Matching probability distributions allows to compare or interpolate them, or model their manifold. Optimal transport is a tool that solves this matching problem. However, despite the development of numerous exact and approximate algorithms, these approaches remain too slow for large datasets due to the inherent challenge of optimizing transport plans. Taking intuitions from recent advances in rectified flows we propose an algorithm that, while not resulting in optimal transport plans, produces transport plans from uniform densities to densities stored on grids that resemble the optimal ones in practice. Our algorithm has linear-time complexity with respect to the problem size and is embarrassingly parallel. It is also trivial to implement, essentially computing three summed-area tables and advecting particles with velocities easily computed from these tables using simple arithmetic. This already allows for applications such as stippling and area-preserving mesh parameterization. Combined with linearized transport ideas, we further extend our approach to match two non-uniform distributions. This allows for wider applications such as shape interpolation or barycenters, matching the quality of more complex optimal or approximate transport solvers while resulting in orders of magnitude speedups. We illustrate our applications in 2D and 3D.

International audience

Abstract We introduce a Delaunay‐based algorithm for reconstructing the underlying surface of a given set of unstructured points in 3D. The implementation is very simple, and it is designed to work in a parameter‐free manner. The solution builds upon the fact that in the continuous case, a closed surface separates the set of maximal empty balls (medial balls) into an interior and exterior. Based on discrete input samples, our reconstructed surface consists of the interface between Voronoi balls, which approximate the interior and exterior medial balls. An initial set of Voronoi balls is iteratively processed, merging Voronoi‐ball pairs if they fulfil an overlapping error criterion. Our complete open‐source reconstruction pipeline performs up to two quick linear‐time passes on the Delaunay complex to output the surface, making it an order of magnitude faster than the state of the art while being competitive in memory usage and often superior in quality. We propose two variants (local and global), which are carefully designed to target two different reconstruction scenarios for watertight surfaces from accurate or noisy samples, as well as real‐world scanned data sets, exhibiting noise, outliers, and large areas of missing data. The results of the global variant are, by definition, watertight, suitable for numerical analysis and various applications (e.g., 3D printing). Compared to classical Delaunay‐based reconstruction techniques, our method is highly stable and robust to noise and outliers, evidenced via various experiments, including on real‐world data with challenges such as scan shadows, outliers, and noise, even without additional preprocessing.

Abstract Denoising is a common, yet critical operation in geometry processing aiming at recovering high‐fidelity models of piecewise‐smooth objects from noise‐corrupted pointsets. Despite a sizable literature on the topic, there is a dearth of approaches capable of processing very noisy and outlier‐ridden input pointsets for which no normal estimates and no assumptions on the underlying geometric features or noise type are provided. In this paper, we propose a new robust‐statistics approach to denoising pointsets based on line processes to offer robustness to noise and outliers while preserving sharp features possibly present in the data. While the use of robust statistics in denoising is hardly new, most approaches rely on prescribed filtering using data‐independent blending expressions based on the spatial and normal closeness of samples. Instead, our approach deduces a geometric denoising strategy through robust and regularized tangent plane fitting of the initial pointset, obtained numerically via alternating minimizations for efficiency and reliability. Key to our variational approach is the use of line processes to identify inliers vs. outliers, as well as the presence of sharp features. We demonstrate that our method can denoise sampled piecewise‐smooth surfaces for levels of noise and outliers at which previous works fall short.

Point patterns are characterized by their density and correlation. While spatial variation of density is well-understood, analysis and synthesis of spatially-varying correlation is an open challenge. No tools are available to intuitively edit such point patterns, primarily due to the lack of a compact representation for spatially varying correlation. We propose a low-dimensional perceptual embedding for point correlations. This embedding can map point patterns to common three-channel raster images, enabling manipulation with off-the-shelf image editing software. To synthesize back point patterns, we propose a novel edge-aware objective that carefully handles sharp variations in density and correlation. The resulting framework allows intuitive and backward-compatible manipulation of point patterns, such as recoloring, relighting to even texture synthesis that have not been available to 2D point pattern design before. Effectiveness of our approach is tested in several user experiments. Code is available at https://github.com/xchhuang/patternshop.

