Smoothing Spline Algorithm Research Articles

A smoothing spline algorithm to interpolate and predict the eigenvalues of matrices extracted from the sequence of preconditioned banded symmetric Toeplitz matrices

<abstract><p>Understanding the eigenvalue distribution of sequence Toeplitz matrices has advanced significantly in recent years. Notable contributors include Bogoya, Grudsky, Böttcher, and Maximenko, who have derived precise asymptotic expansions for these eigenvalues under certain conditions related to the generating function as the matrix size increases. Building on this foundation, the Stefano Serra-Capizzano conjectured that, under certain assumptions about $ \Omega $ and $ \Phi $, a similar expansion may hold for the eigenvalues of a sequence of preconditioned Toeplitz matrices $ T_{n}^{-1}(\Phi) T_n(\Omega) $, given a monotonic ratio $ r = \Omega/\Phi $. In contrast to current eigenvalue solvers, this work presents a novel method for efficiently calculating the eigenvalues of a sequence of large preconditioned banded symmetric Toeplitz matrices (PBST). Our algorithm uses a higher-order spline fitting extrapolation technique to gather spectral data from a smaller sequence of PBST matrices in order to forecast the spectrum of bigger matrices.</p></abstract>

Seasonal Changes in Climate Variables in Rainfed Crop Areas in the Lerma-Chapala-Santiago Basin, Mexico

This paper shows the effects of changes in the spatial-temporal behavior and phase shift of climate variables on rainfed agriculture in the Lerma-Chapala-Santiago Basin in central Mexico. Specifically, changes in rainfall (R), maximum temperature (Tmax), and minimum temperature (Tmin) were analyzed over two 25-year periods (1960 to 1985 and 1986 to 2010). Climate surfaces were generated by interpolation using the thin-plate smoothing spline algorithm in the software ANUSPLIN. Climate data were Fourier-transformed and fitted to a sinusoidal curve model, and changes in amplitude (increase) and phase were analyzed. The temporal behavior (1960–2010) indicated that rainfall was the most stable variable at the monthly level and presented no significant changes. However, Tmax increased by 2°C in the final period, and Tmin increased by 0.7°C at the end of the final period. The basin was discretized into ten rainfed crop areas (RCAs) according to the extent of changes in the amplitude and phase of the climate variables. The central and southern portions (55% of the area) presented more significant changes in amplitude, mainly in Tmin and Tmax. The remaining RCAs were smaller (14.6%) but presented greater variation: the amplitude of the Tmin decreased in addition to showing a phase shift, whereas Tmax increased in addition to showing a phase shift. These results translate into a delay in the characteristic temperatures of the spring and summer seasons, which can impact the rainfed crop cycle. Additionally, rainfall showed an annual decrease of approximately 50 mm in all RCAs, which can affect the phenological development of crops during critical stages (emergence through flowering). These changes represent a significant threat to the regional economy and food security of Mexico.

A smooth tool path planning method on NURBS surface based on the shortest boundary geodesic map

Abstract To fill the development gap between computer-aided design (CAD) and computer-aided manufacturing (CAM) on non-uniform rational b-spline (NURBS), a general tool path planning method on NURBS surface is proposed in this study. The generated tool path is composed of a series of closed paths which are derived by mapping the contours of a shortest boundary geodesic map (SBGM) from parameter space to 3D Euclidean space. SBGM that is created in parameter space depicts the shortest boundary geodesic distance (SBG) from internal points on NURBS surface to surface boundaries. It is calculated in discrete form firstly and then reconstructed by using a bicubic smoothing spline algorithm to improve the smoothness of the generated tool path. The tool path interval is deduced according to the requirement of residual scallop height constraint and determined conveniently based on the SBG difference of the two adjacent paths. It is confirmed that the proposed tool path planning method can be applied on all kinds of NURBS surface without the requirements of any other post-processing procedures. Typical case studies and experiments are presented to verify the effectiveness of the proposed method. Comparing with other traditional approaches, the proposed tool path planning method has excellent performance in improving the smoothness and reducing the total length of tool path.

