We introduce NeuBase , a neural parametric surface representation that both accurately fits target surfaces with fine geometric detail and supports intuitive real time surface deformation. NeuBase consists of a Catmull-Clark subdivision base surface and an offset field defined by a set of neural basis functions encoded via a neural map. By construction, NeuBase surfaces exhibit four fundamental geometric properties, i.e., linearity, locality, smoothness, and affine equivariance, enabling real-time, direct manipulation without retraining the neural network. In addition, we propose a scalable neural map that maintains memory efficiency even for complex shapes with dense control meshes. Experiments on a large-scale dataset demonstrate that our method achieves better fitting accuracy than state-of-the-art neural parametric surface representations.

Piecewise parametric surfaces have long been established as prevalent geometric representations; however, they often require surface refinement or sophisticated quadrangulation to accurately represent complex geometries. Geometric deep learning has shown that neural networks can provide greater representational power than conventional methods. Nevertheless, approaches using a single parametric surface for shape fitting struggle to capture fine-grained geometric details, while multi-patch methods fail to ensure seamless connections between adjacent patches. We present Neural Piecewise Parametric Surfaces ( NeuPPS ), the first piecewise neural surface representation that allows for coarse patch layouts composed of arbitrary n -sided surface patches to model complex surface geometries with high precision, offering enhanced flexibility compared with traditional parametric surfaces. This new surface representation guarantees, by construction, the continuity between adjacent patches, a property that other neural patch-based approaches cannot ensure. Two novel components are introduced: a learnable feature complex and a continuous mapping function approximated by multi-layer perceptrons (MLPs). We apply the proposed NeuPPS to surface fitting and shape space learning tasks. Extensive experiments demonstrate the advantages of NeuPPS over traditional parametric representations and existing patch-based learning approaches.

Cross fields play a critical role in various geometry processing tasks, especially for quad mesh generation. Existing methods for cross field generation often struggle to balance computational efficiency with generation quality, using slow per-shape optimization. We introduce CrossGen , a novel framework that supports both feed-forward prediction and latent generative modeling of cross fields for quad meshing by unifying geometry and cross field representations within a joint latent space. Our method enables extremely fast computation of high-quality cross fields of general input shapes, typically within one second without per-shape optimization. Our method assumes a point-sampled surface, also called a point-cloud surface , as input, so we can accommodate various surface representations by a straightforward point sampling process. Using an auto-encoder network architecture, we encode input point-cloud surfaces into a sparse voxel grid with fine-grained latent spaces, which are decoded into both SDF-based surface geometry and cross fields (see the teaser figure). We also contribute a dataset of models with both high-quality signed distance fields (SDFs) representations and their corresponding cross fields, and use it to train our network. Once trained, the network is capable of computing a cross field of an input surface in a feed-forward manner, ensuring high geometric fidelity, noise resilience, and rapid inference. Furthermore, leveraging the same unified latent representation, we incorporate a diffusion model for computing cross fields of new shapes generated from partial input, such as sketches. To demonstrate its practical applications, we validate CrossGen on the quad mesh generation task for a large variety of surface shapes. Experimental results demonstrate that CrossGen generalizes well across diverse shapes and consistently yields high-fidelity cross fields, thus facilitating the generation of high-quality quad meshes.

We propose a method, named DualMesh-UDF, to extract a surface from unsigned distance functions (UDFs), encoded by neural networks, or neural UDFs. Neural UDFs are becoming increasingly popular for surface representation because of their versatility in presenting surfaces with arbitrary topologies, as opposed to the signed distance function that is limited to representing a closed surface. However, the applications of neural UDFs are hindered by the notorious difficulty in extracting the target surfaces they represent. Recent methods for surface extraction from a neural UDF suffer from significant geometric errors or topological artifacts due to two main difficulties: (1) A UDF does not exhibit sign changes; and (2) A neural UDF typically has substantial approximation errors.DualMesh-UDF addresses these two difficulties. Specifically, given a neural UDF encoding a target surface $\bar S$ to be recovered, we first estimate the tangent planes of $\bar S$ at a set of sample points close to $\bar S$. Next, we organize these sample points into local clusters, and for each local cluster, solve a linear least squares problem to determine a final surface point. These surface points are then connected to create the output mesh surface, which approximates the target surface. The robust estimation of the tangent planes of the target surface and the subsequent minimization problem constitute our core strategy, which contributes to the favorable performance of DualMesh-UDF over other competing methods. To efficiently implement this strategy, we employ an adaptive Octree. Within this framework, we estimate the location of a surface point in each of the octree cells identified as containing part of the target surface. Extensive experiments show that our method outperforms existing methods in terms of surface reconstruction quality while maintaining comparable computational efficiency.

