling effects with porous media. We develop a physics-driven model that captures elasticity and dynamic pore-structure evolution under capillary action, enabling realistic simulation of softening and swelling. We derive a saturation-aware pressure Poisson equation to enforce fluid incompressibility within and around the porous medium, ensuring accurate capillary-driven flow while preserving mass and momentum. Finally, we propose a representative elementary volume-based formulation to unify the modeling of homogeneous macro-porous media and cavity-embedded structures, enhancing the representation of pore-scale PEC effects. Comparisons with prior work and real footage show the advantages of our approach in achieving visually realistic fluid-porous media interactions.

ABSTRACT High‐accuracy point cloud (self‐) reconstruction is crucial for point cloud editing, translation, and unsupervised representation learning. However, existing point cloud reconstruction methods often sacrifice many geometric details. Altough many techniques have proposed how to construct better point cloud decoders, only a few have designed point cloud encoders from a reconstruction perspective. We propose an autodecoder architecture to achieve detail‐preserving point cloud reconstruction while bypassing the performance bottleneck of the encoder. Our architecture is theoretically applicable to any existing point cloud decoder. For training, both the weights of the decoder and the pre‐initialised latent codes, corresponding to the input points, are updated simultaneously. Experimental results demonstrate that our autodecoder achieves an average reduction of 24.62% in Chamfer Distance compared to existing methods, significantly improving reconstruction quality on the ShapeNet dataset. Furthermore, we verify the effectiveness of our autodecoder in point cloud generation, upsampling, and unsupervised representation learning to demonstrate its performance on downstream tasks, which is comparable to the state‐of‐the‐art methods. We will make our code publicly available after peer review.

A component tree is a graph representation that encodes the connected components of the upper or lower level sets of a grayscale image . Consequently, the nodes of a component tree represent binary images of the encoded connected components. There exist various algorithms that efficiently extract information and attributes of nodes of a component tree by incrementally exploiting the subset relation encoding in the tree. However, to the best of our knowledge, there is no such incremental approach to extract the contours of the nodes. In this paper, we propose an efficient incremental method to compute the contours of the nodes of a component tree by counting the edges (sides) of contour pixels. In addition, we discuss our method’s time complexity. We also experimentally show that our proposed method is faster than the standard approach based on node reconstruction.

We present a Lagrangian dynamic importance Monte Carlo method without non-trivial random walks for solving the Velocity-Vorticity Poisson Equation (VVPE) in Smoothed Particle Hydrodynamics (SPH) for vortical flows. Key to our approach is the use of the Kinematic Vorticity Number (KVN) to detect vortex cores and to compute the KVN-based importance of each particle when solving the VVPE. We use Adaptive Kernel Density Estimation (AKDE) to extract a probability density distribution from the KVN for the the Monte Carlo calculations. Even though the distribution of the KVN can be non-trivial, AKDE yields a smooth and normalized result which we dynamically update at each time step. As we sample actual particles directly, the Lagrangian attributes of particle samples ensure that the continuously evolved KVN-based importance, modeled by the probability density distribution extracted from the KVN by AKDE, can be closely followed. Our approach enables effective vortical flow simulations with significantly reduced computational overhead and comparable quality to the classic Biot-Savart law that in contrast requires expensive global particle querying.

Bone cement filling is an important method for preventing osteoporosis and treating fractures. In bone cement filling surgery, the preparation and dosage of the cement usually depend on specific product manuals and the doctor’s experience. If bone cement is not used properly, it may cause additional damage. For teaching and auxiliary medical purposes, for example, assisting doctors to observe the possible flow of bone cement, this paper proposes a multiphase non-Newtonian fluid simulation method to simulate and visualize the flow behavior during the wet sand phase of bone cement blending and polymerization. Our method enables showing intuitively the application process of bone cement under different scene settings to obtain dynamic bone cement effects with high stability and performance. Compared with other methods, our method can simulate highly viscous mixed fluids efficiently and robustly, which supports our method’s usage in the aforementioned training and experimentation scenarios.

