Bicubic Spline Research Articles

Valuation of Capital Protection Options

This paper presents numerical algorithm and results for pricing a capital protection option offered by many asset managers for investment portfolios to take advantage of market growth and protect savings. Under optimal withdrawal policyholder behaviour the pricing of such a product is an optimal stochastic control problem that cannot be solved using Monte Carlo method. In low dimension case, it can be solved using PDE based methods such as finite difference. In this paper, we develop a much more efficient Gauss-Hermite quadrature method with a one-dimensional cubic spline for calculation of the expectation between withdrawal/reset dates, and a bi-cubic spline interpolation for applying the jump conditions across withdrawal/reset dates. We show results for both static and dynamic withdrawals and for both the asset accumulation and the pension phases (different penalties for any excessive withdrawal) in the retirement investment cycle. To evaluate products with capital protection option, it is common industry practice to assume static withdrawals and use Monte Carlo method. As a result, the fair fee is underpriced if policyholder behaves optimally. We found that extra fee that has to be charged to counter the optimal policyholder behaviour is most significant at smaller interest rate and higher volatility levels, and it is sensitive to the penalty threshold. At low interest rate and a moderate penalty threshold level (15% of the portfolio value per annum) typically set in practice, the extra fee due to optimal withdrawal can be as high as 40% and more on top of the base case of no withdrawal or the case of fixed withdrawals at the penalty threshold.

Positivity Preserving Interpolation Using Rational Bicubic Spline

This paper discusses the positivity preserving interpolation for positive surfaces data by extending theC1rational cubic spline interpolant of Karim and Kong to the bivariate cases. The partially blended rational bicubic spline has 12 parameters in the descriptions where 8 of them are free parameters. The sufficient conditions for the positivity are derived on every four boundary curves network on the rectangular patch. Numerical comparison with existing schemes also has been done in detail. Based on Root Mean Square Error (RMSE), our partially blended rational bicubic spline is on a par with the established methods.

A data porting tool for coupling models with different discretization needs

The presented work is part of a larger research program dealing with developing tools for coupling biogeochemical models in contaminated landscapes. The specific objective of this article is to provide researchers with a data porting tool to build hexagonal raster using information from a rectangular raster data (e.g. GIS format). This tool involves a computational algorithm and an open source software (written in C). The method of extending the reticulated functions defined on 2D networks is an essential key of this algorithm and can also be used for other purposes than data porting. The algorithm allows one to build the hexagonal raster with a cell size independent from the geometry of the rectangular raster. The extended function is a bi-cubic spline which can exactly reconstruct polynomials up to degree three in each variable. We validate the method by analyzing errors in some theoretical case studies followed by other studies with real terrain elevation data. We also introduce and briefly present an iterative water routing method and use it for validation on a case with concrete terrain data.

Positivity-preserving rational bi-cubic spline interpolation for 3D positive data

This paper deals with the shape preserving interpolation problem for visualization of 3D positive data. A required display of 3D data looks smooth and pleasant. A rational bi-cubic function involving six shape parameters is presented for this objective which is an extension of piecewise rational function in the form of cubic/quadratic involving three shape parameters. Simple data dependent constraints for shape parameters are derived to conserve the inherited shape feature (positivity) of 3D data. Remaining shape parameters are left free for designer to modify the shape of positive surface as per industrial needs. The interpolant is not only local, C 1 but also it is a computationally economical in comparison with existing schemes. Several numerical examples are supplied to support the worth of proposed interpolant.