Soft computing models for predicting blast-induced air over-pressure: A novel artificial intelligence approach

Applying soft computing models for solving real-life problems has yielded many significant benefits, especially in the mining industry. This study proposed a novel soft computing model for estimating blast-induced air over-pressure (AOp) with high accuracy. Accordingly, the boosted smoothing spline (BSTSM) and genetic algorithm (GA) were considered and combined, namely GA-BSTSM model. One hundred twenty-one blasts were collected at the Coc Sau open-pit coal mine (Vietnam) for this aim. The explosive capacity used (W) and distance (R) were considered as the primary input variables for the aiming of AOp prediction. Also, the meteorological conditions such as temperature (T), relative humidity (RH), atmospheric pressure (AP), wind speed (WS), and wind direction (WD) were taken into account to predict AOp in this study. To confirm the performance of the proposed GA-BSTSM model, an empirical model and six other artificial intelligence models were also developed to predict AOp based on the same dataset, including CART (classification and regression tree), KNN (k-nearest neighbors), ANN (artificial neural network), BRR (Bayesian ridge regression), SVR (support vector regression), and Gaussian process (GP). The developed models were then evaluated through three performance indices (i.e., RMSE, R2, and VAF) using the testing dataset. In addition, a Taylor diagram was also developed to evaluate the quality of the models. The evaluation results showed that the proposed GA-BSTSM model yielded a robust performance with high accuracy for AOp prediction herein. The findings also disclosed that meteorological factors have a strong influence on the accuracy of AOp predictive models, especially RH and WS.

Autonomous programming and adaptive filling of lap joint based on three-dimensional welding-seam model by laser scanning

Traditional teaching programing for the welding of large-scale complex structures is extremely difficult, with low efficiency and poor consistency, and it is very difficult to obtain the random change of joint geometry and achieve excellent welding quality along the entire welding path. In this paper, an integrated method of automatic generation of welding paths and adaptive filling of lap welding based on laser-structured light scanning is proposed. First, the point-cloud data of the welding joint on the entire welding path were acquired by laser scanning a workpiece, and the welding-seam model of the lap joint was accurately reconstructed using a cubic smoothing spline algorithm. The key feature points of the welding path were then extracted by Douglas-Peucker algorithm and the corresponding postures calculated and optimized. Furthermore, a neural network model that correlated the optimized process parameters with the gaps in the lap joints was established. During the welding process, the welding parameters were adaptively adjusted according to the variation of the obtained joint geometry based on the model. The results of the experiments showed that the proposed method has excellent adaptability to complex-curve welding seams with gaps randomly distributed between 0 and 2 mm, achieving satisfactory results in both path accuracy and weld appearance.

Simultaneous Estimation of Corneal Topography, Pachymetry, and Curvature.

Identification of objective criteria to correctly diagnose ectatic diseases of the cornea or to detect early stages of corneal ectasia is of great interest in ophthalmology and optometry. Metrics for diagnosis typically employed are curvature maps (axial/sagittal, tangential); elevation map of the anterior surface of the cornea with respect to a reference sphere; and pachymetry (thickness) map of the cornea. We present evidence that currently used curvature maps do not represent the actual curvatures (principal or mean) in a human cornea. A novel contribution of this paper is the computation of the true mean curvature over every point of a central region of the cornea. We show that the true mean curvature can accurately identify the location of the ectasia. We present a quartic smoothing spline algorithm for the simultaneous computation of elevation maps for anterior and posterior corneal surfaces, pachymetry, and true mean curvature. The input to the algorithm is data from a single measurement from imaging devices such as an anterior segment optical coherence tomographer or a Scheimpflug imager. We show that a different combination of metrics is useful for the diagnosis of existing ectasia (true mean curvature and anterior elevation map) as opposed to subclinical ectasia (pachymetry and posterior elevation map). We compare our results with existing algorithms, and present applications to a normal cornea, a forme fruste keratoconic cornea, and an advanced keratoconic cornea.

