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.

Vision guided navigation for a nonholonomic mobile robot

Abstract: Theoretical and analytical aspects of the visual servoing problem have not received much attention. Furthermore, the problem of estimation from the vision measurements has been considered separately from the design of the control strategies. Instead of addressing the pose estimation and control problems separately, we attempt to characterize the types of control tasks which can be achieved using only quantities directly measurable in the image, bypassing the pose estimation phase. We consider the task of navigation for a nonholonomic ground mobile base tracking an arbitrarily shaped continuous ground curve. This tracking problem is formulated as one of controlling the shape of the curve in the image plane. We study the controllability of the system characterizing the dynamics of the image curve, and show that the shape of the image curve is controllable only up to its "linear" curvature parameters. We present stabilizing control laws for tracking piecewise analytic curves, and propose to track arbitrary curves by approximating them by piecewise "linear" curvature curves. Simulation results are given for these control schemes. Observability of the curve dynamics by using direct measurements from vision sensors as the outputs is studied and an extended Kalman filter is proposed to dynamically estimate the image quantities needed for feedback control from the actual noisy images.

A new approach to analyzing gene expression time series 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 non-uniformly 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 the warp through the specification of parameterized functions, which helps to avoid overfitting. We demonstrate that our algorithm produces stable low-error alignments on real expression data and further show a specific application to yeast knockout data that produces biologically meaningful results.

A Weak Galerkin Mixed Finite Element Method for Second-Order Elliptic Problems

Abstract: A new weak Galerkin (WG) method is introduced and analyzed for the second order elliptic equation formulated as a system of two first order linear equations. This method, called WG-MFEM, is designed by using discontinuous piecewise polynomials on finite element partitions with arbitrary shape of polygons/polyhedra. The WG-MFEM is capable of providing very accurate numerical approximations for both the primary and flux variables. Allowing the use of discontinuous approximating functions on arbitrary shape of polygons/polyhedra makes the method highly flexible in practical computation. Optimal order error estimates in both discrete $H^1$ and $L^2$ norms are established for the corresponding weak Galerkin mixed finite element solutions.

A Total Variation Model for Retinex

Abstract: Human vision has the ability to recognize color under varying illumination conditions. Retinex theory is introduced to explain how the human visual system perceives color. The main aim of this paper is to present a total variation model for Retinex. Different from the existing methods, we consider and study two important elements which include illumination and reflection. We assume spatial smoothness of the illumination and piecewise continuity of the reflection, where the total variation term is employed in the model. The existence of the solution of the model is shown in the paper. We employ a fast computation method to solve the proposed minimization problem. Numerical examples are presented to illustrate the effectiveness of the proposed model.

New likelihood test methods for change detection in image sequences

Abstract: Modeling the image as a piecewise linear gray-value function of the pixel coordinates considerably improved a change detection test based previously on a piecewise constant gray-value function. These results encouraged investigations into modeling the picture as a mosaic of patches where the gray-value function within each patch is described as a second-order bivariate polynomial of the pixel coordinates. Such a more appropriate model allowed the assumption to be made that the remaining gray-value variation within each patch can be attributed to noise related to the sensing and digitizing devices, independent of the individual image frames in a sequence. This assumption made it possible to relate the likelihood test for change detection to well-known statistical tests ( t test, F test), facilitating the determination of threshold values related to a priori confidence limits.