B-spline Kernel Research Articles

Wiener–Hopf Difference Equations and Semi-Cardinal Interpolation with Integrable Convolution Kernels

Let Hsubset {mathbb {Z}}^d be a half-space lattice, defined either relative to a fixed coordinate (e.g. H = {mathbb {Z}}^{d-1}!times !{mathbb {Z}}_+), or relative to a linear order preceq on {mathbb {Z}}^d, i.e. H = {jin {mathbb {Z}}^d: 0preceq j}. We consider the problem of interpolation at the points of H from the space of series expansions in terms of the H-shifts of a decaying kernel phi . Using the Wiener–Hopf factorization of the symbol for cardinal interpolation with phi on {mathbb {Z}}^d, we derive some essential properties of semi-cardinal interpolation on H, such as existence and uniqueness, Lagrange series representation, variational characterization, and convergence to cardinal interpolation. Our main results prove that specific algebraic or exponential decay of the kernel phi is transferred to the Lagrange functions for interpolation on H, as in the case of cardinal interpolation. These results are shown to apply to a variety of examples, including the Gaussian, Matérn, generalized inverse multiquadric, box-spline, and polyharmonic B-spline kernels.

B-Spline Parameterized Joint Optimization of Reconstruction and K-Space Trajectories (BJORK) for Accelerated 2D MRI.

Optimizing k-space sampling trajectories is a promising yet challenging topic for fast magnetic resonance imaging (MRI). This work proposes to optimize a reconstruction method and sampling trajectories jointly concerning image reconstruction quality in a supervised learning manner. We parameterize trajectories with quadratic B-spline kernels to reduce the number of parameters and apply multi-scale optimization, which may help to avoid sub-optimal local minima. The algorithm includes an efficient non-Cartesian unrolled neural network-based reconstruction and an accurate approximation for backpropagation through the non-uniform fast Fourier transform (NUFFT) operator to accurately reconstruct and back-propagate multi-coil non-Cartesian data. Penalties on slew rate and gradient amplitude enforce hardware constraints. Sampling and reconstruction are trained jointly using large public datasets. To correct for possible eddy-current effects introduced by the curved trajectory, we use a pencil-beam trajectory mapping technique. In both simulations and in- vivo experiments, the learned trajectory demonstrates significantly improved image quality compared to previous model-based and learning-based trajectory optimization methods for 10× acceleration factors. Though trained with neural network-based reconstruction, the proposed trajectory also leads to improved image quality with compressed sensing-based reconstruction.

A stable SPH with adaptive B-spline kernel

Tensile instability, often observed in smoothed particle hydrodynamics (SPH), is a numerical artifact that manifests itself by unphysical clustering or separation of particles. The instability originates in estimating the derivatives of the smoothing functions which, when interact with material constitution may result in negative stiffness in the discretized system. In the present study, a stable formulation of SPH is developed where the kernel function is continuously adapted at every material point. B-spline basis function with a variable intermediate knot is used as the kernel function. The shape of the kernel function is then modified by changing the intermediate knot position such that the condition associated with instability does not arise. While implementing the algorithm the simplicity and computational efficiency of SPH are not compromised. One-dimensional dispersion analysis is performed to understand the effect of adaptive kernel on the stability. Finally, the efficacy of the algorithm is demonstrated through some benchmark elastic dynamics problems.

Solving 2D Poisson-type equations using meshless SPH method

Abstract In the present study, 2D Poisson-type equation is solved by a meshless Symmetric Smoothed Particle Hydrodynamics (SSPH) method. The influence of the kernel function, smoothing length and particle discretizations of problem domain on the solutions of Poisson-type equations is investigated. Compared with other kernel functions, the cubic B-spline kernel function shows good capacity to reproduce a complicated function value and its first and second derivatives, when the smoothing length is chosen as 1.1 times the particles’ distance. Several types of Poisson equations are solved by SSPH method, and the numerical results exhibit a very good accuracy when comparing with the analytical solutions.