The use of subdivision schemes in applied and real-world contexts requires the development of conceptually simple algorithms that can be converted into fast and efficient implementation procedures. In the domain of interpolatory subdivision schemes, there is a demand for developing an algorithm capable of (i) reproducing all types of conic sections whenever the input data (in our case point-normal pairs) are arbitrarily sampled from them, (ii) generating a visually pleasing limit curve without creating unwanted oscillations, and (iii) having the potential to be naturally and easily extended to the bivariate case. In this paper we focus on the construction of an interpolatory subdivision scheme that meets all these conditions simultaneously. At the center of our construction lies a conic fitting algorithm that requires as few as four point-normal pairs for finding new edge points (and associated normals) in a subdivision step. Several numerical results are included to showcase the validity of our algorithm.

Abstract Gradient meshes are a vector graphics primitive formed by a regular grid of bicubic quad patches. They allow for the creation of complex geometries and colour gradients, with recent extensions supporting features such as local refinement and sharp colour transitions. While many methods exist for recolouring raster images, often achieved by modifying an automatically detected palette of the image, gradient meshes have not received the same amount of attention when it comes to global colour editing. We present a novel method that allows for real‐time palette‐based recolouring of gradient meshes, including gradient meshes constructed using local refinement and containing sharp colour transitions. We demonstrate the utility of our method on synthetic illustrative examples as well as on complex gradient meshes.

Abstract For vortex particle methods relying on SPH‐based simulations, the direct approach of iterating all fluid particles to capture velocity from vorticity can lead to a significant computational overhead during the Biot‐Savart summation process. To address this challenge, we present a Monte Carlo vortical smoothed particle hydrodynamics (MCVSPH) method for efficiently simulating turbulent flows within an SPH framework. Our approach harnesses a Monte Carlo estimator and operates exclusively within a pre‐sampled particle subset, thus eliminating the need for costly global iterations over all fluid particles. Our algorithm is decoupled from various projection loops which enforce incompressibility, independently handles the recovery of turbulent details, and seamlessly integrates with state‐of‐the‐art SPH‐based incompressibility solvers. Our approach rectifies the velocity of all fluid particles based on vorticity loss to respect the evolution of vorticity, effectively enforcing vortex motions. We demonstrate, by several experiments, that our MCVSPH method effectively preserves vorticity and creates visually prominent vortical motions.

Abstract We propose an SPH‐based method for simulating viscoelastic non‐Newtonian fluids within a multiphase framework. For this, we use mixture models to handle component transport and conformation tensor methods to handle the fluid's viscoelastic stresses. In addition, we consider a bonding effects network to handle the impact of microscopic chemical bonds on phase transport. Our method supports the simulation of both steady‐state viscoelastic fluids and discontinuous shear behavior. Compared to previous work on single‐phase viscous non‐Newtonian fluids, our method can capture more complex behavior, including material mixing processes that generate non‐Newtonian fluids. We adopt a uniform set of variables to describe shear thinning, shear thickening, and ordinary Newtonian fluids while automatically calculating local rheology in inhomogeneous solutions. In addition, our method can simulate large viscosity ranges under explicit integration schemes, which typically requires implicit viscosity solvers under earlier single‐phase frameworks.

Mesh-based image vectorization techniques have been studied for a long time, mostly owing to their compactness and flexibility in capturing image features. However, existing methods often lead to relatively dense meshes, especially when applied to images with high-frequency details or textures. We present a novel method that automatically vectorizes an image into a sparse collection of Coons patches whose size adapts to image features. To balance the number of patches and the accuracy of feature alignment, we generate the layout based on a harmonic cross field constrained by image features. We support T-junctions, which keeps the number of patches low and ensures local adaptation to feature density, naturally complemented by varying mesh-color resolution over the patches. Our experimental results demonstrate the utility, accuracy, and sparsity of our method.

Physics-based fluid simulation has played an increasingly important role in the computer graphics community. Recent methods in this area have greatly improved the generation of complex visual effects and its computational efficiency. Novel techniques have emerged to deal with complex boundaries, multiphase fluids, gas–liquid interfaces, and fine details. The parallel use of machine learning, image processing, and fluid control technologies has brought many interesting and novel research perspectives. In this survey, we provide an introduction to theoretical concepts underpinning physics-based fluid simulation and their practical implementation, with the aim for it to serve as a guide for both newcomers and seasoned researchers to explore the field of physics-based fluid simulation, with a focus on developments in the last decade. Driven by the distribution of recent publications in the field, we structure our survey to cover physical background; discretization approaches; computational methods that address scalability; fluid interactions with other materials and interfaces; and methods for expressive aspects of surface detail and control. From a practical perspective, we give an overview of existing implementations available for the above methods.