Structured 2D patterns formed by the anisotropic distribution of arbitrary shapes are ubiquitous in nature and man-made environments. They may include both bounded and unbounded (extended fiber-like) shapes. In this work, we address the problem of interactively generating such patterns from a single exemplar sketched by a user. We build our solution on a new data structure, the Support Structure Hierarchy, computed from a multi-resolution analysis of the input exemplar, that encodes the main anisotropy directions at different scales as well as deviations from them. We propose an efficient method based on this structure to synthesize a similar distribution of shapes in an extended 2D domain. The user can also choose to hybridize multiple input exemplars by combining structural shapes extracted at different scales. As shown in a user study, our multi-scale solution generates structured shape-patterns that perceptually compete with state-of-the-art methods, whether learning-based or not. Moreover, our interactive solution, which requires no pre-calculation, fits well with the needs of an interactive authoring tool, where the user can not only sketch and extend 2D vector textures but also combine them seamlessly.

Abstract Point pattern synthesis requires capturing both local and non‐local correlations from a given exemplar. Recent works employ deep hierarchical representations from VGG‐19 [SZ15] convolutional network to capture the features for both point‐pattern and texture synthesis. In this work, we develop a simplified optimization pipeline that uses more traditional Gabor transform‐based features. These features when convolved with simple random filters gives highly expressive feature maps. The resulting framework requires significantly less feature maps compared to VGG‐19‐based methods [TLH19; RGF∗20], better captures both the local and non‐local structures, does not require any specific data set training and can easily extend to handle multi‐class and multi‐attribute point patterns, e.g., disk and other element distributions. To validate our pipeline, we perform qualitative and quantitative analysis on a large variety of point patterns to demonstrate the effectiveness of our approach. Finally, to better understand the impact of random filters, we include a spectral analysis using filters with different frequency bandwidths.

Abstract We introduce Delaunay Painting, a novel and easy‐to‐use method to flat‐colour contour‐sketches with gaps. Starting from a Delaunay triangulation of the input contours, triangles are iteratively filled with the appropriate colours, thanks to the dynamic update of flow values calculated from colour hints. Aesthetic finish is then achieved, through energy minimisation of contour‐curves and further heuristics enforcing the appropriate sharp corners. To be more efficient, the user can also make use of our colour diffusion framework, which automatically extends colouring to small, internal regions such as those delimited by hatches. The resulting method robustly handles input contours with strong gaps. As an interactive tool, it minimizes user's efforts and enables any colouring strategy, as the result does not depend on the order of interactions. We also provide an automatized version of the colouring strategy for quick segmentation of contours images, that we illustrate with applications to medical imaging and sketch segmentation.

We present a novel method for authoring landscapes with flora and fauna while considering their mutual interactions. Our algorithm outputs a steady-state ecosystem in the form of density maps for each species, their daily circuits, and a modified terrain with eroded trails from a terrain, climatic conditions, and species with related biological information. We introduce the Resource Access Graph, a new data structure that encodes both interactions between food chain levels and animals traveling between resources over the terrain. A novel competition algorithm operating on this data progressively computes a steady-state solution up the food chain, from plants to carnivores. The user can explore the resulting landscape, where plants and animals are instantiated on the fly, and interactively edit it by over-painting the maps. Our results show that our system enables the authoring of consistent landscapes where the impact of wildlife is visible through animated animals, clearings in the vegetation, and eroded trails. We provide quantitative validation with existing ecosystems and a user-study with expert paleontologist end-users, showing that our system enables them to author and compare different ecosystems illustrating climate changes over the same terrain while enabling relevant visual immersion into consistent landscapes.

