Showing papers on "Bilinear interpolation published in 2002"

What is a Good Linear Element? Interpolation, Conditioning, and Quality Measures.

Abstract: When a mesh of simplicial elements (triangles or tetrahedra) is used to form a piecewise linear approximation of a function, the accuracy of the approximation depends on the sizes and shapes of the elements. In finite element methods, the conditioning of the stiffness matrices also depends on the sizes and shapes of the elements. This paper explains the mathematical connections between mesh geometry, interpolation errors, and stiffness matrix conditioning. These relationships are expressed by error bounds and element quality measures that determine the fitness of a triangle or tetrahedron for interpolation or for achieving low condition numbers. Unfortunately, the quality measures for these two purposes do not agree with each other; for instance, small angles are bad for matrix conditioning but not for interpolation. Several of the upper and lower bounds on interpolation errors and element stiffness matrix conditioning given here are tighter than those that have appeared in the literature before, so the quality measures are likely to be unusually precise indicators of element fitness.

Interpolation of Linear Operators

Abstract: The survey is devoted to the modern state of the theory of interpolation of linear operators acting in Banach spaces. Principal attention is devoted to real and complex methods and applications of the theory of interpolation to analysis.

On stabilization of bilinear uncertain time-delay stochastic systems with Markovian jumping parameters

Abstract: In this paper, we investigate the stochastic stabilization problem for a class of bilinear continuous time-delay uncertain systems with Markovian jumping parameters. Specifically, the stochastic bilinear jump system under study involves unknown state time-delay, parameter uncertainties, and unknown nonlinear deterministic disturbances. The jumping parameters considered here form a continuous-time discrete-state homogeneous Markov process. The whole system may be regarded as a stochastic bilinear hybrid system that includes both time-evolving and event-driven mechanisms. Our attention is focused on the design of a robust state-feedback controller such that, for all admissible uncertainties as well as nonlinear disturbances, the closed-loop system is stochastically exponentially stable in the mean square, independent of the time delay. Sufficient conditions are established to guarantee the existence of desired robust controllers, which are given in terms of the solutions to a set of either linear matrix inequalities (LMIs), or coupled quadratic matrix inequalities. The developed theory is illustrated by numerical simulation.

Six boundary elements per wavelength: is that enough?

Abstract: The commonly applied rule of thumb to use six (linear) elements per wavelength in linear time-harmonic acoustics is discussed in this paper. In a survey of related work, rules of element design in computational acoustics are collected. This is followed by a brief review of the boundary element method and a more detailed presentation of boundary element interpolation functions. Constant, bilinear and biquadratic interpolation polynomials are used on triangular and quadrilateral elements. In the investigation of a long duct, the numeric solution of the three dimensional problem is compared with the analytic solution. The performance of triangular and quadrilateral, constant, bilinear and biquadratic elements is compared. The error of the numeric solution is calculated in the maximum norm and the Euclidean norm on the surface and at internal points. It is estimated how many elements per wavelength are required to remain below certain error bounds of the sound pressure magnitude. Finally, a sedan cabin compar...

Bilinear estimates and applications to nonlinear wave equations

Abstract: We undertake a systematic review of results proved in [26, 27, 30-32] concerning local well-posedness of the Cauchy problem for certain systems of nonlinear wave equations, with minimal regularity assumptions on the initial data. Moreover we give a considerably simplified and unified treatment of these results and provide also complete proofs for large data. The paper is also intended as an introduction to and survey of current research in the very active area of nonlinear wave equations. The key ingredients throughout the survey are the use of the null structure of the equations we consider and, intimately tied to it, bilinear estimates.

Person Identification from the EEG using Nonlinear Signal Classification

Abstract: Objectives: This paper focusses on the person identification problem based on features extracted from the ElectroEncephaloGram (EEG). A bilinear rather than a purely linear model is fitted on the EEG signal, prompted by the existence of non-linear components in the EEG signal – a conjecture already investigated in previous research works. The novelty of the present work lies in the comparison between the linear and the bilinear results, obtained from real field EEG data, aiming towards identification of healthy subjects rather than classification of pathological cases for diagnosis. Methods: The EEG signal of a, in principle, healthy individual is processed via (non)linear (AR, bilinear) methods and classified by an artificial neural network classifier. Results: Experiments performed on real field data show that utilization of the bilinear model parameters as features improves correct classification scores at the cost of increased complexity and computations. Results are seen to be statistically significant at the 99.5% level of significance, via the X2 test for contingency. Conclusions: The results obtained in the present study further corroborate existing research, which shows evidence that the EEG carries individual-specific information, and that it can be successfully exploited for purposes of person identification and authentication.