We propose a robust normal estimation method for both point clouds and meshes using a low rank matrix approximation algorithm. First, we compute a local isotropic structure for each point and find its similar, non-local structures that we organize into a matrix. We then show that a low rank matrix approximation algorithm can robustly estimate normals for both point clouds and meshes. Furthermore, we provide a new filtering method for point cloud data to smooth the position data to fit the estimated normals. We show the applications of our method to point cloud filtering, point set upsampling, surface reconstruction, mesh denoising, and geometric texture removal. Our experiments show that our method generally achieves better results than existing methods.

Abstract We construct a family of barycentric coordinates for 2D shapes including non‐convex shapes, shapes with boundaries, and skeletons. Furthermore, we extend these coordinates to 3D and arbitrary dimension. Our approach modifies the construction of the Floater‐Hormann‐Kós family of barycentric coordinates for 2D convex shapes. We show why such coordinates are restricted to convex shapes and show how to modify these coordinates to extend to discrete manifolds of co‐dimension 1 whose boundaries are composed of simplicial facets. Our coordinates are well‐defined everywhere (no poles) and easy to evaluate. While our construction is widely applicable to many domains, we show several examples related to image and mesh deformation.

Retrieving salient structure from textured images is an important but difficult problem in computer vision because texture, which can be irregular, anisotropic, non-uniform and complex, shares many of the same properties as structure. Observing that salient structure in a textured image should be piece-wise smooth, we present a method to retrieve such structures using an minimization of a modified form of the relative total variation metric. Thanks to the characteristics shared by texture and small structures, our method is effective at retrieving structure based on scale as well. Our method outperforms state-of-art methods in texture removal as well as scale-space filtering. We also demonstrate our method's ability in other applications such as edge detection, clip art compression artifact removal, and inverse half-toning.

Abstract This paper presents a new preconditioning technique for large‐scale geometric optimization problems, inspired by applications in mesh parameterization. Our positive (semi‐)definite preconditioner acts on the gradients of optimization problems whose variables are positions of the vertices of a triangle mesh in ℝ2or of a tetrahedral mesh in ℝ3, converting localized distortion gradients into the velocity of a globally near‐rigid motion via a linear solve. We pose our preconditioning tool in terms of the Killing energy of a deformation field and provide new efficient formulas for constructing Killing operators on triangle and tetrahedral meshes. We demonstrate that our method is competitive with state‐of‐the‐art algorithms for locally injective parameterization using a variety of optimization objectives and show applications to two‐ and three‐dimensional mesh deformation.

Bijective maps are commonly used in many computer graphics and scientific computing applications, including texture, displacement, and bump mapping. However, their computation is numerically challenging due to the global nature of the problem, which makes standard smooth optimization techniques prohibitively expensive. We propose to use a scaffold structure to reduce this challenging and global problem to a local injectivity condition. This construction allows us to benefit from the recent advancements in locally injective maps optimization to efficiently compute large scale bijective maps (both in 2D and 3D), sidestepping the need to explicitly detect and avoid collisions. Our algorithm is guaranteed to robustly compute a globally bijective map, both in 2D and 3D. To demonstrate the practical applicability, we use it to compute globally bijective single patch parametrizations, to pack multiple charts into a single UV domain, to remove self-intersections from existing models, and to deform 3D objects while preventing self-intersections. Our approach is simple to implement, efficient (two orders of magnitude faster than competing methods), and robust, as we demonstrate in a stress test on a parametrization dataset with over a hundred meshes.