Rate of approximation for multivariate sampling Kantorovich operators on some functions spaces

In this paper, the problem of the order of approximation for the multivariate sampling Kantorovich operators is studied. The cases of uniform approximation for uniformly continuous and bounded functions/signals belonging to Lipschitz classes and the case of the modular approximation for functions in Orlicz spaces are considered. In the latter context, Lipschitz classes of Zygmund-type which take into account of the modular functional involved are introduced. Applications to $L^p(\R^n)$, interpolation and exponential spaces can be deduced from the general theory formulated in the setting of Orlicz spaces. The special cases of multivariate sampling Kantorovich operators based on kernels of the product type and constructed by means of Fej\'er's and B-spline kernels have been studied in details.

Order of approximation for sampling Kantorovich operators

In this paper, we study the problem of the rate of approximation for the family of sampling Kantorovich operators in the uniform norm, for uniformly continuous and bounded functions belonging to Lipschitz classes (Zygmund-type classes), and for functions in Orlicz spaces. The general setting of Orlicz spaces allows us to directly deduce the results concerning the order of approximation in $L^p$-spaces, $1 \le p \lt \infty$, very useful in applications to Signal Processing, in Zygmund spaces and in exponential spaces. Particular cases of the sampling Kantorovich series based on Fej\'er's kernel and B-spline kernels are studied in detail.

Medical image registration using Edgeworth-based approximation of Mutual Information

We propose a new similarity measure for iconic medical image registration, an Edgeworth-based third order approximation of Mutual Information (MI) and named 3-EMI. Contrary to classical Edgeworth-based MI approximations, such as those proposed for independent component analysis, the 3-EMI measure is able to deal with potentially correlated variables. The performance of 3-EMI is then evaluated and compared with the Gaussian and B-Spline kernel-based estimates of MI, and the validation is leaded in three steps. First, we compare the intrinsic behavior of the measures as a function of the number of samples and the variance of an additive Gaussian noise. Then, they are evaluated in the context of multimodal rigid registration, using the RIRE data. We finally validate the use of our measure in the context of thoracic monomodal non-rigid registration, using the database proposed during the MICCAI EMPIRE10 challenge. The results show the wide range of clinical applications for which our measure can perform, including non-rigid registration which remains a challenging problem. They also demonstrate that 3-EMI outperforms classical estimates of MI for a low number of samples or a strong additive Gaussian noise. More generally, our measure gives competitive registration results, with a much lower numerical complexity compared to classical estimators such as the reference B-Spline kernel estimator, which makes 3-EMI a good candidate for fast and accurate registration tasks.

Order of approximation for nonlinear sampling Kantorovich operators in Orlicz spaces

In this paper, we study the rate of approximation for the nonlinear sampling Kantorovich operators. We consider the case of uniformly continuous and bounded functions belonging to Lipschitz classes of the Zygmund-type, as well as the case of functions in Orlicz spaces. We estimate the aliasing errors with respect to the uniform norm and to the modular functional of the Orlicz spaces, respectively. The general setting of Orlicz spaces allows to deduce directly the results concerning the rate of convergence in \(L^p\)-spaces, \(1 \miu p\) < \(\infty\), very useful in the applications to Signal Processing. Others examples of Orlicz spaces as interpolation spaces and exponential spaces are discussed and the particular cases of the nonlinear sampling Kantorovich series constructed using Fej\'er and B-spline kernels are also considered.

Reproducing kernel triangular B-spline-based FEM for solving PDEs

We propose a reproducing kernel triangular B-spline-based finite element method (FEM) as an improvement to the conventional triangular B-spline element for solving partial differential equations (PDEs). In the latter, unexpected errors can occur throughout the analysis domain mainly due to the excessive flexibilities in defining the B-spline. The performance therefore becomes unstable and cannot be controlled in a desirable way. To address this issue, the proposed improvement adopts the reproducing kernel approximation in the calculation of B-spline kernel function. Three types of PDE problems are tested to validate the present element and compare against the conventional triangular B-spline. It has been shown that the improved triangular B-spline satisfies the partition of unity condition even for extreme conditions including corners and holes.