Seismic trace interpolation in the f-x-y domain

Abstract: Seismic trace interpolation is implemented as a 2‐D (x, y) spatial prediction, performed separately on each frequency (f) slice. This so‐called f‐x‐y domain trace interpolation method is based on the relation that the linear prediction (LP) operator estimated at a given frequency may be used to predict data at a higher frequency but a smaller trace spacing. The relationship originally given for thef‐x domain trace interpolation is successfully extended to the f‐x‐y domain. The extension is achieved by masking the data samples selectively from the input frequency slice to design the LP operators. Two interpolation algorithms using the full‐step and the fractional‐step predictions, respectively, are developed. Both methods use an all‐azimuth prediction in the x‐y domain, but the fractional‐step prediction method is computationally more efficient. While the interpolation method can be applied to a common‐offset cube of 3‐D seismic, it can also be applied to 2‐D seismic traces for prestack data processing. Sy...

Adaptive interlace-to-progressive scan conversion algorithm

Abstract: An interlace-to-progressive scan conversion system comprises: a spatial line averaging prefilter; a motion estimator; a three-stage adaptive recursive filter. The motion estimator comprises: a 3-D recursive search sub-component having a bilinear interpolator; a motion correction sub-component having an error-function including penalties related to the difference between a given candidate vector and a plurality of neighboring vectors; a block erosion sub-component. The motion estimator assumes that motion is constant between fields. The three-stage adaptive recursive filter comprises: a first stage that selects between using static pixels data and moving pixels data from a next field; a second stage that selects a more valid set of data between motion compensated data from a previous field and the pixels selected by the first stage; a third stage that combines an intra-field interpolation with the more valid set of data selected by the second stage.

Adaptive interpolation filter system for motion compensated predictive video coding

Abstract: An adaptive interpolation filter system for searching to obtain an optimized interpolation filter that minimizes prediction error in a video codec includes an interpolation module and a discrete search space. A plurality of discrete interpolation filters are positioned in a three dimensional grid within the search space. The interpolation module may select a current minimum filter. Based on the current minimum filter, a search region within the search space that includes a plurality of candidate filters located adjacent to the current minimum filter may be identified. The interpolation module may interpolate a reference image signal with each of the candidate filters. The candidate filter resulting in the smallest prediction error may be identified as the current minimum filter and the search repeated until the prediction error is minimized.

Bivariate fractal interpolation functions on grids

Abstract: In this paper, a method of construction of fractal interpolation functions (FIF) on random grids in ℝ2 is examined.

Robust filtering for bilinear uncertain stochastic discrete-time systems

Abstract: This paper deals with the robust filtering problem for uncertain bilinear stochastic discrete-time systems with estimation error variance constraints. The uncertainties are allowed to be norm-bounded and enter into both the state and measurement matrices. We focus on the design of linear filters, such that for all admissible parameter uncertainties, the error state of the bilinear stochastic system is mean square bounded, and the steady-state variance of the estimation error of each state is not more than the individual prespecified value. It is shown that the design of the robust filters can be carried out by solving some algebraic quadratic matrix inequalities. In particular, we establish both the existence conditions and the explicit expression of desired robust filters. A numerical example is included to show the applicability of the present method.

Inelastic deformation response of SDOF systems subjected to earthquakes

Abstract: Performance-based seismic design requires reliable methods to predict earthquake demands on structures, and particularly inelastic deformations, to ensure that specific damage-based criteria are met. Several methods based on the response of equivalent linear single-degree-of-freedom (SDOF) systems have been proposed to estimate the response of multi-degree-of-freedom structures. These methods do not offer advantages over the traditional Veletsos–Newmark–Hall (VNH) procedure, indeed, they have been shown to be inaccurate. In this study, the VNH method is revised, considering the inelastic response of elastoplastic, bilinear, and stiffness-degrading systems with 5% damping subjected to two sets of earthquake ground motions. One is an ensemble of 51 earthquake records in the Circumpacific Belt, and the other is a group of 44 records in California. A statistical analysis of the response data provides factors for constructing VNH inelastic spectra. Such factors show that the ‘equal-displacement’ and ‘equal-energy’ rules to relate elastic and inelastic responses are unconservative for high ductilities in the acceleration- and velocity-sensitive regions of the spectrum. It is also shown that, on average, the effect of the type of force–deformation relationship of non-linear systems is not significant, and responses can be conservatively predicted using the simple elastoplastic model. Copyright © 2001 John Wiley & Sons, Ltd.