Isotopic identification using Pulse Shape Analysis of current signals from silicon detectors: Recent results from the FAZIA collaboration

The FAZIA apparatus exploits Pulse Shape Analysis (PSA) to identify nuclear fragments stopped in the first layer of a Silicon-Silicon-CsI(Tl) detector telescope. In this work, for the first time, we show that the isotopes of fragments having atomic number as high as Z∼20 can be identified. Such a remarkable result has been obtained thanks to a careful construction of the Si detectors and to the use of low noise and high performance digitizing electronics. Moreover, optimized PSA algorithms are needed. This work deals with the choice of the best algorithm for PSA of current signals. A smoothing spline algorithm is demonstrated to give optimal results without requiring too much computational resources.

Systematic Evaluation of Amide Proton Chemical Exchange Saturation Transfer at 3 T: Effects of Protein Concentration, pH, and Acquisition Parameters.

The goal of this work was to systematically evaluate the reproducibility of amide proton transfer chemical exchange saturation transfer (APT-CEST) at 3 T and its signal dependence on pH, protein concentration, and acquisition parameters. An in vitro system based on bovine serum albumin (BSA) was used, and its limitations were tested by comparing it to in vivo measurements. The contribution of small endogenous metabolites on the APT-CEST signal at 3 T was also investigated. In addition, the reliability of different z-spectrum interpolations as well as the use of only a few frequency offset data points instead of a whole z-spectrum were tested. We created both a BSA phantom at different concentrations and pH values and a metabolite phantom with different small molecules. Chemical exchange saturation transfer data were acquired using a 2-dimensional fast spoiled gradient-echo sequence with pulsed CEST preparation at different saturation durations and power levels. Healthy volunteer measurements were taken for comparison. Z-spectra were interpolated using a 24th-order polynomial (Poly), an eighth-order Fourier series (Fourier), and a smoothing Spline (sSpline) algorithm. To evaluate reduced data sets, only 6 to 14 frequency offsets of the z-spectrum were used and interpolated via a cubic Spline. Region of interest (ROI) evaluations were used to investigate the reproducibility of amide magnetization transfer ratio asymmetry [MTRasym(3.5 ppm)] and to analyze the MTRasym and z-spectra. Interscan standard deviations of MTRasym(3.5 ppm) were always below 0.3%. MTRasym(3.5 ppm) increased when the BSA concentrations increased and decreased when the pH increased. The amine MTRasym signal of small molecules was very small compared with BSA and was only detectable using short saturation times and higher power levels. The MTRasym(3.5 ppm) between BSA concentration steps and between nearly all pH steps was significantly different for all 3 fitting methods. The Fourier and sSpline methods showed no statistically significant differences; however, the results for the Poly method were significantly higher at some concentrations and pH values. Using only few frequency offsets resulted in less significant differences compared with fitting the complete z-spectrum. In general, MTRasym(3.5 ppm) of gray matter, white matter, and ventricle ROIs from volunteer scans increased with an increase in saturation power and partially decreased with an increase in saturation duration. Intra-ROI covariances of MTRasym(3.5 ppm) revealed the highest variations for Poly, whereas using reduced spectral data resulted in an increased signal variation. Amide proton transfer-CEST imaging is a highly reproducible method in which absolute signal differences of approximately 0.5% are detectable in principle. For in vivo applications, Fourier or sSpline interpolations of z-spectra are preferable. Using reduced data sets delivers similar results but with increased variation and therefore decreased (pH/concentration) differentiation capability. Differentiation capability increases with increases in the saturation duration and power level. The results from the in vitro BSA system cannot be directly transferred to the in vivo situation due to different chemical environments resulting in, for example, higher asymmetric macromolecular cMT effects in vivo. Amine signals from small molecules are unlikely to contribute to APT-CEST at 3 T (except for creatine); however, signals can be enhanced by using short saturation times and higher power levels.

An extraction method of laser stripe centre based on Legendre moment

Stable and efficient extracting the centre of laser stripe in image is a challenging computer vision problem. Existing methods are mainly based on the Gaussian and parabola distribution of laser stripe. However, the new beam-shaping technology is developed to make the light intensity of laser stripes with a uniformly distribution instead of Gaussian distribution. This paper proposes a fast and robust algorithm for laser stripe centre extraction based on Legendre moment theory. The proposed method can be summarized as follows: first, an ideal one-dimensional light intensity model of laser stripe is given based on the uniformly distribution; second, a closed-form solution for extracting the centre from every transversal section of laser stripe is derived based on conservation laws of Legendre moment; finally, a smoothing spline algorithm is used to eliminate noise as well as maintain the original features of the centre points. The experimental results show that the method has good robustness and accuracy.

