Surface representation and processing is one of the key topics in computer graphics and geometric modeling, since it greatly affects the range of possible applications. In this paper we will present ...recent advances in geometry processing that are related to the Laplacian processing framework and differential representations. This framework is based on linear operators defined on polygonal meshes, and furnishes a variety of processing applications, such as shape approximation and compact representation, mesh editing, watermarking and morphing. The core of the framework is the definition of differential coordinates and new bases for efficient mesh geometry representation, based on the mesh Laplacian operator.
We formalize Italian smocking, an intricate embroidery technique that gathers flat fabric into pleats along meandering lines of stitches, resulting in pleats that fold and gather where the stitching ...veers. In contrast to English smocking, characterized by colorful stitches decorating uniformly shaped pleats, and Canadian smocking, which uses localized knots to form voluminous pleats, Italian smocking permits the fabric to move freely along the stitched threads following curved paths, resulting in complex and unpredictable pleats with highly diverse, irregular structures, achieved simply by pulling on the threads. We introduce a novel method for digital previewing of Italian smocking results, given the thread stitching path as input. Our method uses a coarse‐grained mass‐spring system to simulate the interaction between the threads and the fabric. This configuration guides the fine‐level fabric deformation through an adaptation of the state‐of‐the‐art simulator, C‐IPC LKJ21. Our method models the general problem of fabric‐thread interaction and can be readily adapted to preview Canadian smocking as well. We compare our results to baseline approaches and physical fabrications to demonstrate the accuracy of our method.
Developable Approximation via Gauss Image Thinning Binninger, Alexandre; Verhoeven, Floor; Herholz, Philipp ...
Computer graphics forum,
August 2021, 2021-08-00, 20210801, Volume:
40, Issue:
5
Journal Article
Peer reviewed
Open access
Approximating 3D shapes with piecewise developable surfaces is an active research topic, driven by the benefits of developable geometry in fabrication. Piecewise developable surfaces are ...characterized by having a Gauss image that is a 1D object – a collection of curves on the Gauss sphere. We present a method for developable approximation that makes use of this classic definition from differential geometry. Our algorithm is an iterative process that alternates between thinning the Gauss image of the surface and deforming the surface itself to make its normals comply with the Gauss image. The simple, local‐global structure of our algorithm makes it easy to implement and optimize. We validate our method on developable shapes with added noise and demonstrate its effectiveness on a variety of non‐developable inputs. Compared to the state of the art, our method is more general, tessellation independent, and preserves the input mesh connectivity.
Discrete Laplacians for triangle meshes are a fundamental tool in geometry processing. The so‐called cotan Laplacian is widely used since it preserves several important properties of its smooth ...counterpart. It can be derived from different principles: either considering the piecewise linear nature of the primal elements or associating values to the dual vertices. Both approaches lead to the same operator in the two‐dimensional setting. In contrast, for tetrahedral meshes, only the primal construction is reminiscent of the cotan weights, involving dihedral angles. We provide explicit formulas for the lesser‐known dual construction. In both cases, the weights can be computed by adding the contributions of individual tetrahedra to an edge. The resulting two different discrete Laplacians for tetrahedral meshes only retain some of the properties of their two‐dimensional counterpart. In particular, while both constructions have linear precision, only the primal construction is positive semi‐definite and only the dual construction generates positive weights and provides a maximum principle for Delaunay meshes. We perform a range of numerical experiments that highlight the benefits and limitations of the two constructions for different problems and meshes.
Current approaches to skeletally-controlled character articulation range from real-time, closed-form skinning methods to offline, physically-based simulation. In this paper, we seek a closed-form ...skinning method that approximates nonlinear elastic deformations well while remaining very fast. Our contribution is two-fold: (1) we optimize skinning weights for the standard linear and dual quaternion skinning techniques so that the resulting deformations minimize an elastic energy function. We observe that this is not sufficient to match the visual quality of the original elastic deformations and therefore, we develop (2) a new skinning method based on the concept of
joint-based deformers
. We propose a specific deformer which is visually similar to nonlinear variational deformation methods. Our final algorithm is fully automatic and requires little or no input from the user other than a rest-pose mesh and a skeleton. The runtime complexity requires minimal memory and computational overheads compared to linear blend skinning, while producing higher quality deformations than both linear and dual quaternion skinning.