A new direction adaptive scheme for image interpolation

Abstract: We present a novel image interpolation method based on variational models with both smoothing and orientation constraints. By introducing the orientation constraint, we simplify the nonlinear PDE problem into a series of problems with explicit solutions. In our model, the gradient directions for the interpolated pixels are first estimated using a modified orientation diffusion method. Using these estimated gradient directions adaptive directional interpolation is carried out. An effective numerical implementation of the adaptive directional interpolation is presented for the case of upsampling by factors of two. This implementation had very low complexity and is well suited for real-time applications.

Canny edge based image expansion

Abstract: In this paper, a Canny edge-based image expansion method is introduced. Our proposed expansion method outperforms the pixel replication, the bilinear interpolation and the bicubic interpolation methods. It gives crisp and less zigzag pictures. Our method is applied on the image after it has been expanded using bilinear or bicubic interpolation. The edges of such an expanded image are obtained using the Canny edge detector. The values of pixels around the edges are modified to yield a crisper and less zigzagged picture.

Robust array interpolation using second-order cone programming

Abstract: We study Friedlander's (1993) array interpolation technique, whose main shortcoming in multisource scenarios is that it does not provide sufficient robustness against sources arriving outside specified interpolation sectors. In this letter, we develop a new robust interpolation approach by minimizing the interpolation error inside the sectors of interest while setting multiple "stopband" constraints outside these sectors to prevent performance degradation effects caused by out-of-sector sources. Computationally efficient convex formulations of the robust interpolation matrix design problem using second-order cone programming are derived.

Adaptive interpolation filter for motion compensated prediction

Abstract: Standardized hybrid video coding systems are based on motion compensated prediction with fractional-pel displacement vector resolution. In the recent JVT video coding scheme (MPEG-4 part 10, H.264) displacement vector resolutions of 1/4- or 1/8-pel are applied. In order to estimate and compensate these fractional-pel displacements, interpolation filters are used. So far, these interpolation filters are invariant. The same filter coefficients are applied for all sequences and for all images of a sequence. Therefore it is not possible to consider non-stationary statistical properties of video signals in the interpolation process. In this paper an adaptive interpolation scheme is presented. This interpolation scheme uses filter coefficients that are adapted once per image to the non-stationary statistical properties of the video signal. The filter-coefficients are coded and transmitted. Due to the adaptive interpolation filter a coding gain up to 0.8 dB PSNR is obtained in the JVT coding scheme.

Binary morphological shape-based interpolation applied to 3-D tooth reconstruction

Abstract: In this paper, we propose an interpolation algorithm using a mathematical morphology morphing approach. The aim of this algorithm is to reconstruct the n-dimensional object from a group of (n-1)-dimensional sets representing sections of that object. The morphing transformation modifies pairs of consecutive sets such that they approach in shape and size. The interpolated set is achieved when the two consecutive sets are made idempotent by the morphing transformation. We prove the convergence of the morphological morphing. The entire object is modeled by successively interpolating a certain number of intermediary sets between each two consecutive given sets. We apply the interpolation algorithm for three-dimensional tooth reconstruction.

Interpolation for function spaces related to mixed boundary value problems

Abstract: Interpolation theorems are proved for Sobolev spaces of functions on nonsmooth domains with vanishing trace on a part of the boundary.