Lens distortion correction for improving measurement accuracy of digital image correlation

Digital image correlation is widely applied in the measurement of displacement and strain fields through detecting the laser or sprayed speckle on the specimen surface. Generally, the existing digital image correlation methods often met difficulties of low detection accuracy under the influence of lens distortion. To tackle such difficulties, the piecewise spline function is used to describe the distortion, and a continuous smooth distortion model can be obtained after eliminating the noises by means of the mean value filtering and smoothing spline algorithm. Since the proposed method corrects lens distortion on the pixel plane, it will provide opportunity to obtain unbiased displacement in digital image correlation. Experiment results verify that the proposed method eliminates the effect of lens distortion from displacements measurement by means of two-dimensional digital image correlation.

Evaluation of the impact of extended aging duration on visco-elastic properties of asphalt binders

There are variations in the haulage distance between an asphalt mixing plant and the road construction site (field) and this may cause variations in the levels of aging of the asphalt mixture produced. This study evaluates the effects of extended aging durations during mixture transport from the plant to field. The rheological properties of different binders were characterized from the complex modulus, phase angle and Superpave™ rutting parameter. In addition, binder rheological properties changes were calculated using an Arrhenius activation energy equation. A prolonged aging characterization method (PACM) and its gradient (▿PACM) were also proposed to characterize the binder rheology subjected to different aging conditions. The test results showed that the trends in aging differ and depend on binder type, test temperature and aging conditions. The smoothing spline algorithm can be a suitable parameter that can predict the effects of aging on ▿PACM trend.

A robust method to extract a laser stripe centre based on grey level moment

Abstract Extraction of a laser stripe centre from an image is one of the most important measurement procedures when using structured light techniques. However, due to the influence of noise, extracting a light stripe centre accurately and stably remains a challenging problem. In this paper, a robust laser stripe centre extraction method is proposed. The method involves two steps: First, a closed-form solution for extracting the centre from every transversal section of a laser stripe is derived based on conservation laws of grey level moment. Second, a smoothing spline algorithm is used to determine a continuous curve of centre points. This algorithm can eliminate noise and maintain the original features of the discrete points. In the experiment, the proposed method is compared with recently reported techniques by detecting the same image. The results demonstrate that the method is more accurate and robust in the presence of noise.

A robust edge detection method with sub-pixel accuracy

In this paper, a new edge detection approach combining gray-moments operator with smoothing spline algorithm is proposed, which is invariant to additive and multiplicative noises in the image. This approach consists of two steps: firstly, a continuous blurred edge model is obtained using the smoothing spline algorithm in the edge region detected by Sobel operator; then a gray-moment solution is derived for both the one- and two-dimensional situations using the blurred edge model. Testing of this new detection approach demonstrates more robustness against the white Gaussian noise and speckle noise, and run time very close to the gray-moment and space-moment operators. The above advantages indicate this approach is very suitable for on-line accurate detection.

Revision of design rainfalls over Australia: A pilot study

Design rainfalls for standard durations from 24 hours up to 72 hours are derived using all readily available up-to-date rainfall data for an area in southeastern Queensland/northeastern New South Wales. Based on the annual maximum rainfall series, L-moments are derived at each site for a set of standard durations. A “region of influence” approach is employed to produce regionalised estimates of design rainfall. The network of continuously recording rain gauges is much less dense than the network of daily rain gauges. Data at longer durations is used to infer statistics at the shorter durations. For durations between 1 and 12 hours, a Partial Least Squares Regression model is developed to infer L-CV and L-skewness from the independent variables: L-CV and L-skewness at 24 hours, latitude, and distance from the coast. This procedure utilises information at the 24 hour duration to increase data coverage at the shorter durations. Maps of design rainfalls for standard average recurrence intervals from 1 to 100 years, and standard durations are produced using a combination of meteorological experience and a thin-plate smoothing spline algorithm. Results are compared with design rainfalls published in Institution of Engineers Australia (1987).