Multisensor fusion of satellite images for urban information extraction using pseudo-Wigner distribution

A new algorithm has been proposed to fuse high-resolution panchromatic image with a low-resolution multispectral image of World View-2, based on pseudo-Wigner distribution (PWD). Spatial-frequency method, particularly PWD, provides pixel-wise analysis, shift invariant as well as characterization of local spectral properties of nonstationary image, which is indispensable for image fusion. The input images are re-sampled using nearest neighbor (NN), bicubic spline (BCS), and cubic convolution (CC). The comparison of performance of the proposed method and the discrete wavelet transform (DWT) has been evaluated using root mean square error, peak signal-to-noise ratio, correlation coefficient index. It has been found that PWD fusion technique outperforms DWT fusion technique using average as well as maximum selection rule, both quantitatively and qualitatively. All three resampling methods—the NN, the BCS, and the CC—do not have any significant effect on the final visual appearance of the fused images.

The Splicing and Recovery of Shredded Paper

This paper provides an automatic method of splicing shredded printed paper which was shredded by shredder. A splicing degree function was defined by using bicubic spline technology. Based on the splicing degree function, a 0-1 mixed integer programming model was used to solve the shredded paper splicing problem. The whole solving process was automatic, and the result was perfect.

Monotonicity-preserving rational bi-cubic spline surface interpolation

We discuss the problem of monotonicity preservation of surfaces through 3D monotone data. This can be done using a rational bi-cubic blended function that is an extension of a rational cubic function in the form of a cubic numerator and quadratic denominator. The function involves twelve shape parameters in each rectangular patch. Data- dependent constraints are derived on four of these shape parameters to conserve the shape of the data while the other eight are left free to modify the monotone surface as desired. Several numerical examples are presented to show the effectiveness and capability of the scheme. The present scheme isC 1 , flexible, simple, local, and economical.

Effect of interpolation on parameters extracted from seating interface pressure arrays.

Interpolation is a common data processing step in the study of interface pressure data collected at the wheelchair seating interface. However, there has been no focused study on the effect of interpolation on features extracted from these pressure maps, nor on whether these parameters are sensitive to the manner in which the interpolation is implemented. Here, two different interpolation paradigms, bilinear versus bicubic spline, are tested for their influence on parameters extracted from pressure array data and compared against a conventional low-pass filtering operation. Additionally, analysis of the effect of tandem filtering and interpolation, as well as the interpolation degree (interpolating to 2, 4, and 8 times sampling density), was undertaken. The following recommendations are made regarding approaches that minimized distortion of features extracted from the pressure maps: (1) filter prior to interpolate (strong effect); (2) use of cubic interpolation versus linear (slight effect); and (3) nominal difference between interpolation orders of 2, 4, and 8 times (negligible effect). We invite other investigators to perform similar benchmark analyses on their own data in the interest of establishing a community consensus of best practices in pressure array data processing.

New bi-cubic spline wavelet based thin plate finite elements

New bi-cubic spline wavelet based thin plate finite elements

ГЕоінформаційнЕ забезпеченнЯ якісноЇ оцінкИ грунтіВ

The structure of geospatial database (GDB) in the form of a set of models (conceptual and logical) was developed in the study. The conceptual model of GDB is constructed on the basis of the unified modeling language UML (Unified Modeling Language). Physical implementation of the model for ensuring the qualitative evaluation of soil is shown. The paper gives the analysis of the use of various interpolators (distance weighting, bicubic spline, universal kriging) with the purpose to obtain digital continuous surfaces on the example of the humus content in soils. Significant difference in the application of various interpolators was not found. The GRID-model, obtained on the basis of the data of agrochemical inspection of soils was used as the digital image of the link. Interpolation as a component of geoinformation mapping can be effectively used for the reduction of the cost of agrochemical inspections due to the optimization of the number of soil samples

Optimization of the parameters of surfaces by interpolating variational bicubic splines

In this paper we present an interpolation method from a surface or a data set by the optimization of a quadratic functional in a bicubic splines functional space. The existence and the uniqueness of the solution of this problem are shown and as well a convergence result of the method is established. The mentioned functional involves some real non negative parameters; the optimal surface is obtained by a suitable optimization of these parameters. Finally, we analyze some numerical and graphic examples in order to prove the efficiency of the presented method.