Hydrodynamic capabilities of an SPH code incorporating an artificial conductivity term with a gravity-based signal velocity

This paper investigates the hydrodynamic performances of an SPH code incorporating an artificial heat conductivity term in which the adopted signal velocity is applicable when gravity is present. In accordance with previous findings it is shown that the performances of SPH to describe the development of Kelvin-Helmholtz instabilities depend strongly on the consistency of the initial condition set-up and on the leading error in the momentum equation due to incomplete kernel sampling. An error and stability analysis shows that the quartic B-spline kernel (M_5) possesses very good stability properties and we propose its use with a large neighbor number, between ~50 (2D) to ~ 100 (3D), to improve convergence in simulation results without being affected by the so-called clumping instability. SPH simulations of the blob test show that in the regime of strong supersonic flows an appropriate limiting condition, which depends on the Prandtl number, must be imposed on the artificial conductivity SPH coefficients in order to avoid an unphysical amount of heat diffusion. Results from hydrodynamic simulations that include self-gravity show profiles of hydrodynamic variables that are in much better agreement with those produced using mesh-based codes. In particular, the final levels of core entropies in cosmological simulations of galaxy clusters are consistent with those found using AMR codes. Finally, results of the Rayleigh-Taylor instability test demonstrate that in the regime of very subsonic flows the code has still several difficulties in the treatment of hydrodynamic instabilities. These problems being intrinsically due to the way in which in standard SPH gradients are calculated and not to the implementation of the artificial conductivity term.

Stochastic controller as an active queue management based on B-spline kernel observer via particle swarm optimization

Given the fact that the current Internet is getting more difficult in handling the traffic congestion control, the proposed method is compatible with the stochastic nature of network dynamics. Most conventional active queue management is based on the first stochastic moment. In stochastic theory, the first moment is not efficient for non-Gaussian systems that are the same as the network queue size. We propose a new stochastic active queue management technique, based on stochastic control and B-spline window observer, called intelligent probability density function AQM (IPDF-AQM). The IPDF-AQM is based on a PDF control and particle swarm optimization, which not only considers the average queue length at the current time slot, but also takes into consideration the PDF of queue lengths within a round-trip time. We provide a guideline for the selection of the probability of dropping as control input for TCP/AQM system to make the PDF of queue length converge at a certain PDF target based on B-spline approximation and improve the network performance. Simulation results show that the proposed stochastic AQM scheme does improve the end-to-end performance.

Temporal diffeomorphic free-form deformation: Application to motion and strain estimation from 3D echocardiography

This paper presents a new registration algorithm, called Temporal Diffeomorphic Free Form Deformation (TDFFD), and its application to motion and strain quantification from a sequence of 3D ultrasound (US) images. The originality of our approach resides in enforcing time consistency by representing the 4D velocity field as the sum of continuous spatiotemporal B-Spline kernels. The spatiotemporal displacement field is then recovered through forward Eulerian integration of the non-stationary velocity field. The strain tensor is computed locally using the spatial derivatives of the reconstructed displacement field. The energy functional considered in this paper weighs two terms: the image similarity and a regularization term. The image similarity metric is the sum of squared differences between the intensities of each frame and a reference one. Any frame in the sequence can be chosen as reference. The regularization term is based on the incompressibility of myocardial tissue. TDFFD was compared to pairwise 3D FFD and 3D+t FFD, both on displacement and velocity fields, on a set of synthetic 3D US images with different noise levels. TDFFD showed increased robustness to noise compared to these two state-of-the-art algorithms. TDFFD also proved to be more resistant to a reduced temporal resolution when decimating this synthetic sequence. Finally, this synthetic dataset was used to determine optimal settings of the TDFFD algorithm. Subsequently, TDFFD was applied to a database of cardiac 3D US images of the left ventricle acquired from 9 healthy volunteers and 13 patients treated by Cardiac Resynchronization Therapy (CRT). On healthy cases, uniform strain patterns were observed over all myocardial segments, as physiologically expected. On all CRT patients, the improvement in synchrony of regional longitudinal strain correlated with CRT clinical outcome as quantified by the reduction of end-systolic left ventricular volume at follow-up (6 and 12months), showing the potential of the proposed algorithm for the assessment of CRT.