We present a method for constructing almost-everywhere curvature-continuous, piecewise-quadratic curves that interpolate a list of control points and have local maxima of curvature only at the control points. Our premise is that salient features of the curve should occur only at control points to avoid the creation of features unintended by the artist. While many artists prefer to use interpolated control points, the creation of artifacts, such as loops and cusps, away from control points has limited the use of these types of curves. By enforcing the maximum curvature property, loops and cusps cannot be created unless the artist intends for them to be. To create such curves, we focus on piecewise quadratic curves, which can have only one maximum curvature point. We provide a simple, iterative optimization that creates quadratic curves, one per interior control point, that meet with G 2 continuity everywhere except at inflection points of the curve where the curves are G 1 . Despite the nonlinear nature of curvature, our curves only obtain local maxima of the absolute value of curvature only at interpolated control points.

Abstract This paper presents a method to selectively elevate the degree of an S‐Patch of arbitrary dimension. We consider not only S‐Patches with 2D domains but 3D and higher‐dimensional domains as well, of which volumetric cage deformations are a subset. We show how to selectively insert control points of a higher degree patch into a lower degree patch while maintaining the polynomial reproduction order of the original patch. This process allows the user to elevate the degree of only one portion of the patch to add new degrees of freedom or maintain continuity with adjacent patches without elevating the degree of the entire patch, which could create far more degrees of freedom than necessary. Finally we show an application to cage‐based deformations where we increase the number of control points by elevating the degree of a subset of cage faces. The result is a cage deformation with higher degree triangular Bézier functions on a subset of cage faces but no interior control points.

We present a fully automatic method for generating guaranteed bijective surface parameterizations from triangulated 3D surfaces partitioned into charts. We do so by using a distortion metric that prevents local folds of triangles in the parameterization and a barrier function that prevents intersection of the chart boundaries. In addition, we show how to modify the line search of an interior point method to directly compute the singularities of the distortion metric and barrier functions to maintain a bijective map. By using an isometric metric that is efficient to compute and a spatial hash to accelerate the evaluation and gradient of the barrier function for the boundary, we achieve fast optimization times. Unlike previous methods, we do not require the boundary be constrained by the user to a non-intersecting shape to guarantee a bijection, and the boundary of the parameterization is free to change shape during the optimization to minimize distortion.

Abstract Our method approximates exact texture filtering for arbitrary scales and translations of an image while taking into account the performance characteristics of modern GPUs. Our algorithm is fast because it accesses textures with a high degree of spatial locality. Using bilinear samples guarantees that the texels we read are in a regular pattern and that we use a hardware accelerated path. We control the texel weights by manipulating the u, v parameters of each sample and the blend factor between the samples. Our method is similar in quality to Cardinality‐Constrained Texture Filtering [ MS13 ] but runs two times faster.

Abstract We present a method of analytically rasterizing shapes that have curved boundaries and linear color gradients using piecewise polynomial prefilters. By transforming the convolution of filters with the image from an integral over area into a boundary integral, we find closed‐form expressions for rasterizing shapes. We show that a polynomial expression can be used to rasterize any combination of polynomial curves and filters. Our rasterizer also handles rational quadratic boundaries, which allows us to evaluate circles and ellipses. We apply our technique to rasterizing vector graphics and show that our derivation gives an efficient implementation as a scanline rasterizer.

We present an algorithm for denoising triangulated models based on L 0 minimization. Our method maximizes the flat regions of the model and gradually removes noise while preserving sharp features. As part of this process, we build a discrete differential operator for arbitrary triangle meshes that is robust with respect to degenerate triangulations. We compare our method versus other anisotropic denoising algorithms and demonstrate that our method is more robust and produces good results even in the presence of high noise.

We present a method to create high-quality sampling filters by combining a prescribed number of texels from several resolutions in a mipmap. Our technique provides fine control over the number of texels we read per texture sample so that we can scale quality to match a memory bandwidth budget. Our method also has a fixed cost regardless of the filter we approximate, which makes it feasible to approximate higher-quality filters such as a Lánczos 2 filter in real-time rendering. To find the best set of texels to represent a given sampling filter and what weights to assign those texels, we perform a cardinality-constrained least-squares optimization of the most likely candidate solutions and encode the results of the optimization in a small table that is easily stored on the GPU. We present results that show we accurately reproduce filters using few texel reads and that both quality and speed scale smoothly with available bandwidth. When using four or more texels per sample, our image quality exceeds that of trilinear interpolation.

