Spline Surfaces Research Articles

Bicubic splines and biquartic polynomials

Abstract The paper proposes a new efficient approach to computation of interpolating spline surfaces. The generalization of an unexpected property, noticed while approximating polynomials of degree four by cubic ones, confirmed that a similar interrelation property exists between biquartic and bicubic polynomial surfaces as well. We prove that a 2×2-component C1 -class bicubic Hermite spline will be of class C2 if an equispaced grid is used and the coefficients of the spline components are computed from a corresponding biquartic polynomial. It implies that a 2×2 uniform clamped spline surface can be constructed without solving any equation. The applicability of this biquartic polynomials based approach to reducing dimensionalitywhile computing spline surfaces is demonstrated on an example.

Differentiable Piecewise-Bézier Surfaces on Riemannian Manifolds

We generalize the notion of Bézier surfaces and surface splines to Riemannian manifolds. To this end we put forward and compare three possible alternative definitions of Bézier surfaces. We furthermore investigate how to achieve ${\mathcal{C}}^0$- and ${\mathcal{C}}^1$-continuity of Bézier surface splines. Unlike in Euclidean space and for one-dimensional Bézier splines on manifolds, ${\mathcal{C}}^1$-continuity cannot be ensured by simple conditions on the Bézier control points: it requires an adaptation of the Bézier spline evaluation scheme. Finally, we propose an algorithm to optimize the Bézier control points given a set of points to be interpolated by a Bézier surface spline. We show computational examples on the sphere, the special orthogonal group, and two Riemannian shape spaces.

Design of optical elements with TIR freeform surface

A novel method for the design of total internal reflection based (TIR) optical elements generating complex two-dimensional narrow-angle light distributions is presented. The method consists of two parts: computation of a piecewise smooth solution and its subsequent approximation by a spline surface. Examples of TIR optical elements designed using the proposed approach are discussed. The simulation results demonstrate high performance of the proposed method: the luminous efficacy of the optical elements is 91.8% and the relative root-mean-square error is less than 8.6%.

Cubic Trigonometric Nonuniform Spline Curves and Surfaces

A class of cubic trigonometric nonuniform spline basis functions with a local shape parameter is constructed. Their totally positive property is proved. The associated spline curves inherit most properties of usual polynomialB-spline curves and enjoy some other advantageous properties for engineering design. They haveC2continuity at single knots. For equidistant knots, they haveC3continuity andC5continuity for particular choice of shape parameter. They can express freeform curves as well as ellipses. The associated spline surfaces can exactly represent the surfaces of revolution. Thus the curve and surface representation scheme unifies the representation of freeform shape and some analytical shapes, which is popular in engineering.

Topographic relationships for design rainfalls over Australia

Summary Design rainfall statistics are the primary inputs used to assess flood risk across river catchments. These statistics normally take the form of Intensity–Duration–Frequency (IDF) curves that are derived from extreme value probability distributions fitted to observed daily, and sub-daily, rainfall data. The design rainfall relationships are often required for catchments where there are limited rainfall records, particularly catchments in remote areas with high topographic relief and hence some form of interpolation is required to provide estimates in these areas. This paper assesses the topographic dependence of rainfall extremes by using elevation-dependent thin plate smoothing splines to interpolate the mean annual maximum rainfall, for periods from one to seven days, across Australia. The analyses confirm the important impact of topography in explaining the spatial patterns of these extreme rainfall statistics. Continent-wide residual and cross validation statistics are used to demonstrate the 100-fold impact of elevation in relation to horizontal coordinates in explaining the spatial patterns, consistent with previous rainfall scaling studies and observational evidence. The impact of the complexity of the fitted spline surfaces, as defined by the number of knots, and the impact of applying variance stabilising transformations to the data, were also assessed. It was found that a relatively large number of 3570 knots, suitably chosen from 8619 gauge locations, was required to minimise the summary error statistics. Square root and log data transformations were found to deliver marginally superior continent-wide cross validation statistics, in comparison to applying no data transformation, but detailed assessments of residuals in complex high rainfall regions with high topographic relief showed that no data transformation gave superior performance in these regions. These results are consistent with the understanding that in areas with modest topographic relief, as for most of the Australian continent, extreme rainfall is closely aligned with elevation, but in areas with high topographic relief the impacts of topography on rainfall extremes are more complex. The interpolated extreme rainfall statistics, using no data transformation, have been used by the Australian Bureau of Meteorology to produce new IDF data for the Australian continent. The comprehensive methods presented for the evaluation of gridded design rainfall statistics will be useful for similar studies, in particular the importance of balancing the need for a continentally-optimum solution that maintains sufficient definition at the local scale.

