Monotone Regions Research Articles

High order numerical methods for flows with hysteretic fluxes

A second order method and a limited variation WENO numerical scheme are proposed for the Scanning Hysteresis Model (SHM) which describes continuum flows with hysteretic looping fluxes with local memory. The SHM is a nonconservative hyperbolic system with discontinuous coefficients. The limited variation WENO method is fifth order accurate in spatial coordinate and third order accurate in time steps in smooth and monotone regions. The proposed high resolution methods produce the desired solutions more efficiently than the first order upwinding method. These methods are total variation diminishing for the flux, and monotonicity-preserving for the flux and the hysteresis state variable. The flux and the hysteresis state variable produced by these two methods satisfy maximum principle. These two methods also have invariant regions.

Application of morphological processing methods in computer stereo vision

Computer vision algorithms are important for many areas of human activity. In particular, the number of applications related to the need to process images of real-world objects with computerized tools and the subsequent use of descriptive information in a variety of interactive and automated decision-making systems is increased. An important tool for analyzing real-world scenes are approaches to the application of stereo vision algorithms. The important step of many stereo matching algorithms is a disparity map. Depending on the content of the observed scene, part of the values on the disparity map can be immediately attributed to background values on a certain basis, or form a "natural" background, which is characterized by loss of informative data due to unacceptable error of subsequent resultant distance values. The calculated disparity map of any algorithm may contain some shortcomings in the form of discontinuities of continuous information areas caused by the complexity of shooting conditions, the impact of noise of various natures, hardware imperfections, and so on. An approach to mitigating the undesirable influence of negative factors on the resulting disparity is the use of mathematical morphology operations to process disparity maps at the post-processing stage. This paper presents information technology for increasing the content of disparity maps based on the mathematical morphology methods. The technology is based on a combination of morphological operations of erosion and dilation, which eliminates the typical problems of discontinuities of monotone regions and erroneous values on disparity maps. The proposed approach allows reducing the impact of common problems that arise during the operation of stereo matching algorithms, as well as increase the overall informativeness of disparity maps for images of real objects in the absence of partial or complete initial data on the characteristics of the observed scene. The results of testing morphological operations with disparity maps for real objects allow us to conclude about the possibility of partial restoration of areas of disparity maps with gaps in continuous information areas, as well as to reduce the impact of random anomalous values on the overall content of the disparity maps.

Simple smoothness indicator WENO-Z scheme for hyperbolic conservation laws

The advantage of WENO-JS scheme (1996) [22] over the WENO-LOC scheme (1994) [27] is that the WENO-LOC non-linear weights do not achieve the desired order of convergence in smooth monotone regions and at regions containing critical points. In this article, this drawback is overcome with the WENO-LOC smoothness indicators by constructing ‘WENO-Z type’ non-linear weights with a novel global smoothness indicator. This novel smoothness indicator measures the derivatives of the reconstructed flux in a global stencil. As a result, the proposed numerical scheme could decrease the dissipation near the discontinuous regions. The theoretical and numerical experiments to achieve the required order of convergence in smooth monotone regions, at critical points, and the essentially non-oscillatory (ENO) property of proposed non-linear weights are studied. Numerical tests for scalar, one and two-dimensional system of Euler equations are presented to show the effective performance of the proposed numerical scheme.

TOPOLOGY-GUIDED TESSELLATION OF QUADRATIC ElEMENTS

Topology-based methods have been successfully used for the analysis and visualization of piecewise-linear functions defined on triangle meshes. This paper describes a mechanism for extending these methods to piecewise-quadratic functions defined on triangulations of surfaces. Each triangular patch is tessellated into monotone regions, so that existing algorithms for computing topological representations of piecewise-linear functions may be applied directly to the piecewise-quadratic function. In particular, the tessellation is used for computing the Reeb graph, a topological data structure that provides a succinct representation of level sets of the function.