Abstract We provide a method for improving the parameterization of patching schemes that approximate Catmull‐Clark subdivision surfaces, such that the new parameterization conforms better to that of the original subdivision surface. We create this reparameterization in real‐time using a method that only depends on the topology of the surface and is independent of the surface's geometry. Our method can handle patches with more than one extraordinary vertex and avoids the combinatorial increase in both complexity and storage associated with multiple extraordinary vertices. Moreover, the reparameterization function is easy to implement and fast.

Abstract We present a method of generating mipmaps that takes into account the distortions due to the parameterization of a surface. Existing algorithms for generating mipmaps assume that the texture is isometrically mapped to the surface and ignore the actual surface parameterization. Our method correctly downsamples warped textures by assigning texels weights proportional to their area on a surface. We also provide a least‐squares approach to filtering over these warped domains that takes into account the postfilter used by the GPU. Our method improves texture filtering for most models but only modifies mipmap generation, requires no modification of art assets or rasterization algorithms, and does not affect run‐time performance.

Abstract We present a method for analytically calculating an anti‐aliased rasterization of arbitrary polygons or fonts bounded by Bézier curves in 2D as well as oriented triangle meshes in 3D. Our algorithm rasterizes multiple resolutions simultaneously using a hierarchical wavelet representation and is robust to degenerate inputs. We show that using the simplest wavelet, the Haar basis, is equivalent to performing a box‐filter to the rasterized image. Because we evaluate wavelet coefficients through line integrals in 2D, we are able to derive analytic solutions for polygons that have Bézier curve boundaries of any order, and we provide solutions for quadratic and cubic curves. In 3D, we compute the wavelet coefficients through analytic surface integrals over triangle meshes and show how to do so in a computationally efficient manner.

Abstract We propose a method for calculating deformations of models by deforming a low‐resolution mesh and adding details while ensuring that the details we add satisfy a set of constraints. Our method builds a low‐resolution representation of a mesh by using edge collapses and performs an as‐rigid‐as‐possible deformation on the simplified mesh. We then add back details by reversing edge‐collapses so that the shape of the mesh is locally preserved. While adding details, we deform the mesh to match the predicted positions of constraints so that constraints on the full‐resolution mesh are met. Our method operates on meshes with arbitrary triangulations, satisfies constraints over the full‐resolution mesh and converges quickly.

Abstract We present a method for calculating the boundary of objects from Discrete Indicator Functions that store 2‐material volume fractions with a high degree of accuracy. Although Marching Cubes and its derivatives are effective methods for calculating contours of functions sampled over discrete grids, these methods perform poorly when contouring non‐smooth functions such as Discrete Indicator Functions. In particular, Marching Cubes will generate surfaces that exhibit aliasing and oscillations around the exact surface. We derive a simple solution to remove these problems by using a new function to calculate the positions of vertices along cell edges that is efficient, easy to implement, and does not require any optimization or iteration. Finally, we provide empirical evidence that the error introduced by our contouring method is significantly less than is introduced by Marching Cubes.

Abstract We present a method for generating scales and scale‐like structures on a polygonal mesh through surface replacement. As input, we require a triangular mesh that will be covered with scales and one or more proxy‐models to be used as the scale's shape. A user begins scale generation by drawing a lateral line on the model to control the distribution and orientation of scales on the surface. We then create a vector field over the surface to control an anisotropic Voronoi tessellation, which represents the region occupied by each scale. Next we replace these regions by cutting the proxy model to match the boundary of the Voronoi region and deform the cut model onto the surface. The result is a fully connected 2‐manifold that is suitable for subsequent post‐processing applications like surface subdivision.

