The sparsest packing problem emerges in manufacturing multi-hole extrusion dies to obtain small and precise products in the automotive, aviation, food, and medical industries. The goal is to maximize the minimum Euclidean distance between the objects and between the objects to the boundaries of the container. Additionally, this task might be subject to balancing constraints that determine that the deviation of gravity center of the system should stay within a threshold. We present a novel custom environment that encompasses the constraints present in this task. We experiment with the proposed environment using Proximal Policy Optimization to assess the applicability of reinforcement learning for the sparsest packing problem with circular objects in a circular container. Our results indicate that the proposed agent learns efficiently, demonstrating promising results in both finding feasible solutions and optimizing the placement of objects. Our approach not only shows the potential of reinforcement learning for solving the sparsest packing problem but also provides insights into its effectiveness in environments with complex spatial and balancing constraints.

While humans show remarkable proficiency in solving visual puzzles, machines often fall short due to the complex combinatorial nature of such tasks. Consequently, there is a growing interest in developing computational methods for the automatic solution of different puzzles, especially through deep learning approaches. The Tangram, an ancient Chinese puzzle, challenges players to arrange seven polygonal pieces to construct different patterns. Despite its apparent simplicity, solving the Tangram is considered an NP-complete problem, being a challenge even for the most sophisticated algorithms. Moreover, ensuring the generality and adaptability of machine learning models across different Tangram arrangements and complexities is an ongoing research problem. In this paper, we introduce a generative model specifically designed to solve the Tangram. Our model competes favorably with previous methods regarding accuracy while delivering fast inferences. It incorporates a novel loss function that integrates pixel-based information with geometric features, promoting a deeper understanding of the spatial relationships between pieces. Unlike previous approaches, our model takes advantage of the geometric properties of the Tangram to formulate a solving strategy, exploiting its inherent properties only through exposure to training data rather than through direct instruction. Extending the proposed loss function, we present a novel evaluation metric as a better fitting measure for assessing Tangram solutions than previous metrics. We further provide a new dataset containing more samples than others reported in the literature. Our findings highlight the potential of deep learning approaches in geometric problem domains.

Seeded segmentation methods have gained a lot of attention due to their good performance in fragmenting complex images, easy usability and synergism with graph-based representations. These methods usually rely on sophisticated computational tools whose performance strongly depends on how good the training data reflect a sought image pattern. Moreover, poor adherence to the image contours, lack of unique solution, and high computational cost are other common issues present in most seeded segmentation methods. In this work we introduce Laplacian Coordinates, a quadratic energy minimization framework that tackles the issues above in an effective and mathematically sound manner. The proposed formulation builds upon graph Laplacian operators, quadratic energy functions, and fast minimization schemes to produce highly accurate segmentations. Moreover, the presented energy functions are not prone to local minima, i.e., the solution is guaranteed to be globally optimal, a trait not present in most image segmentation methods. Another key property is that the minimization procedure leads to a constrained sparse linear system of equations, enabling the segmentation of high-resolution images at interactive rates. The effectiveness of Laplacian Coordinates is attested by a comprehensive set of comparisons involving nine state-of-the-art methods and several benchmarks extensively used in the image segmentation literature.

SUMMARY: iTUPA is a free online application for automatizing the Topographic-Unit Parsimony Analysis (TUPA), which identifies areas of endemism based on topography. iTUPA generates species-occurrences matrices based on user-defined topographic units (TUs) and provides a parsimony analysis of the generated matrix. We tested iTUPA after a proposal of regionalization for the Brazilian Atlantic Forest. iTUPA can handle millions of species registers simultaneously and uses Google Earth high-definition maps to visually explore the endemism data. We believe iTUPA is a useful tool for further discussions on biodiversity conservation. AVAILABILITY AND IMPLEMENTATION: iTUPA is hosted on Google cloud and freely available at http://nuvem.ufabc.edu.br/itupa. iTUPA is implemented using R (version 3.5.1), with RStudio 1.1.453 used as the implementation IDE, Shiny 1.1.0 web framework, and Google Maps® API version 3.36.

Abstract The Hermite radial basis functions (HRBF) implicits reconstruct an implicit function which interpolates or approximates scattered multivariate Hermite data (i.e. unstructured points and their corresponding normals). Experiments suggest that HRBF implicits allow the reconstruction of surfaces rich in details and behave better than previous related methods under coarse and/or non‐uniform samplings, even in the presence of close sheets. HRBF implicits theory unifies a recently introduced class of surface reconstruction methods based on radial basis functions (RBF), which incorporate normals directly in their problem formulation. Such class has the advantage of not depending on manufactured offset‐points to ensure existence of a non‐trivial implicit surface RBF interpolant. In fact, we show that HRBF implicits constitute a particular case of Hermite–Birkhoff interpolation with radial basis functions, whose main results we present here. This framework not only allows us to show connections between the present method and others but also enable us to enhance the flexibility of our method by ensuring well‐posedness of an interesting combined interpolation/regularization approach.

Abstract Moving‐least‐squares (MLS) surfaces undergoing large deformations need periodic regeneration of the point set (point‐set resampling) so as to keep the point‐set density quasi‐uniform. Previous work by the authors dealt with algebraic MLS surfaces, and proposed a resampling strategy based on defining the new points at the intersections of the MLS surface with a suitable set of rays. That strategy has very low memory requirements and is easy to parallelize. In this article new resampling strategies with reduced CPU‐time cost are explored. The basic idea is to choose as set of rays the lines of a regular, Cartesian grid, and to fully exploit this grid: as data structure for search queries, as spatial structure for traversing the surface in a continuation‐like algorithm, and also as approximation grid for an interpolated version of the MLS surface. It is shown that in this way a very simple and compact resampling technique is obtained, which cuts the resampling cost by half with affordable memory requirements.

Abstract We propose a novel vortex core line extraction method based on the λ2vortex region criterion in order to improve the detection of vortex features for 3D flow visualization. The core line is defined as a curve that connects λ2minima restricted to planes that are perpendicular to the core line. The basic algorithm consists of the following stages: (1) λ2field construction and isosurface extraction; (2) computation of the curve skeleton of the λ2isosurface to build an initial prediction for the core line; (3) correction of the locations of the prediction by searching for λ2minima on planes perpendicular to the core line. In particular, we consider the topology of the vortex core lines, guaranteeing the same topology as the initial curve skeleton. Furthermore, we propose a geometry‐guided definition of vortex bifurcation, which represents the split of one core line into two parts. Finally, we introduce a user‐guided approach in order to narrow down vortical regions taking into account the graph of λ2along the computed vortex core line. We demonstrate the effectiveness of our method by comparing our results to previous core line detection methods with both simulated and experimental data; in particular, we show robustness of our method for noise‐affected data.

Point-based models are gaining lately considerable attention as an alternative to traditional surface meshes. In this context, Point Set Surfaces (PSS) were proposed as a modeling and rendering method with important topological and approximation properties. However, ray-tracing PSS is computationally expensive. Therefore, we propose an interactive ray-tracing algorithm for PSS implemented completely on commodity graphics hardware. We also exploit the advantages of PSS to propose a novel technique for extracting surfaces directly from volumetric data. This technique is based on the well known predictor-corrector principle from the numerical methods for solving ordinary differential equations. Our technique provides good approximations to surfaces defined by a certain property in the volume, such as iso-surfaces or surfaces located at regions of high gradient magnitude. Also, local details of the surfaces could be manipulated by changing the local polynomial approximation and the smoothing parameters used. Furthermore, the surfaces generated are smooth and low frequency noise is naturally handled.