Minimal bi-6 G2 completion of bicubic spline surfaces

This paper addresses a gap in the arsenal of curvature continuous tensor-product spline constructions: an algorithm is provided to fill n-sided holes in C2 bi-3 spline surfaces using n patches of degree bi-6. Numerous experiments illustrate good highlight line and curvature distribution on the resulting surfaces.

Usage of multivariate geostatistics in interpolation processes for meteorological precipitation maps

Long-term meteorological data are very important both for the evaluation of meteorological events and for the analysis of their effects on the environment. Prediction maps which are constructed by different interpolation techniques often provide explanatory information. Conventional techniques, such as surface spline fitting, global and local polynomial models, and inverse distance weighting may not be adequate. Multivariate geostatistical methods can be more significant, especially when studying secondary variables, because secondary variables might directly affect the precision of prediction. In this study, the mean annual and mean monthly precipitations from 1984 to 2014 for 268 meteorological stations in Turkey have been used to construct country-wide maps. Besides linear regression, the inverse square distance and ordinary co-Kriging (OCK) have been used and compared to each other. Also elevation, slope, and aspect data for each station have been taken into account as secondary variables, whose use has reduced errors by up to a factor of three. OCK gave the smallest errors (1.002 cm) when aspect was included.

Can bi‐cubic surfaces be class A?

Abstract'Class A surface’ is a term in the automotive design industry, describing spline surfaces with aesthetic, non‐oscillating highlight lines. Tensor‐product B‐splines of degree bi‐3 (bicubic) are routinely used to generate smooth design surfaces and are often the de facto standard for downstream processing. To bridge the gap, this paper explores and gives a concrete suggestion, how to achieve good highlight line distributions for irregular bi‐3 tensor‐product patch layout by allowing, along some seams, a slight mismatch of normals below the industry‐accepted tolerance of one tenth of a degree. Near the irregularities, the solution can be viewed as transforming a higher‐degree, high‐quality formally smooth surface into a bi‐3 spline surface with few pieces, sacrificing formal smoothness but qualitatively retaining the shape.

Isogeometric segmentation: Construction of auxiliary curves

In the context of segmenting a boundary represented solid into topological hexahedra suitable for isogeometric analysis, it is often necessary to split an existing face by constructing auxiliary curves. We consider solids represented as a collection of trimmed spline surfaces, and design a curve which can split the domain of a trimmed surface into two pieces satisfying the following criteria: the curve must not intersect the boundary of the original domain, it must not intersect itself, the two resulting pieces should have good shape, and the endpoints and the tangents of the curve at the endpoints must be equal to specified values.

Research of correcting distorted X-ray images based on least squares and biharmonic spline surface

Owing to the geometric distortions of the X-ray images, C-arm CT (computed tomography) imaging system usually suffers from adverse effects. To overcome the difficulties, an integrated approach has been put forward to correct distorted X-ray images in the paper. The approach firstly uses OTSU algorithm and morphological open operation to extract the control points coordinates in the distorted X-ray images. To obtain more coordinates and improve the correction precision, least squares is used to fit the polynomial passing by the extracted points. More coordinates are calculated by using the fitted polynomial. Then, a biharmonic spline surface is interpolated by the extracted and calculated coordinates. Finally, the distorted X-ray images are corrected by using the interpolated biharmonic spline surface. Experiment on the acquired distorted X-ray images demonstrates that the proposed approach can effectively correct the distorted X-ray images. A comparison with classical approaches further shows that the approach proposed in the paper performs better in visual effects and correction accuracy.

