Cardinal Spline Research Articles

Nasal and temporal curvatures of lamina CRIBROSA in myopic eyes

Little is known about the myopic characteristics of lamina cribrosa (LC) curvature. As such, we investigated nasal and temporal LC curvatures in myopia. In this retrospective, cross-sectional study, 144 myopic eyes (refraction < − 2D) and 88 non-myopic eyes (refraction > − 0.5D) underwent swept-source optical coherence tomography scanning of the LC. The anterior border of LC curvature was delineated with 17 points and interpolated with the “cardinal spline” curve-fitting method. The average curvature indices of the temporal and nasal sides were presented as the temporal and nasal curvatures. Myopic eyes had a mean refraction of − 6.7 ± 2.8D, while for non-myopic eyes, the value was 0.3 ± 1.0D. Nasal LC curvature was visible in 54 myopia (37.5%) and 42 non-myopia (47.7%) cases (P = 0.126), and temporal LC curvature was visible in 142 myopia (98.6%) and 68 non-myopia (77.3%) cases (P = 0.001). The nasal LC curvature was significantly larger in myopia than in non-myopia (P < 0.001). Contrastingly, the temporal LC curvature was significantly smaller in myopia than in non-myopia (P < 0.001). Axial length was associated with larger nasal LC curvature, smaller temporal LC curvature, and larger nasal–temporal LC curvature difference (all P’s < 0.05). In myopic relative to non-myopic eyes, LC curvature was decreased temporally and increased nasally.

EEG analysis and classification based on cardinal spline empirical mode decomposition and synchrony features.

Dementia is a major cause of disability and dependency among older adults. Diagnosis is most effective at an early stage of the disease, as patients can start early treatment to delay progressive cognitive decline. While other diagnostic methods for dementia are available, electroencephalography (EEG) is noninvasive, more accessible, and less complicated than other biomarker measurements. Moreover, it may be orders of magnitude less expensive, thereby offering the possibility of low-cost mass screening. This paper presents a novel digital signal processing method called cardinal spline empirical mode decomposition (CS-EMD) for EEG processing. This new method uses a different signal envelope interpolation algorithm to separate EEG signals into constituent components, called intrinsic mode functions (IMFs), with better signal decomposition properties than classical empirical mode decomposition (EMD). The IMFs obtained from the new method are then used to compute longitudinal and transversal synchrony measures, which are explored as features for healthy and dementia classification using a support vector machine (SVM). The performance of the proposed method is studied on a publicly available EEG dataset. The results show that using multiple synchrony measures of both longitudinal and transversal EEG channels on five IMFs produces the best classification result of 90% accuracy, 96.67% specificity, 83.33% sensitivity, and 96.15% precision, outperforming the classical EMD method and various other approaches. This new, data-driven CS-EMD method shows good potential as a dementia screening tool. CS-EMD may also be applied in processing other nonlinear and nonstationary biosignals.

Lithium battery attenuation estimation method based on curvature analysis and segmented high-order Gaussian fitting

Lithium battery has been widely used in various areas, and accurate estimation of battery states is vital to efficient and safe application of batteries. Of all the states, life attenuation is essential to batteries. To improve the estimation accuracy of lithium battery life attenuation, a battery attenuation estimation method based on curvature analysis and segmented Gaussian fitting is designed. The designed method firstly utilizes Cardinal spline curve to smooth the battery attenuation curve. Then, a curvature analysis method is used to segment the life attenuation curve into different sections. Then, a least square linear fitting is used to fit the segmented curves. Except the segmented curve with better linear fitting performance, the remaining segmented curves are fitted with high-order Gaussian fitting. Experiments indicates that the designed method can accurately express the characteristics of different battery life stages.

Efficient spline regression for neural spiking data.

Point process generalized linear models (GLMs) provide a powerful tool for characterizing the coding properties of neural populations. Spline basis functions are often used in point process GLMs, when the relationship between the spiking and driving signals are nonlinear, but common choices for the structure of these spline bases often lead to loss of statistical power and numerical instability when the signals that influence spiking are bounded above or below. In particular, history dependent spike train models often suffer these issues at times immediately following a previous spike. This can make inferences related to refractoriness and bursting activity more challenging. Here, we propose a modified set of spline basis functions that assumes a flat derivative at the endpoints and show that this limits the uncertainty and numerical issues associated with cardinal splines. We illustrate the application of this modified basis to the problem of simultaneously estimating the place field and history dependent properties of a set of neurons from the CA1 region of rat hippocampus, and compare it with the other commonly used basis functions. We have made code available in MATLAB to implement spike train regression using these modified basis functions.