Abstract While animation using barycentric coordinates or other automatic weight assignment methods has become a popular method for shape deformation, the global nature of the weights limits their use for real‐time applications. We present a method that reduces the number of control points influencing a vertex to a user‐specified number such that the deformations created by the reduced weight set resemble that of the original deformation. To do so we show how to set up a Poisson minimization problem to solve for a reduced weight set and illustrate its advantages over other weight reduction methods. Not only does weight reduction lower the amount of storage space necessary to deform these models but also allows GPU acceleration of the resulting deformations. Our experiments show that we can achieve a factor of 100 increase in speed over CPU deformations using the full weight set, which makes real‐time deformations of large models possible.

Abstract We provide a simple method that extracts an isosurface that is manifold and intersection‐free from a function over an arbitrary octree. Our method samples the function dual to minimal edges, faces, and cells, and we show how to position those samples to reconstruct sharp and thin features of the surface. Moreover, we describe an error metric designed to guide octree expansion such that flat regions of the function are tiled with fewer polygons than curved regions to create an adaptive polygonalization of the isosurface. We then show how to improve the quality of the triangulation by moving dual vertices to the isosurface and provide a topological test that guarantees we maintain the topology of the surface. While we describe our algorithm in terms of extracting surfaces from volumetric functions, we also show that our algorithm extends to generating manifold level sets of co‐dimension 1 of functions of arbitrary dimension.

Abstract We propose a new family of barycentric coordinates that have closed‐forms for arbitrary 2D polygons. These coordinates are easy to compute and have linear precision even for open polygons. Not only do these coordinates have linear precision, but we can create coordinates that reproduce polynomials of a set degree m as long as degree m polynomials are specified along the boundary of the polygon. We also show how to extend these coordinates to interpolate derivatives specified on the boundary.

We present a method for parameterizing subdivision surfaces in an as-rigid-as-possible fashion. While much work has concentrated on parameterizing polygon meshes, little if any work has focused on subdivision surfaces despite their popularity. We show that polygon parameterization methods produce suboptimal results when applied to subdivision surfaces and describe how these methods may be modified to operate on subdivision surfaces. We also describe a method for creating extended charts to further reduce the distortion of the parameterization. Finally we demonstrate how to take advantage of the multi-resolution structure of subdivision surfaces to accelerate convergence of our optimization.

We propose a technique that takes a triangulated surface as input and outputs a surface with the same topology but altered geometry such that each polygon falls into a set of discrete equivalence classes. We begin by describing an error function that measures how close the polygons are to satisfying this criteria. To optimize this error function, we first cluster triangles into discrete sets such that the assignment of sets minimizes our error. We then find canonical polygons for each set using nonlinear optimization. Next, we solve a Poisson equation to find positions of vertices such that the surface polygons match the canonical polygons as close as possible. We also describe how to incorporate a fairness criteria into the optimization to avoid oscillations of the surface. We iterate this entire process until we reach a user specified tolerance, possibly adding clusters during iteration to guarantee convergence. We have been able to successfully reduce the number of unique triangles to lie within a small percentage of the total number of triangles in the surface and demonstrate our technique on various examples.

The behavior of Catmull-Rom curves heavily depends on the choice of parameter values at the control points. We analyze a class of parameterizations ranging from uniform to chordal parameterization and show that, within this class, curves with centripetal parameterization contain properties that no other curves in this family possess. Researchers have previously indicated that centripetal parameterization produces visually favorable curves compared to uniform and chordal parameterizations. However, the mathematical reasons behind this behavior have been ambiguous. In this paper we prove that, for cubic Catmull-Rom curves, centripetal parameterization is the only parameterization in this family that guarantees that the curves do not form cusps or self-intersections within curve segments. Furthermore, we provide a formulation that bounds the distance of the curve to the control polygon and explain how globally intersection-free Catmull-Rom curves can be generated using these properties.

Abstract We present a method for simplifying a polygonal character with an associated skeletal deformation such that the simplified character approximates the original shape well when deformed. As input, we require a set of example poses that are representative of the types of deformations the character undergoes and we produce a multi‐resolution hierarchy for the simplified character where all simplified vertices also have associated skin weights. We create this hierarchy by minimizing an error metric for a simplified set of vertices and their skin weights, and we show that this quartic error metric can be effectively minimized using alternating quadratic minimization for the vertices and weights separately. To enable efficient GPU accelerated deformations of the simplified character, we also provide a method that guarantees the maximum number of bone weights per simplified vertex is less than a user specified threshold at all levels of the hierarchy.