The impact of raindrop drift in a three-dimensional wind field on a radar–gauge rainfall comparison

There are many causes for the discrepancies between weather radar and rain gauges, and among these, displacement of raindrops due to wind drift – which is especially a problem with high-spatial resolution weather radar – is largely ignored in the published literature. This is mainly due to the lack of high-resolution three-dimensional wind fields and feasible treatment of the raindrop size distribution (DSD). In this study, a new systematic approach is proposed to explore the radar–gauge relationship under the wind influence. The mass-weighted mean diameter of raindrops is derived for each radar grid from the DSD data. The reanalysis project ERA-40 data of the European Centre for Medium-range Weather Forecasts (ECMWF) are used to drive the numerical weather research and forecasting (WRF) model to generate high-resolution hourly three-dimensional wind fields. Trajectories and displacements of raindrops are then computed using a three-dimensional motion equation from the given radar beam height to the ground surface. Based on the radar rainfall surface interpolated by the bicubic spline method, the correlation of the radar–gauge pairs is used to validate the results. A case study with 20 storm events in the Brue catchment in South West England is chosen to evaluate the proposed scheme. It has been found that when wind drift is taken into account, the correlation coefficient in hourly gauge–radar comparisons can be enhanced by up to 30% and the average correlation coefficient for an event can be improved by 10%. However, there are still some situations in which the scheme fails to work, indicating the complexity and uncertainties in tackling this challenging problem. Further studies are needed to explore why those cases cause problems to the scheme and how it could be improved to cope with them.

The AutoQual ultrasound elastography method for quantitative assessment of lateral strain in post-rupture Achilles tendons

This paper presents the AutoQual elastography method: a novel algorithm that improves the quality of 2D displacement field calculation from ultrasound radio frequency (RF) sequences of acutely ruptured Achilles tendons to determine image-lateral strain fields and has potential use for ligaments and muscles. This method uses 2D bicubic spline interpolation of the RF signal, Quality Determined Search, Automatic Search Range and Adaptive Block Size components as a novel combination that is designed to improve continuity and decrease displacement field noise, especially in areas of low signal strength. We present a simple experiment for quantitatively comparing the AutoQual method to a multiscale (MS) elastography method from ultrasound RF sequences of a 5% agar phantom for rigid body motion and known lateral strain loads with speeds up to 5mm/s. We finally present examples of four in vivo Achilles tendons in various damage states and with manual or artificially controlled passive flexion of the foot. Results show that the AutoQual method offers a substantial improvement on the MS method, achieving similar performance for rigid body tracking at all speeds, a lower normalized square error at all strains induced and a more continuous strain field at higher compression rates. AutoQual also showed a greater average normalized cross correlation for image blocks in the area of interest, a lower standard deviation of the strain field and a visually more acceptable point tracking for in vivo examples. This work demonstrates lateral ultrasound elastography which is robust to the complex passive motion of the Achilles and to various imaging artifacts associated with imaging tendon rupture. This method potentially has a wide clinical application for assessing in vivo strains in and hence mechanical function of any near skin surface tissues that are longitudinally loaded.

Ronchigram simulations for free-form concave reflective surfaces

A new algorithm to simulate ronchigrams and bironchigrams for free-form reflective concave surfaces is reported. In order to know the surface saggita and its partial derivatives at any point within mirror domain a bicubic spline interpolation is done over a finite set of points where the free surface is defined. To simulate the ronchigram irradiances we used ruling with cosinusoidal transmittance instead of Ronchi ruling. Several examples of bironchigrams are shown for spherical surface affected by local and/or radial errors; and surfaces used in ophthalmic progressive lenses.

Spline-based and stress-monitored patterning of fabric structures

The shape of tensioned fabric structures is realised via a process of patterning, which involves ‘flattening’ of sections, or panels, of a 3D membrane, onto a plane. The patterning method proposed here uses bicubic spline descriptions of surfaces undergoing flattening, followed by elastic analysis of strains arising from this process and from membrane pre-stress. The results, which ensure stress compatibility along panel edges, produce a cutting pattern and stress variation within it when the panel is assembled into the 3D form. They confirm a general relationship between panel width and stress deviation from the intended pre-stress reported in [1].