Development and optimization of immersive virtual drilling equipment teaching environment

In view of the phenomenon of scene jumps, slow model refresh speed and fixed visual direction during virtual roaming, in the virtual drilling equipment teaching platform developed based on Unity 3D, the Cardinal spline curve interpolation algorithm is used to optimize the roaming path and improve the smoothness of the roaming. It uses rendering optimization algorithm to improve the fluency of the scene and adjusts the visual angle of observation through orientation change technology algorithm for design and optimization. The results show that the optimized drilling equipment teaching platform improves the comfort of roaming learning, the scene loading speed is increased by 28.73%, and the visual orientation is more convenient to observe the experimental objects. On the whole, it provides students with a comfortable and realistic teaching environment.

Creating a bridge between cardinal B[formula omitted]-spline fundamental functions for interpolation and subdivision

This paper presents innovative contributions to the fields of cardinal spline interpolation and subdivision. In particular, it unifies cardinal Br-spline fundamental functions for interpolation that are made of r=ML+1 (L∈N∪{0}) distinct pieces between each pair of interpolation nodes and are featured by the properties of C2M−2 smoothness, approximation order 2M and support width 2M(r+1)r, with the basic limit functions of a special class of non-stationary subdivision schemes of arity M.After introducing a general result, we focus our attention on the subclass of fourth-order accurate, C2 smooth Br-splines with maximum width of the compact support 6. The binary subdivision scheme yielding these fundamental functions outperforms the existing interpolatory schemes and seems to be the most adequate starting point to obtain compactly supported fundamental (spline) functions for local interpolation over quadrilateral and triangular meshes.

The Cardinal Spline Methods for the Numerical Solution of Nonlinear Integral Equations

In this study, an effective technique is presented for solving nonlinear Volterra integral equations. The method is based on application of cardinal spline functions on small compact supports. The integral equation is reduced to a system of algebra equations. Since the matrix for the system is triangular, it is relatively straightforward to solve for the unknowns and an approximation of the original solution with high accuracy is accomplished. Several cardinal splines are employed in the paper to enhance the accuracy. The sufficient condition for the existence of the inverse matrix is examined, and the convergence rate is analyzed. We compare our method with other methods proposed in recent papers and demonstrated the advantage of our method with several examples.

The Numerical Solutions of Systems of Nonlinear Integral Equations with the Spline Functions

The main goal of this work is to develop an effective technique for solving nonlinear systems of Volterra integral equations. The main tools are the cardinal spline functions on small compact supports. We solve a system of algebra equations to approximate the solution of the system of integral equations. Since the matrix for the algebraic system is nearly triangular, It is relatively painless to solve for the unknowns and an approximation of the original solution with high precision is accomplished. In order to enhance the accuracy, several cardinal splines are employed in the paper. Our schemes were compared with other techniques proposed in recent papers and the advantage of our method was exhibited with several numerical examples.

A New Class of Explicit Interpolatory Splines and Related Measurement Estimation

This paper proposes a new and rich class of cardinal splines called generalized MK-splines (GMK-splines), which manifest several beneficial properties (especially the interpolation property) and the ability to obtain a construction with a desired frequency response. The GMK-spline basis functions of given degree can be constructed by a linear combination of a finite number of shifted B-splines of that degree in a unified framework, the GMK-spline bases form Riesz bases of their generating spaces, and the GMK-spline frequency responses meet Strang-Fix conditions. Furthermore, we provide search methods for minimal-support GMK-splines and local optimal GMK-spline frequency responses under different given passband frequencies. Several related estimates are theoretically assessed, including energy magnitude, maximum, and approximation error of GMK-spline interpolated signals. In signal processing applications, the GMK-splines behave as interpolating FIR filters with simple implementation, and some of them attain lower error than other traditional interpolation system responses, such as cubic cardinal B-spline and linear spline, when approximating to an ideal lowpass filter. Experimental results show that the GMK-spline filters can be implemented efficiently and have significant characteristics in the process of bandlimited signal reconstruction.

The limiting behavior of certain sampling series and cardinal splines

Many families of cardinal sampling kernels {Φα(x):α∈I} consist of members that decay rapidly as x→±∞ and enjoy the property that, as the parameter α tends to its limiting value, Φα(x) tends to the classical cardinal sine, sinc(x). We study the limiting behavior of the sampling series ∑ncnΦα(x−n) when the data sequence {cn} remains fixed. In the case when the sampling kernels are damped cardinal sines, appropriate conditions on the damping functions allow us to characterize the limiting behavior of the corresponding sampling series for a wide class of data sequences {cn}.These results lead to conclusions regarding the limiting behavior of classical piecewise polynomial cardinal splines when the fixed data sequence {cn} is of polynomial growth and the order tends to ∞. One application of this development shows that the classical spline summability method is regular. Another consequence, perhaps more interesting and unexpected, is a resolution of I. J. Schoenberg’s conjecture concerning the recovery, in terms of their samples {f(n)}, of members f(x) of the Bernstein class via the spline summability procedure.