Motivated by the needs of a scalable out-of-core surface reconstruction algorithm available on the cloud, this paper addresses the computation of distributed Delaunay triangulations of massive point sets. The proposed algorithm takes as input a point cloud and first partitions it across multiple processing elements into tiles of relatively homogeneous point sizes. The distributed computation and communication between processing elements is orchestrated so that each one discovers the Delaunay neighbors of its input points within the theoretical overall Delaunay triangulation of all points and computes locally a partial view of this triangulation. This approach prevents memory limitations by never materializing the global triangulation. This efficiency is due to our proposed uncentralized model to represent, manage and locally construct the triangulation corresponding to each tile. The point set is first partitioned into non-overlapping tiles, then we construct within each tile the Delaunay triangulation of the local points and a minimal set of replicated foreign points in order to capture the simplices spanning multiple tiles. Inspired by the star splaying approach for Delaunay triangulation computation/repair, communication is limited to exchanging points of potential Delaunay neighbors across tiles. Therefore, our method is guaranteed to reconstruct, within each tile, a triangulation that contains the star of its local points, as though it were computed within the Delaunay triangulation of all points. The proposed algorithm is implemented with Spark for the scheduling and C++ for the geometric computations. This allows both an optimal scheduling on multiple machines and efficient low-level computation. The results show the efficiency of our algorithm in terms of speedup and strong scaling on a classical Spark configuration with both synthetic and real use case datasets.

Abstract While analysing and synthesising 2D distributions of points has been applied both to the generation of textures with discrete elements and for populating virtual worlds with 3D objects, the results are often inaccurate since the spatial extent of objects cannot be expressed. We introduce three improvements enabling the synthesis of more general distributions of elements. First, we extend continuous pair correlation function (PCF) algorithms to multi‐class distributions using a dependency graph, thereby capturing interrelationships between distinct categories of objects. Second, we introduce a new normalised metric for disks, which makes the method applicable to both point and possibly overlapping disk distributions. The metric is specifically designed to distinguish perceptually salient features, such as disjoint, tangent, overlapping, or nested disks. Finally, we pay particular attention to convergence of the mean PCF as well as the validity of individual PCFs, by taking into consideration the variance of the input. Our results demonstrate that this framework can capture and reproduce real‐life distributions of elements representing a variety of complex semi‐structured patterns, from the interaction between trees and the understorey in a forest to droplets of water. More generally, it applies to any category of 2D object whose shape is better represented by bounding circles than points.

Abstract Animating herds of animals while achieving both convincing global shapes and plausible distributions within the herd is difficult, using simulation methods. In this work, we allow users to rely on photos of real herds, which are widely available, for keyframing their animation. More precisely, we learn global and local distribution features in each photo of the input set (which may depict different numbers of animals) and transfer them to the group of animals to be animated, thanks to a new statistical learning method enabling to analyze distributions of ellipses, as well as their density and orientation fields. The animated herd reconstructs the desired distribution at each keyframe while avoiding obstacles. As our results show, our method offers both high‐level user control and help toward realism, enabling to easily author herd animations.

Abstract Dictionaries are very useful objects for data analysis, as they enable a compact representation of large sets of objects through the combination of atoms. Dictionary‐based techniques have also particularly benefited from the recent advances in machine learning, which has allowed for data‐driven algorithms to take advantage of the redundancy in the input dataset and discover relations between objects without human supervision or hard‐coded rules. Despite the success of dictionary‐based techniques on a wide range of tasks in geometric modeling and geometry processing, the literature is missing a principled state‐of‐the‐art of the current knowledge in this field. To fill this gap, we provide in this survey an overview of data‐driven dictionary‐based methods in geometric modeling. We structure our discussion by application domain: surface reconstruction, compression, and synthesis. Contrary to previous surveys, we place special emphasis on dictionary‐based methods suitable for 3D data synthesis, with applications in geometric modeling and design. Our ultimate goal is to enlight the fact that these techniques can be used to combine the data‐driven paradigm with design intent to synthesize new plausible objects with minimal human intervention. This is the main motivation to restrict the scope of the present survey to techniques handling point clouds and meshes, making use of dictionaries whose definition depends on the input data, and enabling shape reconstruction or synthesis through the combination of atoms.