Particle-based simulations have become increasingly popular in real-time applications due to their efficiency and adaptability, especially for generating highly dynamic fluid effects. However, the swift and stable simulation of interactions among distinct fluids continues to pose challenges for current mixture model techniques. When using a single-mixture flow field to represent all fluid phases, numerical discontinuities in phase fields can result in significant losses of dynamic effects and unstable conservation of mass and momentum. To tackle these issues, we present an advanced implicit mixture model for smoothed particle hydrodynamics. Instead of relying on an explicit mixture field for all dynamic computations and phase transfers between particles, our approach calculates phase momentum sources from the mixture model to derive explicit and continuous velocity phase fields. We then implicitly obtain the mixture field using a phase-mixture momentum-mapping mechanism that ensures conservation of incompressibility, mass, and momentum. In addition, we propose a mixture viscosity model and establish viscous effects between the mixture and individual fluid phases to avoid instability under extreme inertia conditions. Through a series of experiments, we show that, compared to existing mixture models, our method effectively improves dynamic effects while reducing critical instability factors. This makes our approach especially well-suited for long-duration, efficiency-oriented virtual reality scenarios.

Existing flank milling path-planning methods typically lead to tiny gaps or overlaps between neighboring paths, which causes artifacts and imperfections in the workpiece. We propose a new multi-strip path-planning method for 5-axis flank milling of free-form surfaces which targets G 1 (tangent-plane) continuity of the neighboring strips along shared boundaries. While for some geometries one cannot achieve G 1 continuity and high approximation quality at the same time, our optimization framework offers a good trade-off between machining accuracy in terms of distance error and the G 1 connection of neighboring strips. We demonstrate our algorithm on synthetic free-form surfaces as well as on industrial benchmark datasets, showing that we are able to meet fine industrial tolerances and simultaneously significantly reduce the kink angle of adjacent strips, and consequently to improve the surface finish in terms of smoothness.

The use of social media has become so popular that people share photos every day on them. Automatic face recognition and tagging of people's photos have caused privacy preservation issues and some methods have been proposed for hiding the identity of presented people in these images. Blurring and blacking the face area, adding physical adversarial patches to the face, and adding adversarial masks are some proposed methods for this purpose. However, these methods particularly suffer from dissimilarity of the input and output images and inadequate performance in identity concealment from automatic face recognition (AFR) systems. In this paper, we propose the Generative Mask-guided Face Image Manipulation (GMFIM) model based on Generative Adversarial Networks (GANs) to apply imperceptible edits to the input face image to preserve the identity of the person in the image. Our model consists of a face mask module, a GAN-based optimization module, and a merge module. Different criteria are considered in the objective function of the optimization step to produce high-quality images that are as similar as possible to the input image while they cannot be recognized by AFR systems. The results of the experiments on different datasets show that our model provides promising results in terms of the quality of the generated images and the identity concealment performance.

Building on our original Virtual Ray Tracer tool, we present Virtual Ray Tracer 2.0, an interactive and gamified application that allows students/users to view and explore the ray tracing process in real-time. The application shows a scene containing a camera casting rays which interact with objects in the scene. Users are able to modify and explore ray properties such as their animation speed, the number of rays and their visual style, as well as the material properties of the objects in the scene. The goal of the application is to help the users – students of Computer Graphics and the general public – to better understand the ray tracing process and its characteristics. This includes not only the basics of ray tracing, but also more advanced concepts such as soft shadows. To invite users to learn and explore, various explanations and scenes are provided by the application at different levels of complexity, each with a step-by-step tutorial. Several user studies showed the effectiveness of the tool in supporting the understanding and teaching of ray tracing. The educational tool is built with the cross-platform engine Unity, and we make it fully available to be extended and/or adjusted to fit the requirements of courses at other institutions, educational tutorials, or of enthusiasts from the general public.

This paper describes a method for adding surface details to existing CAD models. Our approach is based on truncated powers, which allows us to align the added details with curved knot lines on the surface. Additionally, (truncated) powers give us precise control over the continuity of the perturbed surface across the (curved) knot lines. Our representation is compatible with current CAD/CAM practise and standards, and we showcase it on several examples.