Genetic algorithm based support vector machine regression in predicting wave transmission of horizontally interlaced multi-layer moored floating pipe breakwater

Planning and design of coastal protection works like floating pipe breakwater require information about the performance characteristics of the structure in reducing the wave energy. Several researchers have carried out analytical and numerical studies on floating breakwaters in the past but failed to give a simple mathematical model to predict the wave transmission through floating breakwaters by considering all the boundary conditions. Computational intelligence techniques, such as, Artificial Neural Networks (ANN), fuzzy logic, genetic programming and Support Vector Machine (SVM) are successfully used to solve complex problems. In the present paper, a hybrid Genetic Algorithm Tuned Support Vector Machine Regression (GA-SVMR) model is developed to predict wave transmission of horizontally interlaced multilayer moored floating pipe breakwater (HIMMFPB). Furthermore, optimal SVM and kernel parameters of GA-SVMR models are determined by genetic algorithm. The GA-SVMR model is trained on the data set obtained from experimental wave transmission of HIMMFPB using regular wave flume at Marine Structure Laboratory, National Institute of Technology, Karnataka, Surathkal, Mangalore, India. The results are compared with ANN and Adaptive Neuro-Fuzzy Inference System (ANFIS) models in terms of correlation coefficient, root mean square error and scatter index. Performance of GA-SVMR is found to be reliably superior. b-spline kernel function performs better than other kernel functions for the given set of data.

A geometric nonuniform fast Fourier transform

An efficient algorithm is presented for the computation of Fourier coefficients of piecewise-polynomial densities on flat geometric objects in arbitrary dimension and codimension. Applications range from standard nonuniform FFTs of scattered point data, through line and surface potentials in two and three dimensions, to volumetric transforms in three dimensions. Input densities are smoothed with a B-spline kernel, sampled on a uniform grid, and transformed by a standard FFT, and the resulting coefficients are unsmoothed by division. Any specified accuracy can be achieved, and numerical experiments demonstrate the efficiency of the algorithm for a gallery of realistic examples.

Variational B-Spline Level-Set: A Linear Filtering Approach for Fast Deformable Model Evolution

In the field of image segmentation, most level-set-based active-contour approaches take advantage of a discrete representation of the associated implicit function. We present in this paper a different formulation where the implicit function is modeled as a continuous parametric function expressed on a B-spline basis. Starting from the active-contour energy functional, we show that this formulation allows us to compute the solution as a restriction of the variational problem on the space spanned by the B-splines. As a consequence, the minimization of the functional is directly obtained in terms of the B-spline coefficients. We also show that each step of this minimization may be expressed through a convolution operation. Because the B-spline functions are separable, this convolution may in turn be performed as a sequence of simple 1-D convolutions, which yields an efficient algorithm. As a further consequence, each step of the level-set evolution may be interpreted as a filtering operation with a B-spline kernel. Such filtering induces an intrinsic smoothing in the algorithm, which can be controlled explicitly via the degree and the scale of the chosen B-spline kernel. We illustrate the behavior of this approach on simulated as well as experimental images from various fields.