Software measurement standards for areal surface texture parameters: part 1—algorithms

Reference software for the evaluation of a set of areal surface texture parameters is described, focusing on the definitions of the parameters and giving details of the numerical algorithms employed in the software to implement those definitions. The main consideration in the design and development of reference software is its numerical correctness, and the algorithms chosen and the implementations of those algorithms reflect this consideration. The surface for which parameters are to be evaluated is a bicubic spline interpolant to the available discretely-sampled data, and parameters are evaluated for that continuous surface either to a high numerical precision or to a numerical precision that is under the control of the user.

Free-form splines combining NURBS and basic shapes

We show how to automatically join, into one unified spline surface, C2 tensor-product bi-cubic NURBS and G2 bi-cubic rational splines. The G2 splines are capable of exactly representing basic shapes such as (pieces of) quadrics and surfaces of revolution, including tori and cyclides. The main challenge is to transition between the differing forms of continuity. We transform the G2 splines to splines that are C2 in homogeneous space. This yields Hermite data for a transitional strip of tensor-product splines of degree (6,5) that guarantees overall curvature continuity. We also explain the simpler G1 to C1 transition. Key to the constructions is the C2 parameterization of circles in homogeneous space.

Nonlinear Model Predictive Control of a Vapor Compression Cycle based on First Principle Models

AbstractVapor compression cycles are broadly used for air-conditioning, refrigeration and heat-pump applications and are responsible for a large part of primary energy consumptions. Increasing availability of electrical actuators leads to a high number of possible control inputs of vapor compression cycles. Due to their highly nonlinear behavior and strong cross coupling of inputs and outputs modeling and control of this systems is a nontrivial task. Classical control design methods based on linear plant models seem not to be able to achieve the best possible control in terms of energy efficiency and disturbances rejection. In this contribution we present a Nonlinear Model Predictive Control scheme for vapor compression cycles, which takes nonlinearities and cross coupling explicitly into account. First principle models are used to formulate a system of Ordinary Differential Equations (ODE) describing the dynamic behavior of the underlying process. Inside these models a new highly efficient method for refrigerant fluid properties computation based on bicubic spline interpolation is used. Based on the derived ODE an Optimal Control Problem is formulated and solved by a fast Direct Multiple Shooting algorithm. Finally a Nonlinear Model Predictive Control (NMPC) scheme with a sampling rate of only 0.5 s is derived. Simulation experiments show the ability of the method to immediately react to disturbances and at the same time keeping the system at the most energy efficient working point.

Bilinear and bicubic interpolation methods for division of focal plane polarimeters

This paper presents bilinear and bicubic interpolation methods tailored for the division of focal plane polarization imaging sensor. The interpolation methods are targeted for a 1-Mega pixel polarization imaging sensor operating in the visible spectrum. The five interpolation methods considered in this paper are: bilinear, weighted bilinear, bicubic spline, an approximated bicubic spline and a bicubic interpolation method. The modulation transfer function analysis is applied to the different interpolation methods, and test images as well as numerical error analyses are also presented. Based on the comparison results, the full frame bicubic spline interpolation achieves the best performance for polarization images.

An approximation problem of noisy data by cubic and bicubic splines

Smoothing of noisy data has always been a topic of interest in many areas where computer simulations have been performed, including natural sciences as well as economics and social sciences. In this paper we present an approximation method of explicit curves or surfaces from exact and noisy data by fairness cubic or bicubic splines. A variational problem of explicit curves or surfaces is obtained by minimizing a quadratic functional in a space of cubic or bicubic splines from noisy data. We show the existence and uniqueness of this problem as long as a convergence result especially for noisy data is carefully established. We analyze some numerical and graphical examples using fictional noisy data in order to prove the validity of our method.