We propose the use of curved triangles and mesh colours as a vector primitive for image vectorisation. We show that our representation has clear benefits for rendering performance, texture detail, as well as further editing of the resulting vector images. The proposed method focuses on efficiency, but it still leads to results that compare favourably with those from previous work. We show results over a variety of input images ranging from photos, drawings, paintings, all the way to designs and cartoons. We implemented several editing workflows facilitated by our representation: interactive user-guided vectorisation, and novel raster-style feature-aware brushing capabilities.

The ability to edit an image using intuitive commands and primitives is a desired feature for any image editing software. In this paper, we combine recent results in medial axes with the well-established morphological tree representations to develop an interactive image editing tool that provides global and local image manipulation using high-level primitives. We propose a new way to render interactive morphological trees using icicle plots and introduce different ways of manipulating spline-based medial axis transforms for grayscale and colored image editing. Different applications of the tool, such as watermark removal, image deformation, dataset augmentation for machine learning, artistic illumination manipulation, image rearrangement, and clothing design, are described and showcased on examples.

Abstract We propose USTNet, a novel deep learning approach designed for learning shape‐to‐shape translation from unpaired domains in an unsupervised manner. The core of our approach lies in disentangled representation learning that factors out the discriminative features of 3D shapes into content and style codes. Given input shapes from multiple domains, USTNet disentangles their representation into style codes that contain distinctive traits across domains and content codes that contain domain‐invariant traits. By fusing the style and content codes of the target and source shapes, our method enables us to synthesize new shapes that resemble the target style and retain the content features of source shapes. Based on the shared style space, our method facilitates shape interpolation by manipulating the style attributes from different domains. Furthermore, by extending the basic building blocks of our network from two‐class to multi‐class classification, we adapt USTNet to tackle multi‐domain shape‐to‐shape translation. Experimental results show that our approach can generate realistic and natural translated shapes and that our method leads to improved quantitative evaluation metric results compared to 3DSNet. Codes are available at https://Haoran226.github.io/USTNet .

Abstract Gradient meshes, an advanced vector graphics primitive, are widely used by designers for creating scalable vector graphics. Traditional variants require a regular rectangular topology, which is a severe design restriction. The more advanced subdivision gradient mesh allows for an arbitrary manifold topology and is based on subdivision techniques to define the resulting colour surface. This also allows the artists to manipulate the geometry and colours at various levels of subdivision. Recent advances allow for the interpolation of both geometry and colour, local detail following edits at coarser subdivision levels and sharp colour transitions. A shortcoming of all existing methods is their dependence on global refinement, which makes them unsuitable for real‐time (commercial) design applications. We present a novel method that incorporates the idea of feature‐adaptive subdivision and uses approximating patches suitable for hardware tessellation with real‐time performance. Further novel features include multiple interaction mechanisms and self‐intersection prevention during interactive design/editing.

Medial axes are well-known descriptors used for representing, manipulating, and compressing binary images. In this paper, we present a full pipeline for computing a stable and accurate piece-wise B-spline representation of Medial Axis Transforms (MATs) of binary images. A comprehensive evaluation on a benchmark shows that our method, called Spline-based Medial Axis Transform (SMAT), achieves very high compression ratios while keeping quality high. Compared with the regular MAT representation, the SMAT yields a much higher compression ratio at the cost of a slightly lower image quality. We illustrate our approach on a multi-scale SMAT representation, generating super-resolution images, and free-form binary image deformation.

Abstract A major issue in smoothed particle hydrodynamics (SPH) approaches is the numerical dissipation during the projection process, especially under coarse discretizations. High‐frequency details, such as turbulence and vortices, are smoothed out, leading to unrealistic results. To address this issue, we introduce a vorticity refinement (VR) solver for SPH fluids with negligible computational overhead. In this method, the numerical dissipation of the vorticity field is recovered by the difference between the theoretical and the actual vorticity, so as to enhance turbulence details. Instead of solving the Biot‐Savart integrals, a stream function, which is easier and more efficient to solve, is used to relate the vorticity field to the velocity field. We obtain turbulence effects of different intensity levels by changing an adjustable parameter. Since the vorticity field is enhanced according to the curl field, our method can not only amplify existing vortices, but also capture additional turbulence. Our VR solver is straightforward to implement and can be easily integrated into existing SPH methods.