A uniform estimate

Abstract: In this article we study bilinear operators whose input consists of two functions on the real line and whose output is again a function on the real line. We shall be concerned with a priori estimates for these operators. For simplicity we initially require the two input functions to be Schwartz functions and allow the output function to be a tempered distribution. By abstract generalities such a bilinear operation B can be formally written as

Correction methods for missing data in sinograms of the HRRT PET scanner

Abstract: The High Resolution Research Tomograph (HRRT) is a 3D PET scanner designed for brain imaging and small animal imaging. The HRRT consists of 8 panel detector heads that are separated by a gap of 17 mm resulting in data gaps in the sinogram. Furthermore, data gaps can result from detector-block failure. To prevent artifacts in the reconstruction when using FORE, filling in of the data gaps is required. The purpose of this study was to evaluate the accuracy of several gap filling methods. Two gap-filling methods were investigated: a) bilinear interpolation, b) a model-based method: an intermediate volume is reconstructed (2D) based on only direct planes, after which this image is forward projected towards the data gaps. In addition, an improved model-based method is introduced: c) first fill the gaps using interpolation, then reconstructing using FORE and forward projecting to fill the gaps. Detector gaps and block failures were mimicked by zeroing LORs in simulated and experimentally acquired sinograms. The gaps were filled using the different methods, reconstructed using FORE+2DOSEM and compared with reconstruction of the original sinogram. From the variance of the reconstructions and from difference images it could be concluded that for homogeneous objects which are large as compared to the extent of data gaps all methods give similar results, although the interpolation methods requires significant less computation time. For objects with dimensions comparable to the size of a data gap the interpolation method falls short. The simple model-based method however suffers from artifacts in the intermediate direct planes reconstruction. The latter is overcome by the improved model-based method. In conclusion, the improved model-based method might outperform the interpolation method, but due to the long computation times usage of this method is only justified in case of small objects.

Optimal Domains for Kernel Operators via Interpolation

Abstract: The problem of finding optimal lattice domains for kernel operators with values in rearrangement invariant spaces on the interval (0,1) is considered. The techniques used are based on interpolation theory and integration with respect to C((0,1))-valued measures.

Method and apparatus for pixel filtering using shared filter resource between overlay and texture mapping engines

Abstract: A configurable filter module for providing shared filter resource between an overlay engine and a texture mapping engine of a graphics system. The configurable filter may comprise a plurality of linear blend units each of which receives data input from one of the overlay engine and a mapping engine cache, and generates a linear blend filter output respectively; and a filter output multiplexer which receives data output from the linear blend units and selects a proper byte ordering output, wherein the linear blend units serve as an overlay interpolator filter to perform linear blending of the data input from the overlay engine during a linear blend mode, and serve as a texture bilinear filter to perform bilinear filtering of the data input from the mapping engine cache during a bilinear filtering mode.

Feature-guided shape-based image interpolation

Abstract: A feature-guided image interpolation scheme is presented. It is an effective and improved, shape-based interpolation method used for interpolating image slices in medical applications. The proposed method integrates feature line-segments to guide the shape-based method for better shape interpolation. An automatic method for finding these line segments is given. The proposed feature-guided shape-based method can manage translation, rotation and scaling situations when the slices have similar shapes. It can also interpolate intermediate shapes when the successive slices do not have similar shapes. This method is experimentally evaluated using artificial and real two-dimensional and three-dimensional data. The proposed method generated satisfactory interpolated results in these experiments. We demonstrate the practicality, effectiveness and reproducibility of the proposed method for interpolating medical images.

Reducing Dispersion of Linear Triangular Elements for the Helmholtz Equation

Abstract: The Galerkin/least squares (GLS) modification improves the performance of finite-element computations of time-harmonic acoustics at high wave numbers. The design of the GLS resolution-dependent method parameter for two-dimensional computation in previous work was based on dispersion analysis of one-dimensional and square bilinear elements. We analyze the dispersion of linear triangular finite elements, and define method parameters that eliminate dispersion on a hexagonal patch. Numerical tests compare the performance of the proposed method with established techniques on structured and unstructured triangular meshes. Based on this work, we propose a method parameter that may be used for computation with both linear triangular and bilinear quadrilateral elements.

Linear filtering for bilinear stochastic differential systems with unknown inputs

Abstract: Investigates the problem of state estimation for bilinear stochastic multivariable differential systems in the presence of an additional disturbance, whose statistics are completely unknown.. A linear filter is proposed, based on a suitable decomposition of the state of the bilinear system into two components. The first one is a computable function of the observations while the second component is estimated via a suitable linear filtering algorithm. No a priori information on the disturbance is required for the filter implementation. The proposed filter is robust with respect to the unknown input, in that the covariance of the estimation error is not affected by such input. Numerical simulations show the effectiveness of the proposed filter.