Efficient construction of multi-block volumetric spline parameterization by discrete mask method

Isogeometric analysis attempts to unify the mathematical languages in design and analysis to realize the seamless integration of CAD and CAE. In three-dimensional isogeometric analysis, parametric volume is employed as the computational domain for a given set of boundary information. In this paper, we propose a discrete mask method for the efficient construction of multi-block volumetric parameterization based on a set of given boundary spline surfaces. Given the block-partition information of a model, the interior control points can be obtained efficiently by solving a sparse linear system. The existence and uniqueness of the solution for the linear system are also proved. After performing a pre-process on the non-compatible boundary surfaces, the discrete Coons mask is generalized to a unified form, which provides more choices for the construction of inner control points. The proposed method is not only suitable for the multi-block case with C1 continuity, but can also be used for the case in which the boundary surfaces are of different degrees, and with different number of control points and knot vectors. Several examples are presented to illustrate the effectiveness of the proposed method.

An improved source-subtracted and destriped 408-MHz all-sky map

The all-sky 408 MHz map of Haslam et al. is one the most important total-power radio surveys. It has been widely used to study diffuse synchrotron radiation from our Galaxy and as a template to remove foregrounds in cosmic microwave background data. However, there are a number of issues associated with it that must be dealt with, including large-scale striations and contamination from extragalactic radio sources. We have re-evaluated and reprocessed the rawest data available to produce a new and improved 408-MHz all-sky map. We first quantify the positional accuracy (≈7 arcmin) and effective beam (56.0 ± 1.0 arcmin) of the four individual surveys from which it was assembled. Large-scale striations associated with 1/f noise in the scan direction are reduced to a level ≪1 K using a Fourier-based filtering technique. The most important improvement results from the removal of extragalactic sources. We have used an iterative combination of two techniques – two-dimensional Gaussian fitting and minimum curvature spline surface inpainting – to remove the brightest sources (≳2 Jy), which provides a significant improvement over previous versions of the map. We quantify the impact with power spectra and a template fitting analysis of foregrounds to the WMAP data. The new map is publicly available and is recommended as the template of choice for large-scale diffuse Galactic synchrotron emission. We also provide a higher resolution map with small-scale fluctuations added, assuming a power-law angular power spectrum down to the pixel scale (1.7 arcmin). This should prove useful in simulations used for studying the feasibility of detecting H i fluctuations from the Epoch of Reionization.

Radially symmetric thin plate splines interpolating a circular contour map

Profiles of radially symmetric thin plate spline surfaces minimizing the Beppo Levi energy over a compact annulus R1≤r≤R2 have been studied by Rabut via reproducing kernel methods. Motivated by our recent construction of Beppo Levi polyspline surfaces, we focus here on minimizing the radial energy over the full semi-axis 0<r<∞. Using a L-spline approach, we find two types of minimizing profiles: one is the limit of Rabut’s solution as R1→0 and R2→∞ (identified as a ‘non-singular’ L-spline), the other has a second-derivative singularity and matches an extra data value at 0. For both profiles and p∈[2,∞], we establish the Lp-approximation order 3/2+1/p in the radial energy space. We also include numerical examples and obtain a novel representation of the minimizers in terms of dilates of a basis function.

Improved shape for multi-surface blends

For many design applications, where multiple primary surface pieces meet, the distribution of curvature is more important than formally achieving exact curvature continuity. For parametric spline surfaces, when constructing a multi-sided surface cap, we demonstrate a strong link between the uniform variation of the re-parameterization between (boundary) data of the joining pieces and a desirable distribution of curvature. We illustrate this interdependence between parameterization quality and surface quality by developing a G1 bi-quintic surface cap consisting of n pieces that smoothly fills holes in a piecewise bi-cubic tensor-product spline complex. These bi-5 surface caps have arguably better shape than higher-degree, formally curvature continuous alternatives.

Adapting forest management to climate change using bioclimate models with topographic drivers

Bioclimate models incorporating topographic predictors as surrogates for microclimate effects are developed for Populus tremuloides and Picea engelmannii to provide the fine-grained specificity to local terrain required for adapting management of three Colorado (USA) national forests (1.28 million ha) and their periphery to climate change. Models were built with the Random Forests classification tree using presence–absence observations obtained by overlaying species distribution maps on data points gridded at ∼225 m within the forests and from ground plot observations from adjacent areas. Topographic effects derived from 90-m elevation grids were expressed by weighting aspect by slope angle. Climate estimates were obtained from spline surfaces. Out-of-bag errors were ∼17 per cent, and classification errors for an independent sample from within the forest were ∼13 per cent. Topographic variables were second in importance to climate variables for predicting species distributions; their inclusion captured well-known topographic effects on vegetation in mountainous terrain. Predictions made for future climates described by three General Circulation Models and three emissions scenarios were used to map on 90-m grids the habitat expected to be lost, threatened, persistent or emergent. The habitat categories are used to identify those areas where treatments should have highest likelihood of success.