We propose a generalised B-spline construction that extends uniform bicubic B-splines to multisided regions spanned over extraordinary vertices in quadrilateral meshes. We show how the structure of the generalised Bézier patch introduced by Várady et al. can be adjusted to work with B-spline basis functions. We create ribbon surfaces based on B-splines using special basis functions. The resulting multisided surfaces are C2 continuous internally and connect with G2 continuity to adjacent regular and other multisided B-splines patches. We visually assess the quality of these surfaces and compare them to Catmull–Clark limit surfaces on several challenging geometrical configurations.

Existing techniques that convert B-rep (boundary representation) patches into Clough-Tocher splines guarantee watertight, that is C0, conversion results across B-rep edges. In contrast, our approach ensures global tangent-plane, that is G1, continuity of the converted B-rep CAD models. We achieve this by careful boundary curve and normal vector management, and by converting the input models into Shirman-Séquin macro-elements near their (trimmed) B-rep edges. We propose several different variants and compare them with respect to their locality, visual quality, and difference with the input B-rep CAD model. Although the same global G1 continuity can also be achieved by conversion techniques based on subdivision surfaces, our approach uses triangular splines and thus enjoys full compatibility with CAD.

We present co-skeletons, a new method that computes consistent curve skeletons for 3D shapes from a given family. We compute co-skeletons in terms of sampling density and semantic relevance, while preserving the desired characteristics of traditional, per-shape curve skeletonization approaches. We take the curve skeletons extracted by traditional approaches for all shapes from a family as input, and compute semantic correlation information of individual skeleton branches to guide an edge-pruning process via skeleton-based descriptors, clustering, and a voting algorithm. Our approach achieves more concise and family-consistent skeletons when compared to traditional per-shape methods. We show the utility of our method by using co-skeletons for shape segmentation and shape blending on real-world data.

Among the bivariate subdivision schemes available, spline-based schemes, such as Catmull-Clark and Loop, are the most commonly used ones. These schemes have known continuity and can be evaluated at arbitrary parameter values. In this work, we develop a C1 spline-based scheme based on cubic half-box splines. Although the individual surface patches are triangular, the associated control net is three-valent and thus consists in general of mostly hexagons. In addition to introducing stencils that can be applied in extraordinary regions of the mesh, we also consider boundaries. Moreover, we show that the scheme exhibits ineffective eigenvectors. Finally, we briefly consider architectural geometry and isogeometric analysis as selected applications.

We extend the gradient mesh vector graphics primitive with procedural noise functions. Specifically, we couple Perlin, Worley and Gabor noise to the gradient mesh. We allow local parameters controlling the noise functions to be defined at the vertices of the mesh. The parameters are interpolated along with the geometry similarly to how colour is interpolated in an ordinary gradient mesh, allowing for spatially varying noise patterns. These noisy gradient meshes facilitate a sparse representation of high frequency regions along with underlying smooth colour gradients. The meshes are easy to edit and efficient to evaluate on graphics hardware , making them a suitable candidate for inclusion in modern vector graphics authoring tools. We demonstrate the utility of our method on gradient meshes with added noise functions. Additionally, we show that the approach can be used in combination with regular surface meshes where noise functions are used to govern their displacement mapping.

The free-form deformation (FFD) method deforms geometry in n-dimensional space by employing an n-variate function to deform (parts of) the ambient space. The original method pioneered by Sederberg and Parry in 1986 uses trivariate tensor-product Bernstein polynomials in R3 and is controlled as a Bézier volume. We propose an extension based on truncated hierarchical B-splines (THB-splines). This offers hierarchical and local refinability, an efficient implementation due to reduced supports of THB-splines, and intuitive control point hiding during FFD interaction. Additionally, we address the issue of fold-overs by efficiently checking the injectivity of the hierarchical deformation in real-time.