Evaluation of Methods of Interpolation (Existing and Proposed) using Monte Carlo Simulation Technique and Delete-D Jackknife Method

It is possible to encounter missing value in a research. Missing value may be as a result of none response in primary data collection or unavailability of data in the case of secondary data. It may occur within a set of observations or at the tail end of the observations. The paper addresses missing value within a set of observations (interpolation). Existing methods considered are linear, log-linear, Catmull-Rom spline and cardinal spline and a method of estimating missing value is presented. For evaluation of the methods, random data are simulated using Monte Carlo simulation approach and analytical approach is used to determine the most effective method. Among the findings, linear, log-linear and the proposed method give high precision estimate compare to the CS and CSR methods. With the use of tension parameter, CS is better than CSR method.

Geometric convergence rates for cardinal spline subdivision with general integer arity

A rigorous convergence analysis is presented for arbitrary order cardinal spline subdivision with general integer arity, for which the binary case, with arity two, is a well-studied subject. Explicit geometric convergence rates are derived, and particular attention is devoted to the rendering of cardinal spline graphs and parametric curves.

Cardinal MK-spline signal processing: Spatial interpolation and frequency domain filtering

Altering the sampling rate of a digital signal by interpolation is a very important task in many digital signal processing applications. MK-spline is a kind of promising cardinal spline or explicit interpolation spline, which can achieve the purpose of interpolation without solving systems of coefficient equations, and which can inherit almost all good properties of B-splines, such as local support, high order smoothness, and central symmetry. For using MK-spline as a tool in signal and image processing, we first provide elementary theory which is composed of MK-spline convolution representation, related Fourier transform in continuous domain, and z transform of expanded MK-spline in discrete domain. Then, a cubic MK-spline interpolating filter (MKIF) is designed as a physically realizable FIR digital filter based on the discrete convolution operation which outputs directly a discrete smoothed signal from an input discrete signal. Next, frequency domain MK-spline lowpass and highpass filters are also devised in this study for image smoothing, sharpening, and denoising. Finally, simulated experiments are presented in different dimensional data formats in order to verify that the convolution-based MKIF is faster than those based on traditional continuous method, and that the proposed MK-spline filters have significantly good characteristics in the processes of signal reconstruction, image enlargement, and image denoising, when compared with traditional linear filters and classical B-spline filters.

On the convergence of cardinal splines

This note contains an outline of a solution to the following conjecture formulated by I. J. Schoenberg in 1976: If the function f is in the Bernstein class B π and S k ( { f ( n ) } , x ) is the piecewise polynomial cardinal spline of order 2 k that interpolates ( n , f ( n ) ) , n = 0 , ± 1 , ± 2 , … , then for some constant c lim k → ∞ ⁡ S k ( { f ( n ) } , x ) = f ( x ) − c sin ⁡ π x uniformly on compact subsets of the real axis R . The techniques involved lead to a regularity result for cardinal spline summability and a characterization of the behavior of S k ( { f ( n ) } , x ) as k tends to ∞ for all sample sequences { f ( n ) } of polynomial growth. Precise formulations of these results are also included here.

Beyond Wiener’s Lemma: Nuclear Convolution Algebras and the Inversion of Digital Filters

A convolution algebra is a topological vector space $\mathcal{X}$ that is closed under the convolution operation. It is said to be inverse-closed if each element of $\mathcal{X}$ whose spectrum is bounded away from zero has a convolution inverse that is also part of the algebra. The theory of discrete Banach convolution algebras is well established with a complete characterization of the weighted $\ell_1$ algebras that are inverse-closed and referred to as the Gelfand-Raikov-Shilov (GRS) spaces. Our starting point here is the observation that the space $\mathcal{S}(\mathbb{Z}^d)$ of rapidly decreasing sequences, {which is not Banach but nuclear}, is an inverse-closed convolution algebra. This property propagates to the more constrained space of exponentially decreasing sequences $\mathcal{E}(\mathbb{Z}^d)$ that we prove to be nuclear as well. Using a recent extended version of the GRS condition, we then show that $\mathcal{E}(\mathbb{Z}^d)$ is actually the smallest inverse-closed convolution algebra. This allows us to describe the hierarchy of the inverse-closed convolution algebras from the smallest, $\mathcal{E}(\mathbb{Z}^d)$, to the largest, $\ell_{1}(\mathbb{Z}^d)$. In addition, we prove that, in contrast to $\mathcal{S}(\mathbb{Z}^d)$, all members of $\mathcal{E}(\mathbb{Z}^d)$ admit well-defined convolution inverses in $\mathcal{S}'(\mathbb{Z}^d)$ with the "unstable" scenario (when some frequencies are vanishing) giving rise to inverse filters with slowly-increasing impulse responses. Finally, we use those results to reveal the decay and reproduction properties of an extended family of cardinal spline interpolants.