Research on XRII Image Distortion Correction Based on Biharmonic Spline Surface Interpolation

Research on XRII Image Distortion Correction Based on Biharmonic Spline Surface Interpolation

An assessment of three satellite-based precipitation data sets as applied to the Thailand region

The study examines three satellite-based data sets to estimate long-term precipitation for the Thailand region: the Tropical Rainfall Mapping Mission (TRMM) 3B43, the Climate Prediction Centre morphing technique (CMORPH), and a locally developed regression model using the geostationary meteorological satellite (GMS) covering the Thailand region. Data for the first two sets were available at a spatial resolution of 0.25° × 0.25°, while the local regression model used data from the GMS at a resolution of 5 km × 5 km. The statistical regression model was developed by relating long-term monthly average precipitation from 27 rain gauge stations with concurrent satellite data in the visible and thermal infrared bands. The model was then tested against independent data from 27 rain gauge stations. Satellite/rain gauge ratios were estimated, and a smooth spline surface was used to correct the error from the model. Data from the three approaches were compared with a rain gauge network. The TRMM relation performed better than CMORPH, and the performance for GMS was comparable to TRMM with root mean square different and mean bias difference of 33.6 and 4.2%, respectively. The locally developed regression model was used to produce monthly and yearly total rainfall maps from the GMS data for the entire country.

Polynomial spline surfaces with rational linear transitions

We explore a class of polynomial tensor-product spline surfaces on 3-6 polyhedra, whose vertices have valence n=3 or n=6. This restriction makes it possible to exclusively use rational linear transition maps between the pieces: transitions between the bi-cubic tensor-product spline pieces are either C1 or they are G1 (tangent continuous) based on one single rational linear reparameterization. The simplicity of the transition functions yields simple formulas for a hierarchy of splines on subdivided 3-6 polyhedra.

Computer Simulations Using Pattern Specific Loss Coefficients for Cross Junctions

AbstractDetailed K-factors measured for a symmetrical nonreducing steel cross junction were published in this journal in 2010. This article describes a variable K-factor model for cross junctions that can take advantage of the published data and be incorporated into most existing hydraulic simulation software. The raw data are transformed and represented with bicubic spline surfaces. Newton’s method is used to solve the system of equations resulting from linearization of the equations. Despite the variable junction K-factors, solutions converge quadratically or nearly so. For each iteration step, flow patterns must be detected, cross legs identified, and K-factors updated. Inherent complexity of flow through cross junctions creates challenges: K-factor surfaces have steep gradients and the values are often negative, branches with laminar flow occur alongside branches with turbulent flow, and flow stoppage is recurrent. In addition to comparing solutions based on constant and variable K-factors, intriguing...

A sexually transmitted infection screening algorithm based on semiparametric regression models.

Sexually transmitted infections (STIs) with Chlamydia trachomatis, Neisseria gonorrhoeae, and Trichomonas vaginalis are among the most common infectious diseases in the United States, disproportionately affecting young women. Because a significant portion of the infections present no symptoms, infection control relies primarily on disease screening. However, universal STI screening in a large population can be expensive. In this paper, we propose a semiparametric model-based screening algorithm. The model quantifies organism-specific infection risks in individual subjects and accounts for the within-subject interdependence of the infection outcomes of different organisms and the serial correlations among the repeated assessments of the same organism. Bivariate thin-plate regression spline surfaces are incorporated to depict the concurrent influences of age and sexual partners on infection acquisition. Model parameters are estimated by using a penalized likelihood method. For inference, we develop a likelihood-based resampling procedure to compare the bivariate effect surfaces across outcomes. Simulation studies are conducted to evaluate the model fitting performance. A screening algorithm is developed using data collected from an epidemiological study of young women at increased risk of STIs. We present evidence that the three organisms have distinct age and partner effect patterns; for C. trachomatis, the partner effect is more pronounced in younger adolescents. Predictive performance of the proposed screening algorithm is assessed through a receiver operating characteristic analysis. We show that the model-based screening algorithm has excellent accuracy in identifying individuals at increased risk, and thus can be used to assist STI screening in clinical practice.