Space coordinates offer an elegant, scalable and versatile framework to propagate (multi-)scalar functions from the boundary vertices of a 3-manifold, often called a cage , within its volume. These generalizations of the barycentric coordinate system have progressively expanded the range of eligible cages to triangle and planar polygon surface meshes with arbitrary topology, concave regions and a spatially-varying sampling ratio, while preserving a smooth diffusion of the prescribed on-surface functions. In spite of their potential for major computer graphics applications such as freeform deformation or volume texturing, current space coordinate systems have only found a moderate impact in applications. This follows from the constraint of having only triangles in the cage most of the time, while many application scenarios favor arbitrary (non-planar) quad meshes for their ability to align the surface structure with features and to naturally cope with anisotropic sampling. In order to use space coordinates with arbitrary quad cages currently, one must triangulate them, which results in large propagation distortion. Instead, we propose a generalization of a popular coordinate system - Mean Value Coordinates - to quad and tri-quad cages, bridging the gap between high-quality coarse meshing and volume diffusion through space coordinates. Our method can process non-planar quads, comes with a closed-form solution free from global optimization and reproduces the expected behavior of Mean Value Coordinates, namely smoothness within the cage volume and continuity everywhere. As a result, we show how these coordinates compare favorably to classical space coordinates on triangulated quad cages, in particular for freeform deformation.

In this article we investigate the use of weighted triangulations as discrete, augmented approximations of surfaces for digital geometry processing. By incorporating a scalar weight per mesh vertex, we introduce a new notion of discrete metric that defines an orthogonal dual structure for arbitrary triangle meshes and thus extends weighted Delaunay triangulations to surface meshes. We also present alternative characterizations of this primal-dual structure (through combinations of angles, areas, and lengths) and, in the process, uncover closed-form expressions of mesh energies that were previously known in implicit form only. Finally, we demonstrate how weighted triangulations provide a faster and more robust approach to a series of geometry processing applications, including the generation of well-centered meshes, self-supporting surfaces, and sphere packing.

We introduce Hodge-optimized triangulations (HOT), a family of well-shaped primal-dual pairs of complexes designed for fast and accurate computations in computer graphics. Previous work most commonly employs barycentric or circumcentric duals; while barycentric duals guarantee that the dual of each simplex lies within the simplex, circumcentric duals are often preferred due to the induced orthogonality between primal and dual complexes. We instead promote the use of weighted duals ("power diagrams"). They allow greater flexibility in the location of dual vertices while keeping primal-dual orthogonality, thus providing a valuable extension to the usual choices of dual by only adding one additional scalar per primal vertex. Furthermore, we introduce a family of functionals on pairs of complexes that we derive from bounds on the errors induced by diagonal Hodge stars, commonly used in discrete computations. The minimizers of these functionals, called HOT meshes, are shown to be generalizations of Centroidal Voronoi Tesselations and Optimal Delaunay Triangulations, and to provide increased accuracy and flexibility for a variety of computational purposes.

We consider the problem of reconstructing a compact 3-manifold (with boundary) embedded in ℜ3 from its crosssections with a given set of cutting planes having arbitrary orientations. Under appropriate sampling conditions that are satisfied when the set of cutting planes is dense enough, we prove that the algorithm presented by Liu et al. in [LBD+08] preserves the homotopy type of the original object. Using the homotopy equivalence, we also show that the reconstructed object is homeomorphic (and isotopic) to the original object. This is the first time that shape reconstruction from cross-sections comes with such theoretical guarantees.

Abstract This paper deals with the reconstruction of 2‐dimensional geometric shapes from unorganized 1‐dimensional cross‐sections. We study the problem in its full generality following the approach of Boissonnat and Memari [ BM07 ] for the analogous 3D problem. We propose a new variant of this method and provide sampling conditions to guarantee that the output of the algorithm has the same topology as the original object and is close to it (for the Hausdorff distance).