Very high resolution interpolated climate surfaces for global land areas

We developed interpolated climate surfaces for global land areas (excluding Antarctica) at a spatial resolution of 30 arc s (often referred to as 1-km spatial resolution). The climate elements considered were monthly precipitation and mean, minimum, and maximum temperature. Input data were gathered from a variety of sources and, where possible, were restricted to records from the 1950–2000 period. We used the thin-plate smoothing spline algorithm implemented in the ANUSPLIN package for interpolation, using latitude, longitude, and elevation as independent variables. We quantified uncertainty arising from the input data and the interpolation by mapping weather station density, elevation bias in the weather stations, and elevation variation within grid cells and through data partitioning and cross validation. Elevation bias tended to be negative (stations lower than expected) at high latitudes but positive in the tropics. Uncertainty is highest in mountainous and in poorly sampled areas. Data partitioning showed high uncertainty of the surfaces on isolated islands, e.g. in the Pacific. Aggregating the elevation and climate data to 10 arc min resolution showed an enormous variation within grid cells, illustrating the value of high-resolution surfaces. A comparison with an existing data set at 10 arc min resolution showed overall agreement, but with significant variation in some regions. A comparison with two high-resolution data sets for the United States also identified areas with large local differences, particularly in mountainous areas. Compared to previous global climatologies, ours has the following advantages: the data are at a higher spatial resolution (400 times greater or more); more weather station records were used; improved elevation data were used; and more information about spatial patterns of uncertainty in the data is available. Owing to the overall low density of available climate stations, our surfaces do not capture of all variation that may occur at a resolution of 1 km, particularly of precipitation in mountainous areas. In future work, such variation might be captured through knowledge-based methods and inclusion of additional co-variates, particularly layers obtained through remote sensing. Copyright © 2005 Royal Meteorological Society.

AnL 1 smoothing spline algorithm with cross validation

We propose an algorithm for the computation ofL1 (LAD) smoothing splines in the spacesWM(D), with\([0, 1]^n \subseteq D\). We assume one is given data of the formyi=(f(ti) +ei, i=1,...,N with {itti}i=1N ⊂D, theei are errors withE(ei)=0, andf is assumed to be inWM. The LAD smoothing spline, for fixed smoothing parameterλ⩾0, is defined as the solution,sλ, of the optimization problem\(\min _{g \in W_M }\) (1/N)∑i=1N¦yi−g(ti¦+λJM(g), whereJM(g) is the seminorm consisting of the sum of the squaredL2 norms of theMth partial derivatives ofg. Such an LAD smoothing spline,sλ, would be expected to give robust smoothed estimates off in situations where theei are from a distribution with heavy tails. The solution to such a problem is a “thin plate spline” of known form. An algorithm for computingsλ is given which is based on considering a sequence of quadratic programming problems whose structure is guided by the optimality conditions for the above convex minimization problem, and which are solved readily, if a good initial point is available. The “data driven” selection of the smoothing parameter is achieved by minimizing aCV(λ) score of the form\((1/N)[\sum\nolimits_{i = 1}^N {\left| {y_i - s_\lambda (t_i )} \right| + \sum\nolimits_{res_i = 0} 1 } ]\).The combined LAD-CV smoothing spline algorithm is a continuation scheme in λ↘0 taken on the above SQPs parametrized inλ, with the optimal smoothing parameter taken to be that value ofλ at which theCV(λ) score first begins to increase. The feasibility of constructing the LAD-CV smoothing spline is illustrated by an application to a problem in environment data interpretation.

A method of background determination in quantitative auger spectroscopy

A method for automatically determining a structureless background in a measured line spectrum is presented. The model assumptions for the background are given by the sign of the first three derivatives only, and the whole procedure is based on a smoothing spline algorithm. The method is especially designed for background determination in quantitative Auger spectroscopy, but may be modified to cover other topics. Application to measured Auger spectra is demonstrated and the validity of the method discussed.