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.

Sampling and reconstruction of signals with finite rate of innovation in the presence of noise

Abstract: Recently, it was shown that it is possible to develop exact sampling schemes for a large class of parametric nonbandlimited signals, namely certain signals of finite rate of innovation. A common feature of such signals is that they have a finite number of degrees of freedom per unit of time and can be reconstructed from a finite number of uniform samples. In order to prove sampling theorems, Vetterli et al. considered the case of deterministic, noiseless signals and developed algebraic methods that lead to perfect reconstruction. However, when noise is present, many of those schemes can become ill-conditioned. In this paper, we revisit the problem of sampling and reconstruction of signals with finite rate of innovation and propose improved, more robust methods that have better numerical conditioning in the presence of noise and yield more accurate reconstruction. We analyze, in detail, a signal made up of a stream of Diracs and develop algorithmic tools that will be used as a basis in all constructions. While some of the techniques have been already encountered in the spectral estimation framework, we further explore preconditioning methods that lead to improved resolution performance in the case when the signal contains closely spaced components. For classes of periodic signals, such as piecewise polynomials and nonuniform splines, we propose novel algebraic approaches that solve the sampling problem in the Laplace domain, after appropriate windowing. Building on the results for periodic signals, we extend our analysis to finite-length signals and develop schemes based on a Gaussian kernel, which avoid the problem of ill-conditioning by proper weighting of the data matrix. Our methods use structured linear systems and robust algorithmic solutions, which we show through simulation results.

Brunn–Minkowski inequality for multiplicities

Abstract: This is a convex subset of the real vector space P⊗ZR. In fact, it is a convex polytope; see [Br], where this polytope is discussed from an algebraic point of view. It is known that the total mass of m is a polynomial in m for su ciently large m (denote by k its degree) and m −k m(m · ) weak −→ ( )d ; where d is the Lebesgue measure supported on and the density ( ) is a piecewise-polynomial function; we will not use this piecewise polynomiality in this paper. Recall that a real function f de ned on a convex subset U of a vector space V is called concave, if

Continuous Representations of Time-Series Gene Expression Data

Abstract: We present algorithms for time-series gene expression analysis that permit the principled estimation of unobserved time points, clustering, and dataset alignment. Each expression profile is modeled as a cubic spline (piecewise polynomial) that is estimated from the observed data and every time point influences the overall smooth expression curve. We constrain the spline coefficients of genes in the same class to have similar expression patterns, while also allowing for gene specific parameters. We show that unobserved time points can be reconstructed using our method with 10-15% less error when compared to previous best methods. Our clustering algorithm operates directly on the continuous representations of gene expression profiles, and we demonstrate that this is particularly effective when applied to nonuniformly sampled data. Our continuous alignment algorithm also avoids difficulties encountered by discrete approaches. In particular, our method allows for control of the number of degrees of freedom of...

Piecewise planar and non-planar stereo for urban scene reconstruction

Abstract: Piecewise planar models for stereo have recently become popular for modeling indoor and urban outdoor scenes. The strong planarity assumption overcomes the challenges presented by poorly textured surfaces, and results in low complexity 3D models for rendering, storage, and transmission. However, such a model performs poorly in the presence of non-planar objects, for example, bushes, trees, and other clutter present in many scenes. We present a stereo method capable of handling more general scenes containing both planar and non-planar regions. Our proposed technique segments an image into piecewise planar regions as well as regions labeled as non-planar. The non-planar regions are modeled by the results of a standard multi-view stereo algorithm. The segmentation is driven by multi-view photoconsistency as well as the result of a color-and texture-based classifier, learned from hand-labeled planar and non-planar image regions. Additionally our method links and fuses plane hypotheses across multiple overlapping views, ensuring a consistent 3D reconstruction over an arbitrary number of images. Using our system, we have reconstructed thousands of frames of street-level video. Results show our method successfully recovers piecewise planar surfaces alongside general 3D surfaces in challenging scenes containing large buildings as well as residential houses.

An intuitive framework for real-time freeform modeling

Abstract: We present a freeform modeling framework for unstructured triangle meshes which is based on constraint shape optimization. The goal is to simplify the user interaction even for quite complex freeform or multiresolution modifications. The user first sets various boundary constraints to define a custom tailored (abstract) basis function which is adjusted to a given design task. The actual modification is then controlled by moving one single 9-dof manipulator object. The technique can handle arbitrary support regions and piecewise boundary conditions with smoothness ranging continuously from C0 to C2. To more naturally adapt the modification to the shape of the support region, the deformed surface can be tuned to bend with anisotropic stiffness. We are able to achieve real-time response in an interactive design session even for complex meshes by precomputing a set of scalar-valued basis functions that correspond to the degrees of freedom of the manipulator by which the user controls the modification.