We propose a method for the construction of a planar curve based on piecewise clothoids and straight lines that intuitively interpolates a given sequence of control points. Our method has several ...desirable properties that are not simultaneously fulfilled by previous approaches: Our interpolating curves are C2 continuous, their computation does not rely on global optimization and has local support, enabling fast evaluation for interactive modeling. Further, the sign of the curvature at control points is consistent with the control polygon; the curvature attains its extrema at control points and is monotone between consecutive control points of opposite curvature signs. In addition, we can ensure that the curve has self‐intersections only when the control polygon also self‐intersects between the same control points. For more fine‐grained control, the user can specify the desired curvature and tangent values at certain control points, though it is not required by our method. Our local optimization can lead to discontinuity w.r.t. the locations of control points, although the problem is limited by its locality. We demonstrate the utility of our approach in generating various curves and provide a comparison with the state of the art.
Cosserat Rods with Projective Dynamics Soler, Carlota; Martin, Tobias; Sorkine‐Hornung, Olga
Computer graphics forum,
December 2018, 2018-12-00, 20181201, Volume:
37, Issue:
8
Journal Article
Peer reviewed
We present a novel method to simulate Cosserat rods with Projective Dynamics (PD). The proposed method is both numerically robust and accurate with respect to the underlying physics, making it ...suitable for a variety of applications in computer graphics and related disciplines. Cosserat theory assigns an orientation frame to each point and is thus able to realistically simulate stretching and shearing effects, in addition to bending and twisting. Within the PD framework, it is possible to obtain accurate simulations given the implicit integration over time and its decoupling of the local‐global solve. In the proposed method, we start from the continuous formulation of the Cosserat theory and derive the constraints for the PD solver. We extend the standard definition of PD and add body orientations as system variables. Thus, we include the preservation of angular momentum, so that twisting and bending can be accurately simulated. Our formulation allows the simulation of different bending behaviors with respect to a user‐defined Young's modulus, the radius of the rod's cross‐section, and material density. We show how different material specifications in our simulations converge within a few iterations to a reference solution, generated with a high‐precision finite element method. Furthermore, we demonstrate mesh independence of our formulation: Refining the simulation mesh still results in the same characteristic motion, which is in contrast to previous position based methods.
Mappings and deformations are ubiquitous in geometry processing, shape modeling, and animation. Numerous deformation energies have been proposed to tackle problems like mesh parameterization and ...volumetric deformations. We present an algorithm that modifies any deformation energy to guarantee a locally injective mapping, i.e., without inverted elements. Our formulation can be used to compute continuous planar or volumetric piecewise‐linear maps and it uses a barrier term to prevent inverted elements. Differently from previous methods, we carefully design both the barrier term and the associated numerical techniques to be able to provide immediate feedback to the user, enabling interactive manipulation of inversion‐free mappings. Stress tests show that our method robustly handles extreme deformations where previous techniques converge very slowly or even fail. We demonstrate that enforcing local injectivity increases fidelity of the results in applications such as shape deformation and parameterization.
We present a scalable approach for the optimization of flip-preventing energies in the general context of simplicial mappings and specifically for mesh parameterization. Our iterative minimization is ...based on the observation that many distortion energies can be optimized indirectly by minimizing a family of simpler proxy energies. Minimization of these proxies is a natural extension of the local/global minimization of the ARAP energy. Our algorithm is simple to implement and scales to datasets with millions of faces. We demonstrate our approach for the computation of maps that minimize a conformal or isometric distortion energy, both in two and three dimensions. In addition to mesh parameterization, we show that our algorithm can be applied to mesh deformation and mesh quality improvement.
Data driven and learning based solutions for modeling dynamic garments have significantly advanced, especially in the context of digital humans. However, existing approaches often focus on modeling ...garments with respect to a fixed parametric human body model and are limited to garment geometries that were seen during training. In this work, we take a different approach and model the dynamics of a garment by exploiting its local interactions with the underlying human body. Specifically, as the body moves, we detect local garment‐body collisions, which drive the deformation of the garment. At the core of our approach is a mesh‐agnostic garment representation and a manifold‐aware transformer network design, which together enable our method to generalize to unseen garment and body geometries. We evaluate our approach on a wide variety of garment types and motion sequences and provide competitive qualitative and quantitative results with respect to the state of the art.
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