Topic
Piecewise
About: Piecewise is a research topic. Over the lifetime, 21064 publications have been published within this topic receiving 432096 citations. The topic is also known as: piecewise-defined function & hybrid function.
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TL;DR: It is shown that, for most types of radial basis functions that are considered in this paper, convergence can be achieved without further assumptions on the objective function.
Abstract: We introduce a method that aims to find the global minimum of a continuous nonconvex function on a compact subset of \dRd It is assumed that function evaluations are expensive and that no additional information is available Radial basis function interpolation is used to define a utility function The maximizer of this function is the next point where the objective function is evaluated We show that, for most types of radial basis functions that are considered in this paper, convergence can be achieved without further assumptions on the objective function Besides, it turns out that our method is closely related to a statistical global optimization method, the P-algorithm A general framework for both methods is presented Finally, a few numerical examples show that on the set of Dixon-Szego test functions our method yields favourable results in comparison to other global optimization methods
793 citations
TL;DR: The approach exploits the gain-scheduling nature of fuzzy systems and results in stability conditions that can be verified via convex optimization over linear matrix inequalities, and special attention is given to the computational aspects of the approach.
Abstract: Presents an approach to stability analysis of fuzzy systems. The analysis is based on Lyapunov functions that are continuous and piecewise quadratic. The approach exploits the gain-scheduling nature of fuzzy systems and results in stability conditions that can be verified via convex optimization over linear matrix inequalities. Examples demonstrate the many improvements over analysis based on a single quadratic Lyapunov function. Special attention is given to the computational aspects of the approach and several methods to improve the computational efficiency are described.
775 citations
TL;DR: A class of fourth-order partial differential equations (PDEs) are proposed to optimize the trade-off between noise removal and edge preservation, and speckles are more visible in images processed by the proposed PDEs, because piecewise planar images are less likely to mask speckling.
Abstract: A class of fourth-order partial differential equations (PDEs) are proposed to optimize the trade-off between noise removal and edge preservation. The time evolution of these PDEs seeks to minimize a cost functional which is an increasing function of the absolute value of the Laplacian of the image intensity function. Since the Laplacian of an image at a pixel is zero if the image is planar in its neighborhood, these PDEs attempt to remove noise and preserve edges by approximating an observed image with a piecewise planar image. Piecewise planar images look more natural than step images which anisotropic diffusion (second order PDEs) uses to approximate an observed image. So the proposed PDEs are able to avoid the blocky effects widely seen in images processed by anisotropic diffusion, while achieving the degree of noise removal and edge preservation comparable to anisotropic diffusion. Although both approaches seem to be comparable in removing speckles in the observed images, speckles are more visible in images processed by the proposed PDEs, because piecewise planar images are less likely to mask speckles than step images and anisotropic diffusion tends to generate multiple false edges. Speckles can be easily removed by simple algorithms such as the one presented in this paper.
772 citations
TL;DR: In this article, a class of piecewise continuous, piecewise C transforma-tions on the interval [0, 1] has been shown to have absolutely continuous invariant measures.
Abstract: A class of piecewise continuous, piecewise C transformations on the interval [0,1] is shown to have absolutely continuous invariant measures.
772 citations
TL;DR: An improved model is presented, constructed by adding a nonlinear fourth order diffusive term to the Euler--Lagrange equations of the variational TV model, which substantially reduces the staircase effect, while preserving sharp jump discontinuities.
Abstract: The total variation (TV) denoising method is a PDE-based technique that preserves edges well but has the sometimes undesirable staircase effect, namely, the transformation of smooth regions ( ramps) into piecewise constant regions ( stairs). In this paper we present an improved model, constructed by adding a nonlinear fourth order diffusive term to the Euler--Lagrange equations of the variational TV model. Our technique substantially reduces the staircase effect, while preserving sharp jump discontinuities (edges). We show numerical evidence of the power of resolution of this novel model with respect to the TV model in some 1D and 2D numerical examples.
769 citations