Localization and spreading of contact discontinuity layers in simulations of compressible dissipationless flows

We introduce a systematic approach to examine the localization and width of contact discontinuity layers (CDLs) in simulations of the compressible Euler equations. We study the Piecewise Parabolic Method (PPM) [4] and WENO [3], by simulating a single diffusing or shock-accelerated CDL in 1D and 2D (inclined planar interface). Here the density jump η is greater or less than unity (i.e., fast/slow {f/s} and slow/fast {s/f}, respectively). We advocate a point-wise algorithm for width extraction when the flows are nearly-discontinuous. We examine the CDL localization under mesh refinement and the CDL width spreading at long times. We find that, PPM has an asymmetrically developing CDL width. PPM introduces an artificial steepening for the f/s case and a spreading width CDL for s/f∝ t 1/3. The median point of the interface depends on the order of accuracy r. For PPM, r=2, it is located at a density, ( ρ 1+2 ρ 2)/3, consistent with analysis in [14]. For WENO, r=3, (5th order accurate in smooth monotone regions) the width of the CDL increases as t 1/4, although there is a slight dependence on translation speed of the CDL. These observations are essential for establishing the validity of simulations of accelerated flows of high-gradient stratified and compressible media (Rayleigh–Taylor and Richtmyer–Meshkov environments), particularly for reshock configurations in 2D and 3D.

Reporting Red--Blue Intersections between Two Sets of Connected Line Segments

Abstract. We present a new line sweep algorithm, HEAPSWEEP, for reporting bichromatic (``purple'') intersections between a red and a blue family of line segments. If the union of the segments in each family is connected as a point set, HEAPSWEEP reports all k purple intersections in time O((n+k) α(n) log 3 n) , where n is the total number of input segments and α(n) is the nearly constant inverse Ackermann function. To achieve these bounds, the algorithm maintains only partial information about the vertical ordering between curves of the same color, using a new data structure called a kinetic queue . In order to analyze the running time of HEAPSWEEP, we also show that a simple polygon containing a set of n line segments can be partitioned into monotone regions by a set of vertical threads cutting these segments O(n log n) times.

Block-based unsupervised natural image segmentation

We propose a block-based unsupervised image segmentation algorithm in which the basic unit for the region labeling is an image block rather than a pixel. Adopting the image block, we can introduce a new paradigm for the unsupervised image segmentation, namely, an edge-and-region based segmentation. That is, in our block-based segmentation, we exploit both edge blocks in the region boundary and the homogeneous blocks inside the region. The interior texture and monotone blocks are used to identify regions and the edge blocks are used to find an accurate contour. To obtain a pixel level segmentation, we divide the edge and unlabeled blocks into quadrant and relabel those subblocks to one of the neighboring homogeneous regions. We repeat this process until we obtain a pixel-based segmentation. Experimental results show that the proposed segmentation yields accurate segmentation for natural images even if they contain some texture regions as well as monotone regions.

Stack filters

The median and other rank-order operators possess two properties called the threshold decomposition and the stacking properties. The first is a limited superposition property which leads to a new architecture for these filters; the second is an ordering property which allows an efficient VLSI implementation of the threshold decomposition architecture. Motivated by the success of rank-order filters in a wide variety of applications and by the ease with which they can now be implemented, we consider in this paper a new class of filters called stack filters. They share the threshold decomposition and stacking properties of rank-order filters but are otherwise unconstrained. They are shown to form a very large class of easily implemented nonlinear filters which includes the rank-order operators as well as all compositions of morphological operators. The convergence properties of these filters are investigated using techniques similar to those used to determine root signal behavior of median filters. The results obtained include necessary conditions for a stack filter to preserve monotone regions or edges in signals. The output distribution for these filters is also found. All the stack filters of window width 3 are determined along with their convergence properties. Among these filters are found two which we have named asymmetric median filters. They share all the properties of median filters except that they remove impulses of one sign only; that is, one removes only positive going edges, the other removes only negative going edges, while the median filter removes impulses of both signs. This investigation of the properties of stack filters thus produces several new, useful, and easily implemented filters.

Optimal Point Location in a Monotone Subdivision

Point location, often known in graphics as “hit detection,” is one of the fundamental problems of computational geometry. In a point location query we want to identify which of a given collection of geometric objects contains a particular point. Let $\mathcal{S}$ denote a subdivision of the Euclidean plane into monotone regions by a straight-line graph of m edges. In this paper we exhibit a substantial refinement of the technique of Lee and Preparata [SIAM J. Comput., 6 (1977), pp. 594–606] for locating a point in $\mathcal{S}$ based on separating chains. The new data structure, called a layered dag, can be built in $O(m)$ time, uses $O(m)$ storage, and makes possible point location in $O(\log m)$ time. Unlike previous structures that attain these optimal bounds, the layered dag can be implemented in a simple and practical way, and is extensible to subdivisions with edges more general than straight-line segments.