Browsing by Author "Chen, Renjie"

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    Approximating Planar Conformal Maps Using Regular Polygonal Meshes
    (© 2017 The Eurographics Association and John Wiley & Sons Ltd., 2017) Chen, Renjie; Gotsman, Craig; Chen, Min and Zhang, Hao (Richard)
    Continuous conformal maps are typically approximated numerically using a triangle mesh which discretizes the plane. Computing a conformal map subject to user‐provided constraints then reduces to a sparse linear system, minimizing a quadratic ‘conformal energy’. We address the more general case of non‐triangular elements, and provide a complete analysis of the case where the plane is discretized using a mesh of regular polygons, e.g. equilateral triangles, squares and hexagons, whose interiors are mapped using barycentric coordinate functions. We demonstrate experimentally that faster convergence to continuous conformal maps may be obtained this way. We provide a formulation of the problem and its solution using complex number algebra, significantly simplifying the notation. We examine a number of common barycentric coordinate functions and demonstrate that superior approximation to harmonic coordinates of a polygon are achieved by the Moving Least Squares coordinates. We also provide a simple iterative algorithm to invert barycentric maps of regular polygon meshes, allowing to apply them in practical applications, e.g. for texture mapping.Continuous conformal maps are typically approximated numerically using a triangle mesh which discretizes the plane. Computing a conformal map subject to user‐provided constraints then reduces to a sparse linear system, minimizing a quadratic ‘conformal energy’. We address the more general case of non‐triangular elements, and provide a complete analysis of the case where the plane is discretized using a mesh of regular polygons, e.g. equilateral triangles, squares and hexagons, whose interiors are mapped using barycentric coordinate functions. We demonstrate experimentally that faster convergence to continuous conformal maps may be obtained this way. We examine a number of common barycentric coordinate functions and demonstrate that superior approximation to harmonic coordinates of a polygon are achieved by the Moving Least Squares coordinates. We also provide a simple iterative algorithm to invert barycentric maps of regular polygon meshes, allowing to apply them in practical applications, e.g. for texture mapping.
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    Harmonic Shape Interpolation on Multiply-connected Planar Domains
    (The Eurographics Association and John Wiley & Sons Ltd., 2022) Shi, Dongbo; Chen, Renjie; Campen, Marcel; Spagnuolo, Michela
    Shape interpolation is a fundamental problem in computer graphics. Recently, there have been some interpolation methods developed which guarantee that the results are of bounded amount of geometric distortion, hence ensure high quality interpolation. However, none of these methods is applicable to shapes within the multiply-connected domains. In this work, we develop an interpolation scheme for harmonic mappings, that specifically addresses this limitation. We opt to interpolate the pullback metric of the input harmonic maps as proposed by Chen et al. [CWKBC13]. However, the interpolated metric does not correspond to any planar mapping, which is the main challenge in the interpolation problem for multiply-connected domains. We propose to solve this by projecting the interpolated metric into the planar harmonic mapping space. Specifically, we develop a Newton iteration to minimize the isometric distortion of the intermediate mapping, with respect to the interpolated metric. For more efficient Newton iteration, we further derived a simple analytic formula for the positive semidefinite (PSD) projection of the Hessian matrix of our distortion energy. Through extensive experiments and comparisons with the state-of-the-art, we demonstrate the efficacy and robustness of our method for various inputs.
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    Parallel Loop Subdivision with Sparse Adjacency Matrix
    (The Eurographics Association, 2023) Wang, Kechun; Chen, Renjie; Babaei, Vahid; Skouras, Melina
    Subdivision surface is a popular technique for geometric modeling. Recently, several parallel implementations have been developed for Loop subdivision on the GPU. However, these methods are built on complex data structures which complicate the implementation and affect the performance, especially on the GPU. In this work, we propose to simply use the sparse adjacency matrix which enables us to implement the Loop subdivision scheme in the most straightforward manner. Our implementation run entirely on the GPU and achieves high performance in runtime with significantly lower memory consumption than the state-of-the-art. Through extensive experiments and comparisons, we demonstrate the efficacy and efficiency of our method.
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    Piecewise Linear Mapping Optimization Based on the Complex View
    (The Eurographics Association and John Wiley & Sons Ltd., 2018) Golla, Björn; Seidel, Hans-Peter; Chen, Renjie; Fu, Hongbo and Ghosh, Abhijeet and Kopf, Johannes
    We present an efficient modified Newton iteration for the optimization of nonlinear energies on triangle meshes. Noting that the linear mapping between any pair of triangles is a special case of harmonic mapping, we build upon the results of Chen and Weber [CW17]. Based on the complex view of the linear mapping, we show that the Hessian of the isometric energies has a simple and compact analytic expression. This allows us to analytically project the per-element Hessians to positive semidefinite matrices for efficient Newton iteration. We show that our method outperforms state-of-the-art methods on 2D deformation and parameterization. Further, we inspect the spectra of the per triangle energy Hessians and show that given an initial mapping, simple global scaling can shift the energy towards a more convex state. This allows Newton iteration to converge faster than starting from the given initial state. Additionally, our formulations support adding an energy smoothness term to the optimization with little additional effort, which improves the mapping results such that concentrated distortions are reduced.