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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.


Papers
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Journal ArticleDOI
TL;DR: A level-set based approach for the determination of a piecewise constant density function from data of its Radon transform is presented and it is shown that the method works especially well for large data noise (~10% noise).

122 citations

Journal ArticleDOI
Charles Loop1, James F. Blinn1
01 Jul 2006
TL;DR: This work considers the problem of real-time GPU rendering of algebraic surfaces defined by Bezier tetrahedra and compute univariate polynomial coefficients in Bernstein form to maximize the stability of root finding, and to provide shader instances with an early exit test based on the sign of these coefficients.
Abstract: We consider the problem of real-time GPU rendering of algebraic surfaces defined by Bezier tetrahedra. These surfaces are rendered directly in terms of their polynomial representations, as opposed to a collection of approximating triangles, thereby eliminating tessellation artifacts and reducing memory usage. A key step in such algorithms is the computation of univariate polynomial coefficients at each pixel; real roots of this polynomial correspond to possibly visible points on the surface. Our approach leverages the strengths of GPU computation and is highly efficient. Furthermore, we compute these coefficients in Bernstein form to maximize the stability of root finding, and to provide shader instances with an early exit test based on the sign of these coefficients. Solving for roots is done using analytic techniques that map well to a SIMD architecture, but limits us to fourth order algebraic surfaces. The general framework could be extended to higher order with numerical root finding.

122 citations

Book ChapterDOI
28 Jul 2007
TL;DR: This paper provides algorithms for translating between the grammars defined here and finite state automata as well as an algorithm for deciding whether a regular language is Strictly Piecewise.
Abstract: In this paper we explore the class of Strictly Piecewise languages, originally introduced to characterize long-distance phonotactic patterns by Heinz [7] as the Precedence Languages. We provide a series of equivalent abstract characterizations, discuss their basic properties, locate them relative to other well-known subregular classes and provide algorithms for translating between the grammars defined here and finite state automata as well as an algorithm for deciding whether a regular language is Strictly Piecewise.

121 citations

Book ChapterDOI
06 Sep 2014
TL;DR: This work proposes a new scene flow approach that exploits the local and piecewise rigidity of real world scenes and gives a general formulation to solve for local and global rigid motions by jointly using intensity and depth data.
Abstract: Scene flow is defined as the motion field in 3D space, and can be computed from a single view when using an RGBD sensor. We propose a new scene flow approach that exploits the local and piecewise rigidity of real world scenes. By modeling the motion as a field of twists, our method encourages piecewise smooth solutions of rigid body motions. We give a general formulation to solve for local and global rigid motions by jointly using intensity and depth data. In order to deal efficiently with a moving camera, we model the motion as a rigid component plus a non-rigid residual and propose an alternating solver. The evaluation demonstrates that the proposed method achieves the best results in the most commonly used scene flow benchmark. Through additional experiments we indicate the general applicability of our approach in a variety of different scenarios.

121 citations

Proceedings Article
01 Nov 2016
TL;DR: This paper showed that the number of neurons needed by a shallow network to approximate a function is exponentially larger than the corresponding number required by a deep network for a given degree of function approximation.
Abstract: Recently there has been much interest in understanding why deep neural networks are preferred to shallow networks. We show that, for a large class of piecewise smooth functions, the number of neurons needed by a shallow network to approximate a function is exponentially larger than the corresponding number of neurons needed by a deep network for a given degree of function approximation. First, we consider univariate functions on a bounded interval and require a neural network to achieve an approximation error of $\varepsilon$ uniformly over the interval. We show that shallow networks (i.e., networks whose depth does not depend on $\varepsilon$) require $\Omega(\text{poly}(1/\varepsilon))$ neurons while deep networks (i.e., networks whose depth grows with $1/\varepsilon$) require $\mathcal{O}(\text{polylog}(1/\varepsilon))$ neurons. We then extend these results to certain classes of important multivariate functions. Our results are derived for neural networks which use a combination of rectifier linear units (ReLUs) and binary step units, two of the most popular type of activation functions. Our analysis builds on a simple observation: the multiplication of two bits can be represented by a ReLU.

121 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20251
2023917
20222,014
20211,089
20201,147
20191,106