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.

Edge-preserving decompositions for multi-scale tone and detail manipulation

Abstract: Many recent computational photography techniques decompose an image into a piecewise smooth base layer, containing large scale variations in intensity, and a residual detail layer capturing the smaller scale details in the image. In many of these applications, it is important to control the spatial scale of the extracted details, and it is often desirable to manipulate details at multiple scales, while avoiding visual artifacts.In this paper we introduce a new way to construct edge-preserving multi-scale image decompositions. We show that current basedetail decomposition techniques, based on the bilateral filter, are limited in their ability to extract detail at arbitrary scales. Instead, we advocate the use of an alternative edge-preserving smoothing operator, based on the weighted least squares optimization framework, which is particularly well suited for progressive coarsening of images and for multi-scale detail extraction. After describing this operator, we show how to use it to construct edge-preserving multi-scale decompositions, and compare it to the bilateral filter, as well as to other schemes. Finally, we demonstrate the effectiveness of our edge-preserving decompositions in the context of LDR and HDR tone mapping, detail enhancement, and other applications.

PLUTO: A Numerical Code for Computational Astrophysics

Abstract: We present a new numerical code, PLUTO, for the solution of hypersonic flows in 1, 2, and 3 spatial dimensions and different systems of coordinates. The code provides a multiphysics, multialgorithm modular environment particularly oriented toward the treatment of astrophysical flows in presence of discontinuities. Different hydrodynamic modules and algorithms may be independently selected to properly describe Newtonian, relativistic, MHD, or relativistic MHD fluids. The modular structure exploits a general framework for integrating a system of conservation laws, built on modern Godunov-type shock-capturing schemes. Although a plethora of numerical methods has been successfully developed over the past two decades, the vast majority shares a common discretization recipe, involving three general steps: a piecewise polynomial reconstruction followed by the solution of Riemann problems at zone interfaces and a final evolution stage. We have checked and validated the code against several benchmarks available in literature. Test problems in 1, 2, and 3 dimensions are discussed.

Sampling signals with finite rate of innovation

Abstract: The authors consider classes of signals that have a finite number of degrees of freedom per unit of time and call this number the rate of innovation. Examples of signals with a finite rate of innovation include streams of Diracs (e.g., the Poisson process), nonuniform splines, and piecewise polynomials. Even though these signals are not bandlimited, we show that they can be sampled uniformly at (or above) the rate of innovation using an appropriate kernel and then be perfectly reconstructed. Thus, we prove sampling theorems for classes of signals and kernels that generalize the classic "bandlimited and sinc kernel" case. In particular, we show how to sample and reconstruct periodic and finite-length streams of Diracs, nonuniform splines, and piecewise polynomials using sinc and Gaussian kernels. For infinite-length signals with finite local rate of innovation, we show local sampling and reconstruction based on spline kernels. The key in all constructions is to identify the innovative part of a signal (e.g., time instants and weights of Diracs) using an annihilating or locator filter: a device well known in spectral analysis and error-correction coding. This leads to standard computational procedures for solving the sampling problem, which we show through experimental results. Applications of these new sampling results can be found in signal processing, communications systems, and biological systems.

Computation of piecewise quadratic Lyapunov functions for hybrid systems

Abstract: This paper presents a computational approach to stability analysis of nonlinear and hybrid systems. The search for a piecewise quadratic Lyapunov function is formulated as a convex optimization problem in terms of linear matrix inequalities. The relation to frequency domain methods such as the circle and Popov criteria is explained. Several examples are included to demonstrate the flexibility and power of the approach.