We propose two generalisations of Gregory patches to faces of any valency by using generalised barycentric coordinates in combination with two kinds of multisided Bézier patches. Our first construction builds on S-patches to generalise triangular Gregory patches. The local construction of Chiyokura and Kimura providing G1 continuity between adjoining Bézier patches is generalised so that the novel Gregory S-patches of any valency can be smoothly joined to one another. Our second construction makes a minor adjustment to the generalised Bézier patch structure to allow for cross-boundary derivatives to be defined independently per side. We show that the corresponding blending functions have the inherent ability to blend ribbon data much like the rational blending functions of Gregory patches. Both constructions take as input a polygonal mesh with vertex normals and provide G1 surfaces interpolating the input vertices and normals. Due to the full locality of the methods, they are well suited for geometric modelling as well as computer graphics applications relying on hardware tessellation.

Abstract Gradient mesh design tools allow users to create detailed scalable images, traditionally through the creation and manipulation of a (dense) mesh with regular rectangular topology. Through recent advances it is now possible to allow gradient meshes to have arbitrary manifold topology, using a modified Catmull‐Clark subdivision scheme to define the resultant geometry and colour [ LKSD17 ]. We present two novel methods to allow local and hierarchical refinement of both colour and geometry for such subdivision gradient meshes. Our methods leverage the mesh properties that the particular subdivision scheme ensures. In both methods, the artists enjoy all the standard capabilities of manipulating the mesh and the associated colour gradients at the coarsest level as well as locally at refined levels. Further novel features include interpolation of both position and colour of the vertices of the input meshes, local detail follows coarser‐level edits, and support for sharp colour transitions, all at any level in the hierarchy offered by subdivision.

We present a local refinement approach for gradient meshes, a primitive commonly used in the design of vector illustrations with complex colour propagation. Local refinement allows the artist to add more detail only in the regions where it is needed, as opposed to global refinement which often clutters the workspace with undesired detail and potentially slows down the workflow. Moreover, in contrast to existing implementations of gradient mesh refinement, our approach ensures mathematically exact refinement. Additionally, we introduce a branching feature that allows for a wider range of mesh topologies, as well as a feature that enables sharp colour transitions similar to diffusion curves, which turn the gradient mesh into a more versatile and expressive vector graphics primitive.

We continue the study of rational envelope (RE) surfaces. Although these surfaces are parametrized with the help of square roots, when considering an RE patch as the medial surface transform in 4D of a spatial domain it yields a rational parametrization of the domain’s boundary, i.e., the envelope of the corresponding 2 -parameter family of spheres. We formulate efficient algorithms for G 1 data interpolation using RE surfaces and apply the developed methods to rational skinning and blending of sets of spheres and cones/cylinders, respectively. Our results are demonstrated on several computed examples of skins and blends with rational parametrizations.

We investigate a class of stream surfaces that expand in time as much as possible. Given a vector field, we look for seed curves that locally propagate in time in a stretch-maximizing manner, i.e., curves that infinitesimally expand most progressively. We show that such a curve is generically unique at every point in an incompressible flow and offers a very good initial guess for a stretch-maximizing stream surface. With the application of efficient fluid advection–diffusion in mind, we optimize fluid injection towards optimal advection and show several examples on benchmark datasets.

Barycentric coordinates were introduced by Möbius in 1827 as an alternative to Cartesian coordinates. They describe points relative to the vertices of a simplex and are commonly used to express the linear interpolant of data given at these vertices. Generalized barycentric coordinates and kernels extend this idea from simplices to polyhedra and smooth domains. In this paper, we focus on Wachspress coordinates and Wachspress kernels with respect to strictly convex planar domains. Since Wachspress kernels can be evaluated analytically only in special cases, a common way to approximate them is to discretize the domain by an inscribed polygon and to use Wachspress coordinates, which have a simple closed form. We show that this discretization, which is known to converge quadratically, is safe in the sense that the Wachspress coordinates used in this process are well-defined not only over the inscribed polygon, but over the entire original domain.

Abstract Gradient meshes are a 2D vector graphics primitive where colour is interpolated between mesh vertices. The current implementations of gradient meshes are restricted to rectangular mesh topology. Our new interpolation method relaxes this restriction by supporting arbitrary manifold topology of the input gradient mesh. Our method is based on the Catmull‐Clark subdivision scheme, which is well‐known to support arbitrary mesh topology in 3D. We adapt this scheme to support gradient mesh colour interpolation, adding extensions to handle interpolation of colours of the control points, interpolation only inside the given colour space and emulation of gradient constraints seen in related closed‐form solutions. These extensions make subdivision a viable option for interpolating arbitrary‐topology gradient meshes for 2D vector graphics.