Modeling and refactoring the indigo patterns

For purpose of reconstruction and innovation of indigo patterns, this study explores modeling, reconstruction, and assembling of the pattern elements by means of mathematics. A model for indigo pattern elements is proposed based on cardinal splines, in which the rigidity of shape is conveyed by the tension coefficient, and the concavity and variety by configuration of the knots. The generalized version of this model is capable of covering any complex element. The contour tracing technique is employed to extract pattern elements from the image, and the closest model instance is selected in virtue of invariance of the improved Hu moments. The selected instances are transformed with respect to the geometric center, the coverage, and the coincidence to match the pattern elements in the image so as to reconstruct the whole pattern. The element filters are conducted on the reconstructed patterns to modify the elements and produce new innovative patterns of constant skeleton. The model is borrowed serving as a skeleton in element assembling. The skeleton properties are investigated to provide basis for skeleton embodiment in which these properties are involved into establishing the placement determiner and element determiner so as to carry out the assembling of elements. Also discussed is the extended skeleton which goes beyond the model and brings about variety and flexibility to element assembling. It is turned out that reconstruction with the model well implements a mathematical copy of the pattern, and the assembling of elements by skeletons provides rich possibilities to innovation of indigo patterns.

Research on Influences of Employment in Manufacturing Industry on that in the Service Industry Based on Bayes Model

Using the data of 285 prefectural and the above-level cities from 2004 to 2016, this thesis reveals the impact of employment in China's urban manufacturing industry on the employment of service industries with the Bayesian model. Under the Bayesian framework, partial linear semi-parametric model is proposed. The nonlinear functions are fitted by using truncation base cardinal spline and considering the random error terms of mixed normal fitting models. The results show that: employment in the urban manufacturing industry in China has significant influence on the employment in the service industry. When the number of employees in the manufacturing industry changes from 0 to 650,000, the manufacturing industry has less influence. When the number of the employees in the manufacturing industry changes from 650,000 to 900,000, the employees of the service industry will dramatically increase. When the number of the employees in the manufacturing industry is more than 900,000, the employees in the service industry will be prone to stable growth.

Generalized Taylor operators and polynomial chains for Hermite subdivision schemes

Hermite subdivision schemes act on vector valued data that is not only considered as functions values of a vector valued function from $$\mathbb {R}$$ to $$\mathbb {R}^r$$ , but as evaluations of r consecutive derivatives of a function. This intuition leads to a mild form of level dependence of the scheme. Previously, we have proved that a property called spectral condition or sum rule implies a factorization in terms of a generalized difference operator that gives rise to a “difference scheme” whose contractivity governs the convergence of the scheme. But many convergent Hermite schemes, for example, those based on cardinal splines, do not satisfy the spectral condition. In this paper, we generalize the property in a way that preserves all the above advantages: the associated factorizations and convergence theory. Based on these results, we can include the case of cardinal splines in a systematic way and are also able to construct new types of convergent Hermite subdivision schemes.

Filter bank properties of envelope modified EMD methods

Envelope modified versions of the empirical mode decomposition (EMD) method such as the B-spline interpolation-based EMD (B-EMD) method and cardinal spline interpolation-based EMD (C-EMD) method have been proposed recently for purpose of improving its effectiveness. To shed further light on their performance, the behaviours of these EMD-type methods in the presence of white Gaussian noises are investigated in this study based on extensive numerical experiments. Similarly to the EMD method, it turns out that the envelope modified EMD methods also act as filter banks essentially. However, the spectra among the first several intrinsic mode functions of the B-EMD method have fewer overlaps than those of the EMD and C-EMD methods, which indicate that the B-EMD method has a better ability to alleviate the mode mixing problem for signals with higher frequencies. On the other hand, the C-EMD method is shown to perform better than the EMD and B-EMD methods on separating tones with lower frequencies.

Reconstruction of Splines from Nonuniform Samples

In this paper, we study the reconstruction of spline functions from their nonuniform samples. We investigate the existence and uniqueness of the solution of the following problem: for given data {(xn, yn): n ∈ ℤ}, find a cardinal spline f(x), of a given degree, satisfying yn = f(xn), n ∈ ℤ. Several necessary and/or sufficient conditions for the existence and uniqueness of the solution of the problem are derived. Finally, an example and some applications are presented to illustrate the main results.