Functional data approximation on bounded domains using polygonal finite elements

Cao, Juan; Xiao, Yanyang; Chen, Zhonggui; Wang, Wenping; Bajaj, Chandrajit L.

doi:10.1016/j.cagd.2018.05.005

Cited by 8 publications

(4 citation statements)

References 48 publications

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“…High-order poly-FEM has been successfully applied to solving partial differential equations [33][34][35] and function approximation [36]. We are motivated by these successes to use the high-order poly-FEM method in the image warping problem.…”

Section: High-order Poly-femmentioning

confidence: 99%

Polygonal finite element-based content-aware image warping

et al. 2023

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Mesh-based image warping techniques typically represent image deformation using linear functions on triangular meshes or bilinear functions on rectangular meshes. This enables simple and efficient implementation, but in turn, restricts the representation capability of the deformation, often leading to unsatisfactory warping results. We present a novel, flexible polygonal finite element (poly-FEM) method for content-aware image warping. Image deformation is represented by high-order poly-FEMs on a content-aware polygonal mesh with a cell distribution adapted to saliency information in the source image. This allows highly adaptive meshes and smoother warping with fewer degrees of freedom, thus significantly extending the flexibility and capability of the warping representation. Benefiting from the continuous formulation of image deformation, our poly-FEM warping method is able to compute the optimal image deformation by minimizing existing or even newly designed warping energies consisting of penalty terms for specific transformations. We demonstrate the versatility of the proposed poly-FEM warping method in representing different deformations and its superiority by comparing it to other existing state-of-the-art methods.

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Section: High-order Poly-femmentioning

confidence: 99%

Polygonal finite element-based content-aware image warping

et al. 2023

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“…[CXC14] to piecewise polynomial approximation with arbitrary degrees, while Cao et al . [CXC*18] adopted barycentric coordinates to construct the approximation. They identified the optimal solution when the image features coincide with the edges of Voronoi cells, forming polylines along these features, which also inevitably occurs in other geometric partitions with straight boundaries.…”

Section: Related Workmentioning

confidence: 99%

Image Representation on Curved Optimal Triangulation

Xiao

Cao

Chen

2022

Computer Graphics Forum

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Image triangulation aims to generate an optimal partition with triangular elements to represent the given image. One bottleneck in ensuring approximation quality between the original image and a piecewise approximation over the triangulation is the inaccurate alignment of straight edges to the curved features. In this paper, we propose a novel variational method called curved optimal triangulation, where not all edges are straight segments, but may also be quadratic Bézier curves. The energy function is defined as the total approximation error determined by vertex locations, connectivity and bending of edges. The gradient formulas of this function are derived explicitly in closed form to optimize the energy function efficiently. We test our method on several models to demonstrate its efficacy and ability in preserving features. We also explore its applications in the automatic generation of stylization and Lowpoly images. With the same number of vertices, our curved optimal triangulation method generates more accurate and visually pleasing results compared with previous methods that only use straight segments.

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“…See, e.g. [14], [20], [15], [25], [26], [21], [23], [44], [7], and etc. In addition, they found their applications in numerical solution of partial differential equations.…”

Section: Introductionmentioning

confidence: 99%

On Locality of Harmonic Generalized Barycentric Coordinates and Their Application to Solution of the Poisson Equation

Deng¹,

Lai²

2022

Preprint

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We first extend the construction of generalized barycentric coordinates (GBC) based on the vertices on the boundary of a polygon Ω to a new kind of GBCs based on vertices inside the Ω of interest. For clarity, the standard GBCs are called boundary GBCs while the new GBCs are called interior GBCs. Then we present an analysis on these two kinds of harmonic GBCs to show that each GBC function whose value is 1 at a vertex (boundary or interior vertex of Ω) decays to zero away from its supporting vertex exponentially fast except for a trivial example. Based on the exponential decay property, we explain how to approximate the harmonic GBC functions locally. That is, due to the locality of these two kinds of GBCs, one can approximate each of these GBC functions by its local versions which is supported over a sub-domain of Ω. The local version of these GBC function will help reduce the computational time for shape deformation in graphical design. Next, with these two kinds of GBC functions at hand, we can use them to approximate the solution of the Dirichlet problem of the Poisson equation. This may provide a more efficient way to solve the Poisson equation by using a computer which has graphical processing unit(GPU) with thousands or more processes than the standard methods using a computer with one or few CPU kernels.

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Functional data approximation on bounded domains using polygonal finite elements

Cited by 8 publications

References 48 publications

Polygonal finite element-based content-aware image warping

Polygonal finite element-based content-aware image warping

Image Representation on Curved Optimal Triangulation

On Locality of Harmonic Generalized Barycentric Coordinates and Their Application to Solution of the Poisson Equation

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