NURBS patches have a serious restriction: they are constrained to a strict rectangular topology. This means that a request to insert a single new control point will cause a row of control points to appear across the NURBS patch, a global refinement of control. We investigate a method that can hide unwanted control points from the user so that the user’s interaction is with local, rather than global, refinement. Our method requires only straightforward modification of the user interface and the data structures that represent the control mesh, making it simpler than alternatives that use hierarchical or T-constructions. Our results show that our method is effective in many cases but has limitations where inserting a single new control point in certain cases will still cause a cascade of new control points to appear across the NURBS patch.

Minkowski Pythagorean hodograph (MPH) curves provide a means for representing domains with rational boundaries via the medial axis transform. Based on the observation that MPH curves are not the only curves that yield rational envelopes, we define and study rational envelope (RE) curves that generalise MPH curves while maintaining the rationality of their associated envelopes. To demonstrate the utility of RE curves, we design a simple interpolation algorithm using RE curves, which is in turn used to produce rational surface blends between canal surfaces. Additionally, we initiate the study of rational envelope surfaces as a surface analogy to RE curves.

We investigate the close correspondence between barycentric coordinates and barycentric kernels from the point of view of the limit process when finer and finer polygons converge to a smooth convex domain. We show that any barycentric kernel is the limit of a set of barycentric coordinates and prove that the convergence rate is quadratic. Our convergence analysis extends naturally to barycentric interpolants and mappings induced by barycentric coordinates and kernels. We verify our theoretical convergence results numerically on several examples.

CAD models generally consist of multiple NURBS patches, both trimmed and untrimmed. There is a long-standing challenge that trimmed NURBS patches cause unavoidable gaps in the model. We address this by converting multiple NURBS patches to a single untrimmed NURBS-compatible subdivision surface in a three stage process. First, for each patch, we generate in domain space a quadrangulation that follows boundary edges of the patch and respects the knot spacings along edges. Second, the control points of the corresponding subdivision patch are computed in model space. Third, we merge the subdivision patches across their common boundaries to create a single subdivision surface. The converted model is gap-free and can maintain inter-patch continuity up to C2.

We propose a cluster-based approach to point set saliency detection, a challenge since point sets lack topological information. A point set is first decomposed into small clusters, using fuzzy clustering. We evaluate cluster uniqueness and spatial distribution of each cluster and combine these values into a cluster saliency function. Finally, the probabilities of points belonging to each cluster are used to assign a saliency to each point. Our approach detects fine-scale salient features and uninteresting regions consistently have lower saliency values. We evaluate the proposed saliency model by testing our saliency-based keypoint detection against a 3D interest point detection benchmark. The evaluation shows that our method achieves a good balance between false positive and false negative error rates, without using any topological information.

Traditionally, modelling using spline curves and surfaces is facilitated by control points. We propose to enhance the modelling process by the use of control vectors. This improves upon existing spline representations by providing such facilities as modelling with local (semi-sharp) creases, vanishing and diagonal features, and hierarchical editing. While our prime interest is in surfaces, most of the ideas are more simply described in the curve context. We demonstrate the advantages provided by control vectors on several curve and surface examples and explore avenues for future research on control vectors in the contexts of geometric modelling and finite element analysis based on splines, and B-splines and subdivision in particular.

The boundary representations (B-reps) that are used to represent shape in Computer-Aided Design systems create unavoidable gaps at the face boundaries of a model. Although these inconsistencies can be kept below the scale that is important for visualisation and manufacture, they cause problems for many downstream tasks, making it difficult to use CAD models directly for simulation or advanced geometric analysis, for example. Motivated by this need for watertight models, we address the problem of converting B-rep models to a collection of cubic C1 Clough–Tocher splines. These splines allow a watertight join between B-rep faces, provide a homogeneous representation of shape, and also support local adaptivity. We perform a comparative study of the most prominent Clough–Tocher constructions and include some novel variants. Our criteria include visual fairness, invariance to affine reparameterisations, polynomial precision and approximation error. The constructions are tested on both synthetic data and CAD models that have been triangulated. Our results show that no construction is optimal in every scenario, with surface quality depending heavily on the triangulation and parameterisation that are used.