We present a new method for approximating subdivision surfaces with hardware accelerated parametric patches. Our method improves the memory bandwidth requirements for patch control points, translating into superior performance compared to existing methods. Our input is general, allowing for meshes that contain both quadrilateral and triangular faces in the input control mesh, as well as control meshes with boundary. We present two implementations of our scheme designed to run on Direct3D 11 class hardware equipped with a tessellator unit.

Despite the visual importance of hair and the attention paid to hair modeling in the graphics research, modeling realistic hair still remains a very challenging task that can be performed by very few artists. In this paper we present hair meshes , a new method for modeling hair that aims to bring hair modeling as close as possible to modeling polygonal surfaces. This new approach provides artists with direct control of the overall shape of the hair, giving them the ability to model the exact hair shape they desire. We use the hair mesh structure for modeling the hair volume with topological constraints that allow us to automatically and uniquely trace the path of individual hair strands through this volume. We also define a set of topological operations for creating hair meshes that maintain these constraints. Furthermore, we provide a method for hiding the volumetric structure of the hair mesh from the end user, thus allowing artists to concentrate on manipulating the outer surface of the hair as a polygonal surface. We explain and show examples of how hair meshes can be used to generate individual hair strands for a wide variety of realistic hair styles.

Abstract We present a second order smooth filling of an n‐valent Catmull‐Clark spline ring with n biseptic patches. While an underdetermined biseptic solution to this problem has appeared previously, we make several advances in this paper. Most notably, we cast the problem as a constrained minimization and introduce a novel quadratic energy functional whose absolute minimum of zero is achieved for bicubic polynomials. This means that for the regular 4‐valent case, we reproduce the bicubic B‐splines. In other cases, the resulting surfaces are aesthetically well behaved. We extend our constrained minimization framework to handle the case of input mesh with boundary.

Abstract We present a streaming method for reconstructing surfaces from large data sets generated by a laser range scanner using wavelets. Wavelets provide a localized, multiresolution representation of functions and this makes them ideal candidates for streaming surface reconstruction algorithms. We show how wavelets can be used to reconstruct the indicator function of a shape from a cloud of points with associated normals. Our method proceeds in several steps. We first compute a low‐resolution approximation of the indicator function using an octree followed by a second pass that incrementally adds fine resolution details. The indicator function is then smoothed using a modified octree convolution step and contoured to produce the final surface. Due to the local, multiresolution nature of wavelets, our approach results in an algorithm over 10 times faster than previous methods and can process extremely large data sets in the order of several hundred million points in only an hour.

We present a simple and computationally efficient algorithm for approximating Catmull-Clark subdivision surfaces using a minimal set of bicubic patches. For each quadrilateral face of the control mesh, we construct a geometry patch and a pair of tangent patches. The geometry patches approximate the shape and silhouette of the Catmull-Clark surface and are smooth everywhere except along patch edges containing an extraordinary vertex where the patches are C 0 . To make the patch surface appear smooth, we provide a pair of tangent patches that approximate the tangent fields of the Catmull-Clark surface. These tangent patches are used to construct a continuous normal field (through their cross-product) for shading and displacement mapping. Using this bifurcated representation, we are able to define an accurate proxy for Catmull-Clark surfaces that is efficient to evaluate on next-generation GPU architectures that expose a programmable tessellation unit.

