Topic

# Piecewise linear function

About: Piecewise linear function is a(n) research topic. Over the lifetime, 8133 publication(s) have been published within this topic receiving 161444 citation(s).

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01 Jul 1992TL;DR: A general method for automatic reconstruction of accurate, concise, piecewise smooth surfaces from unorganized 3D points that is able to automatically infer the topological type of the surface, its geometry, and the presence and location of features such as boundaries, creases, and corners.

Abstract: This thesis describes a general method for automatic reconstruction of accurate, concise, piecewise smooth surfaces from unorganized 3D points. Instances of surface reconstruction arise in numerous scientific and engineering applications, including reverse-engineering--the automatic generation of CAD models from physical objects.
Previous surface reconstruction methods have typically required additional knowledge, such as structure in the data, known surface genus, or orientation information. In contrast, the method outlined in this thesis requires only the 3D coordinates of the data points. From the data, the method is able to automatically infer the topological type of the surface, its geometry, and the presence and location of features such as boundaries, creases, and corners.
The reconstruction method has three major phases: (1) initial surface estimation, (2) mesh optimization, and (3) piecewise smooth surface optimization. A key ingredient in phase 3, and another principal contribution of this thesis, is the introduction of a new class of piecewise smooth representations based on subdivision. The effectiveness of the three-phase reconstruction method is demonstrated on a number of examples using both simulated and real data.
Phases 2 and 3 of the surface reconstruction method can also be used to approximate existing surface models. By casting surface approximation as a global optimization problem with an energy function that directly measures deviation of the approximation from the original surface, models are obtained that exhibit excellent accuracy to conciseness trade-offs. Examples of piecewise linear and piecewise smooth approximations are generated for various surfaces, including meshes, NURBS surfaces, CSG models, and implicit surfaces.

2,963 citations

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TL;DR: A technique to compute the explicit state-feedback solution to both the finite and infinite horizon linear quadratic optimal control problem subject to state and input constraints is presented, and it is shown that this closed form solution is piecewise linear and continuous.

Abstract: We present a technique to compute the explicit state-feedback solution to both the finite and infinite horizon linear quadratic optimal control problem subject to state and input constraints. We show that this closed form solution is piecewise linear and continuous. As a practical consequence of the result, constrained linear quadratic regulation becomes attractive also for systems with high sampling rates, as on-line quadratic programming solvers are no more required for the implementation.

2,899 citations

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TL;DR: In this paper, the conditional standard deviation is a piecewise linear function of past values of the white noise, which allows different reactions of the volatility to different signs of the lagged errors.

Abstract: In this paper we consider a modification of the classical ARCH models introduced by Engle (1982). In this modified model the conditional standard deviation is a piecewise linear function of past values of the white noise. This specific form allows different reactions of the volatility to different signs of the lagged errors. Stationarity conditions are derived. Maximum likelihood and least squares estimation are also considered. Finally an empirical example relating to the French CAC stock index is presented and several specifications are compared.

1,948 citations

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TL;DR: The method is tested by testing its ability to track interfaces through large, controlled topology changes, whereby an initially simple interface configuration is subjected to vortical flows, and numerical results for these strenuous test problems provide evidence for the algorithm's improved solution quality and accuracy.

Abstract: A new algorithm for the volume tracking of interfaces in two dimensions is presented. The algorithm is based upon a well-defined, second-order geometric solution of a volume evolution equation. The method utilizes local discrete material volume and velocity data to track interfaces of arbitrarily complex topology. A linearity-preserving, piecewise linear interface geometry approximation ensures that solutions generated retain second-order spatial accuracy. Second-order temporal accuracy is achieved by virtue of a multidimensional unsplit time integration scheme. We detail our geometrically based solution method, in which material volume fluxes are computed systematically with a set of simple geometric tasks. We then interrogate the method by testing its ability to track interfaces through large, controlled topology changes, whereby an initially simple interface configuration is subjected to vortical flows. Numerical results for these strenuous test problems provide evidence for the algorithm's improved solution quality and accuracy.

1,442 citations

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17 Jun 1990TL;DR: The authors describe how a two-layer neural network can approximate any nonlinear function by forming a union of piecewise linear segments and a method is given for picking initial weights for the network to decrease training time.

Abstract: The authors describe how a two-layer neural network can approximate any nonlinear function by forming a union of piecewise linear segments. A method is given for picking initial weights for the network to decrease training time. The authors have used the method to initialize adaptive weights over a large number of different training problems and have achieved major improvements in learning speed in every case. The improvement is best when a large number of hidden units is used with a complicated desired response. The authors have used the method to train the truck-backer-upper and were able to decrease the training time from about two days to four hours

1,397 citations