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.

Practical Box Splines for Reconstruction on the Body Centered Cubic Lattice

Abstract: We introduce a family of box splines for efficient, accurate, and smooth reconstruction of volumetric data sampled on the body-centered cubic (BCC) lattice, which is the favorable volumetric sampling pattern due to its optimal spectral sphere packing property. First, we construct a box spline based on the four principal directions of the BCC lattice that allows for a linear C0 reconstruction. Then, the design is extended for higher degrees of continuity. We derive the explicit piecewise polynomial representations of the C0 and C2 box splines that are useful for practical reconstruction applications. We further demonstrate that approximation in the shift-invariant space - generated by BCC-lattice shifts of these box splines - is twice as efficient as using the tensor-product B-spline solutions on the Cartesian lattice (with comparable smoothness and approximation order and with the same sampling density). Practical evidence is provided demonstrating that the BCC lattice not only is generally a more accurate sampling pattern, but also allows for extremely efficient reconstructions that outperform tensor-product Cartesian reconstructions.

Fuzzy piecewise multilinear and piecewise linear systems as universal approximators in Sobolev norms

Abstract: A piecewise multilinear and a piecewise linear model for fuzzy systems are introduced and their approximation capabilities investigated. Classical results on the approximation of regular functions are improved showing that fuzzy systems may keep their semantic structure while approximating to any degree of accuracy not only sufficiently regular functions, but also their derivatives.

Non-equilibrium thermodynamics of piecewise deterministic markov processes

Abstract: We consider a class of stochastic dynamical systems, called piecewise deterministic Markov processes, with states (x,σ)∈Ω×Γ, Ω being a region in ℝ d or the d-dimensional torus, Γ being a finite set. The continuous variable x follows a piecewise deterministic dynamics, the discrete variable σ evolves by a stochastic jump dynamics and the two resulting evolutions are fully-coupled. We study stationarity, reversibility and time-reversal symmetries of the process. Increasing the frequency of the σ-jumps, the system behaves asymptotically as deterministic and we investigate the structure of its fluctuations (i.e. deviations from the asymptotic behavior), recovering in a non Markovian frame results obtained by Bertini et al. (Phys. Rev. Lett. 87(4):040601, 2001; J. Stat. Phys. 107(3–4):635–675, 2002; J. Stat. Mech. P07014, 2007; Preprint available online at http://www.arxiv.org/abs/0807.4457 , 2008), in the context of Markovian stochastic interacting particle systems. Finally, we discuss a Gallavotti–Cohen-type symmetry relation with involution map different from time-reversal.

ℓ0 minimization for wavelet frame based image restoration

Abstract: The theory of (tight) wavelet frames has been extensively studied in the past twenty years and they are currently widely used for image restoration and other image processing and analysis problems. The success of wavelet frame based models, including balanced approach [18, 7] and analysis based approach [11, 31, 50], is due to their capability of sparsely approximating piecewise smooth functions like images. Motivated by the balanced approach and analysis based approach, we shall propose a wavelet frame based l0 minimization model, where the l0 “norm” of the frame coefficients is penalized. We adapt the penalty decomposition (PD) method of [40] to solve the proposed optimization problem. Some convergence analysis of the adapted PD method will also be provided. Numerical results showed that the proposed model solved by the PD method can generate images with better quality than those obtained by either analysis based approach or balanced approach in terms of restoring sharp features as well as maintaining smoothness of the recovered images.

An explicit basis for C 1 quartic by various bivariate splines

Abstract: We establish the dimension of the space of $C^1 $ bivariate piecewise quartic polynomials defined on a triangulation of a connected polygonal domain. Our approach is to construct a minimal determining set and an associated explicit basis for the space. For general triangulations, the minimal determining set must be defined globally.