In this paper, we describe a method for exact evaluation of a limit mesh defined via subdivision on a uniform grid of any size. Other exact evaluation technique either restrict the grids to have subdivision sampling and are, hence, exponentially increasing in size or make assumptions about the underlying surface being piecewise polynomial (Stam's method is a widely used technique that makes this assumption). As opposed to Stam's technique, our method works for both polynomial and non-polynomial schemes. The values for this exact evaluation scheme can be computed via a simple system of linear equation derived from the scaling relations associated with the scheme or, equivalently, as the dominant left eigenvector of an upsampled subdivision matrix associated with the scheme. To illustrate one possible application of this method, we demonstrate how to generate adaptive polygonalizations of a non-polynomial quad-based subdivision surfaces using our exact evaluation method. Our method guarantees a water-tight tessellation no matter how the surface is sampled and is quite fast. We achieve tessellation rates of over 33.5 million triangles/ second using a CPU implementation.

Dual Contouring (DC) is a feature-preserving isosurfacing method that extracts crack-free surfaces from both uniform and adaptive octree grids. We present an extension of DC that further guarantees that the mesh generated is a manifold even under adaptive simplification. Our main contribution is an octree-based topology-preserving vertex-clustering algorithm for adaptive contouring. The contoured surface generated by our method contains only manifold vertices and edges, preserves sharp features, and possesses much better adaptivity than those generated by other isosurfacing methods under topologically safe simplification.

We provide an image deformation method based on Moving Least Squares using various classes of linear functions including affine, similarity and rigid transformations. These deformations are realistic and give the user the impression of manipulating real-world objects. We also allow the user to specify the deformations using either sets of points or line segments, the later useful for controlling curves and profiles present in the image. For each of these techniques, we provide simple closed-form solutions that yield fast deformations, which can be performed in real-time.

Abstract We present a method for contouring an implicit function using a grid topologically dual to structured grids such as octrees. By aligning the vertices of the dual grid with the features of the implicit function, we are able to reproduce thin features of the extracted surface without excessive subdivision required by methods such as Marching Cubes or Dual Contouring. Dual Marching Cubes produces a crack‐free, adaptive polygonalization of the surface that reproduces sharp features. Our approach maintains the advantage of using structured grids for operations such as CSG while being able to conform to the relevant features of the implicit function yielding much sparser polygonalizations than has been possible using structured grids.

Constructing a function that interpolates a set of values defined at vertices of a mesh is a fundamental operation in computer graphics. Such an interpolant has many uses in applications such as shading, parameterization and deformation. For closed polygons, mean value coordinates have been proven to be an excellent method for constructing such an interpolant. In this paper, we generalize mean value coordinates from closed 2D polygons to closed triangular meshes. Given such a mesh P , we show that these coordinates are continuous everywhere and smooth on the interior of P . The coordinates are linear on the triangles of P and can reproduce linear functions on the interior of P . To illustrate their usefulness, we conclude by considering several interesting applications including constructing volumetric textures and surface deformation.

In this article, we present a subdivision scheme for mixed triangle/quad meshes that is C2 everywhere except for isolated, extraordinary points. The rules that we describe are the same as Stam and Loop's scheme [2003] except that we perform an unzipping pass prior to subdivision. This simple modification improves the smoothness along the ordinary triangle/quad boundary from C1 to C2, and creates a scheme capable of subdividing arbitrary meshes. Finally, we end with a proof based on Levin and Levin's [2003] joint spectral radius calculation to show our scheme is indeed C2 along the triangle/quad boundary.

We present a method for contouring an implicit function using a grid topologically dual to structured grids such as octrees. By aligning the vertices of the dual grid with the features of the implicit function, we are able to reproduce thin features of the extracted surface without excessive subdivision required by methods such as marching cubes or dual contouring. Dual marching cubes produces a crack-free, adaptive polygonalization of the surface that reproduces sharp features. Our approach maintains the advantage of using structured grids for operations such as CSG while being able to conform to the relevant features of the implicit function yielding much sparser polygonalizations than has been possible using structured grids.

This paper describes a new method for contouring a signed grid whose edges are tagged by Hermite data (i.e; exact intersection points and normals). This method avoids the need to explicitly identify and process "features" as required in previous Hermite contouring methods. Using a new, numerically stable representation for quadratic error functions, we develop an octree-based method for simplifying contours produced by this method. We next extend our contouring method to these simpli£ed octrees. This new method imposes no constraints on the octree (such as being a restricted octree) and requires no "crack patching". We conclude with a simple test for preserving the topology of the contour during simplification.