Parzen-Window Based Normalized Mutual Information for Medical Image Registration

Image Registration can be seen as an optimization problem to find a cost function and then use an optimization method to get its minimum. Normalized mutual information is a widely-used robust method to design a cost function in medical image registration. Its calculation is based on the joint histogram of the fixed and transformed moving images. Usually, only a discrete joint histogram is considered in the calculation of normalized mutual information. The discrete joint histogram does not allow the cost function to be explicitly differentiated, so it can only use non-gradient based optimization methods, such as Powell's method, to seek the minimum. In this paper, a parzen-window based method is proposed to estimate the continuous joint histogram in order to make it possible to derive the close form solution for the derivative of the cost function. With this help, we successfully apply the gradient-based optimization method in registration. We also design a new kernel for the parzen-window based method. Our designed kernel is a second order polynomial kernel with the width of two. Because of good theoretical characteristics, this kernel works better than other kernels, such as a cubic B-spline kernel and a first order B-spline kernel, which are widely used in the parzen-window based estimation. Both rigid and non-rigid registration experiments are done to show improved behavior of our designed kernel. Additionally, the proposed method is successfully applied to a clinical CT-MR non-rigid registration which is able to assist a magnetic resonance (MR) guided microwave thermocoagulation of liver tumors.

N-D $C^k$ B-Spline Scattered Data Approximation

Since the 1970’s B-splines have evolved to become the {} standard for curve and surface representation due to many of their salient properties. Conventional least-squares scattered data fitting techniques for B-splines require the inversion of potentially large matrices. This is time-consuming as well as susceptible to ill-conditioning which leads to undesired results. Lee {} proposed a novel B-spline algorithm for fitting a 2-D cubic B-spline surface to scattered data in . The proposed algorithm utilizes an optional multilevel approach for better fitting results. We generalize this technique to support N-dimensional data fitting as well as arbitrary degree of B-spline. In addition, we generalize the B-spline kernel function class to accommodate this new image filter.

Alternative kernel functions for Smoothed Particle Hydrodynamics in cylindrical symmetry

In this paper we present an alternative generic kernel function for use in Smoothed Particle Hydrodynamics (SPH) based methods. The alternative kernel has been implemented in a standard Cartesian description, and the results for the Sod shock tube problem is presented both with the alternative kernel and the traditional B-spline kernel. Kernels for cylindrically symmetric systems are developed based on fundamental interpolation theory for the cylindrical symmetry. A set of modified equations of motions are further derived through the use of Lagrangian formalism. Previous results have shown that the use of the traditional B-spline kernel function as a generic kernel implies that the cylindrically symmetric solution must be solved by table interpolation. By introducing an alternative generic kernel function, we derive simple analytical solution also for cylindrically symmetric systems. When compared to the B-spline function, the computational efficiency has been increased and the results improved. The results show good agreement with the analytical solution both for the Noh infinite shock problem and the Sedov point source explosion.

Generation of Stratified Samples for B-Spline Pixel Filtering

B-spline filter kernels have proved useful in many pixel-sampling applications. A cubic B-spline filter kernel, having a width of four pixels, is particularly effective. In distribution ray tracing, pixel filters are evaluated implicitly by having the density of sampling proportional to the filter value. In this work we present a simple method to generate random samples having an underlying B-spline density function. To reduce error it is important to stratify the samples, akin to jittering for uniform sampling. We provide an algebraic and a numerical technique for doing this for B-spline kernels of degree 1, 2, and 3.

Fractional derivatives and the inverse Fourier transform of ℓ1-radial functions

In [Berens, H. and Xu, Y. (1997). ℓ − 1 summability of multiple Fourier integrals and positivity. Math. Proc. Camb. Phil. Soc., 122, 149–172] and [Berens, H. and zu Castell, W. (1998). Hypergeometric functions as a tool for summability of the Fourier integral. Result. Math., 34, 69–84], H. Berens, Y. Xu, and the author proved that the inverse Fourier integral of ℓ1-radial functions, i.e., functions which are radial w.r.t. the ℓ1-norm on ℝ d , can be decomposed into a multi-dimensional integral with B-spline kernel, which is independent of the function, and a transformation on ℝ+ the kernel of which is given by a Meijer G-function. The latter can be seen as an analog of the well-known Fourier–Bessel transform. Here, we associate weights to the underlying transform. There is an analog development where the B-spline has to be replaced by a Dirichlet spline, while on ℝ+ the related transforms can again be defined via G-function kernels. We further show that the underlying transform can be interpreted as a fractional derivative.