Piecewise linear function

About: Piecewise linear function is a research topic. Over the lifetime, 8133 publications have been published within this topic receiving 161444 citations.

Curve and surface smoothing without shrinkage

Abstract: For a number of computational purposes, including visualization of scientific data and registration of multimodal medical data, smooth curves must be approximated by polygonal curves, and surfaces by polyhedral surfaces. An inherent problem of these approximation algorithms is that the resulting curves and surfaces appear faceted. Boundary-following and iso-surface construction algorithms are typical examples. To reduce the apparent faceting, smoothing methods are used. In this paper, we introduce a new method for smoothing piecewise linear shapes of arbitrary dimension and topology. This new method is in fact a linear low-pass filter that removes high-curvature variations, and does not produce shrinkage. Its computational complexity is linear in the number of edges or faces of the shape, and the required storage is linear in the number of vertices. >

Optimality of a Standard Adaptive Finite Element Method

Abstract: In this paper an adaptive finite element method is constructed for solving elliptic equations that has optimal computational complexity. Whenever, for some s > 0, the solution can be approximated within a tolerance e > 0 in energy norm by a continuous piecewise linear function on some partition with O(e-1/s) triangles, and one knows how to approximate the right-hand side in the dual norm with the same rate with piecewise constants, then the adaptive method produces approximations that converge with this rate, taking a number of operations that is of the order of the number of triangles in the output partition. The method is similar in spirit to that from [SINUM, 38 (2000), pp. 466-488] by Morin, Nochetto, and Siebert, and so in particular it does not rely on a recurrent coarsening of the partitions. Although the Poisson equation in two dimensions with piecewise linear approximation is considered, the results generalize in several respects.

Fast surface reconstruction using the level set method

Abstract: We describe new formulations and develop fast algorithms for implicit surface reconstruction based on variational and partial differential equation (PDE) methods. In particular we use the level set method and fast sweeping and tagging methods to reconstruct surfaces from a scattered data set. The data set might consist of points, curves and/or surface patches. A weighted minimal surface-like model is constructed and its variational level set formulation is implemented with optimal efficiency. The reconstructed surface is smoother than piecewise linear and has a natural scaling in the regularization that allows varying flexibility according to the local sampling density. As is usual with the level set method we can handle complicated topology and deformations, as well as noisy or highly nonuniform data sets easily. The method is based on a simple rectangular grid, although adaptive and triangular grids are also possible. Some consequences, such as hole filling capability, are demonstrated, as well as the viability and convergence of our new fast tagging algorithm.

Some errors estimates for the box method

Abstract: We define and analyze several variants of the box method for discretizing elliptic boundary value problems in the plane. Our estimates show the error to be comparable to a standard Galerkin finite element method using piecewise linear polynomials.