Abstract A challenge in vector graphics is to define primitives that offer flexible manipulation of colour gradients. We propose a new primitive, called a shading curve, that supports explicit and local gradient control. This is achieved by associating shading profiles to each side of the curve. These shading profiles, which can be manually manipulated, represent the colour gradient out from their associated curves. Such explicit and local gradient control is challenging to achieve via the diffusion curve process, introduced in 2008, because it offers only implicit control of the colour gradient. We resolve this problem by using subdivision surfaces that are constructed from shading curves and their shading profiles.

This paper introduces a novel method to convert trimmed NURBS surfaces to untrimmed subdivision surfaces with Bézier edge conditions. We take a NURBS surface and its trimming curves as input, from this we automatically compute a base mesh, the limit surface of which fits the trimmed NURBS surface to a specified tolerance. We first construct the topology of the base mesh by performing a cross-field based decomposition in parameter space. The number and positions of extraordinary vertices required to represent the trimmed shape can be automatically identified by smoothing a cross field bounded by the parametric trimming curves. After the topology construction, the control point positions in the base mesh are calculated based on the limit stencils of the subdivision scheme and constraints to achieve tangential continuity across the boundary. Our method provides the user with either an editable base mesh or a fine mesh whose limit surface approximates the input within a certain tolerance. By integrating the trimming curve as part of the desired limit surface boundary, our conversion can produce gap-free models. Moreover, since we use tangential continuity across the boundary between adjacent surfaces as constraints, the converted surfaces join with G1 continuity.

Abstract We deal with subdivision schemes based on arbitrary degree B‐splines. We focus on extraordinary knots which exhibit various levels of complexity in terms of both valency and multiplicity of knot lines emanating from such knots. The purpose of truncated multiple knot lines is to model creases which fair out. Our construction supports any degree and any knot line multiplicity and provides a modelling framework familiar to users used to B‐splines and NURBS systems.

Abstract We explore a method for generalising Pixar semi‐sharp creases from the univariate cubic case to arbitrary degree subdivision curves. Our approach is based on solving simple matrix equations. The resulting schemes allow for greater flexibility over existing methods, via control vectors. We demonstrate our results on several high‐degree univariate examples and explore analogous methods for subdivision surfaces.

Our goal is to find subdivision rules at creases in arbitrary degree subdivision for piece-wise polynomial curves, but without introducing new control points e.g. by knot insertion. Crease rules are well understood for low degree (cubic and lower) curves. We compare three main approaches: knot insertion, ghost points, and modifying subdivision rules. While knot insertion and ghost points work for arbitrary degrees for B-splines, these methods introduce unnecessary (ghost) control points. The situation is not so simple in modifying subdivision rules. Based on subdivision and subspace selection matrices, a novel approach to finding boundary and sharp subdivision rules that generalises to any degree is presented. Our approach leads to new higher-degree polynomial subdivision schemes with crease control without introducing new control points.

In a recent paper, Warren, Schaefer, Hirani, and Desbrun proposed a simple method of interpolating a function defined on the boundary of a smooth convex domain, using an integral kernel with properties similar to those of barycentric coordinates on simplexes. When applied to vector-valued data, the interpolation can map one convex region into another, with various potential applications in computer graphics, such as curve and image deformation. In this paper we establish some basic mathematical properties of barycentric kernels in general, including the interpolation property and a formula for the Jacobian of the mappings they generate. We then use this formula to prove the injectivity of the mapping of Warren et al.

In this paper, we describe an algorithm for generating an exact rational envelope of a two-parameter family of spheres given by a quadratic patch in R3, 1, which is considered as a medial surface transform (MST) of a spatial domain. Recently, it has been proved that quadratic triangular Bézier patches in R3, 1 belong to the class of MOS surfaces (i.e., surfaces providing rational envelopes of the associated two-parameter family of spheres). We give a detailed description of the symbolic and numerical steps of the envelope algorithm and study the error involved in the numerical part. The presented method is then demonstrated on several examples. Moreover, since quadratic MOS patches are capable of producing C1 approximations of MSTs, this algorithm offers a good basis for consequent methods, e.g. computing rational approximations of envelopes associated to general (free-form) MSTs and inner offsets trimming.