Combination Of B-spline Basis Functions Research Articles

Multiphase Conductivity Imaging With Electrical Impedance Tomography and B-Spline Level Set Method

This article presents a B-spline level set (BLS)-based reconstruction scheme for multiphase conductivity imaging with electrical impedance tomography (EIT). The EIT inverse problem is recast as a shape reconstruction problem where both the inclusion interface and conductivity values are the unknown. In using this scheme, the unknown conductivity to be reconstructed is assumed to be piecewise constant while the inclusion interface is represented by the zero level set of a BLS surface, which, in turn, is represented as a linear combination of B-spline basis functions. Through such a low-order representation technique, the number of unknowns in the reconstruction problem is significantly reduced compared with the traditional pixel-/voxel-based reconstruction methods. The performance of the BLS scheme is tested with heart-and-lungs phantom studies and the cross section of the industrial pipeline phantom studies. The results demonstrate that the BLS scheme is capable of preserving sharp features of inclusions and providing estimates to both the interface locations and unknown conductivity values with reliable accuracy. In addition, we also found that the BLS scheme is quite robust to the selection of the regularization weighting parameter in the least-square-based optimization regime and the additional parameter in the Heaviside function used for modeling the conductivity profile.

Spatiotemporal Thermal Field Modeling Using Partial Differential Equations With Time-Varying Parameters

Accurate modeling of a thermal field is one of the fundamental requirements in engineering thermal management in numerous industries. Existing studies have shown that using differential equations to model a thermal field delivers good performance when the parameters are predetermined through physical or experimental analysis. However, due to variations of the inner medium affected by certain latent factors, the parameters in differential equation models may not be treated as constants while the thermal field is estimated, and this fact poses a new challenge to field estimation by directly solving the differential equation models. In this study, a novel approach to thermal field modeling is developed by considering the parameters as functional variables that vary temporally in partial differential equations (PDEs). This approach provides a new perspective to model the dynamic thermal field by fully using the collected sensor data from the thermal system. Specifically, time-varying parameters can be constructed through a combination of basis functions whose coefficients can be efficiently estimated through the sensor data. A two-level iterative parameter estimation algorithm is also tailored to obtain the parameters in the PDE model. Both simulation and real case studies show that our proposed approach provides satisfactory estimation performance compared with the benchmark method that uses the constant parameter estimation. Note to Practitioners —The proposed method aims to model a thermal field using PDEs with time-varying parameters. To better implement this method in practice, three things are noteworthy: first, the proposed method models a thermal field by fully considering physics-specific engineering knowledge using PDEs and the collected sensor data from thermal systems. Second, because time-varying parameters in PDEs cannot be estimated directly, the proposed model represents the time-varying parameters by a combination of B-spline basis functions in terms of time. Estimating time-varying parameters is converted into estimating the constant coefficients of the basis functions. Because the derivatives of a thermal field might not have an analytical expression, the proposed model represents the thermal field by a combination of B-spline basis functions. Taking the derivatives of the thermal field is converted into taking the derivatives of the corresponding basis functions. Third, the proposed method can not only model a thermal field but can also be applied in other physics-specific engineering cases.

Robot path planning in a dynamic environment with stochastic measurements

We study the problem of trajectory planning for autonomous vehicles designed to minimize the travel distance while avoiding moving obstacles whose position and speed are not known. Because, in practice, observations from sensors have measurement errors, the stochasticity of the data is modeled using maximum likelihood estimators, which are shown to be consistent as the sample size increases. To comply with the kinematic constraints of the vehicle, we consider smooth trajectories that can be represented by a linear combination of B-spline basis functions, transforming the infinite-dimensional problem into a finite-dimensional one. Moreover, a smooth penalty function is added to the travel distance, transforming the constrained optimization problem into an unconstrained one. The planned stochastic trajectory, obtained from the minimization problem with stochastic confidence regions, is shown to converge almost surely to the deterministic one as the number of sensor observations increases. Finally, we present two simulation studies to demonstrate the proposed method.

A lack-of-fit test for generalized linear models via single-index techniques

A generalized partially linear single-index model (GPLSIM) is proposed in which the unknown smooth function of single index is approximated by a spline function that can be expressed as a linear combination of B-spline basis functions. The regression coefficients and the unknown smooth function are estimated simultaneously via a modified Fisher-scoring method. It can be shown that the estimators of regression parameters are asymptotically normally distributed. The asymptotic covariance matrix of the estimators can be estimated directly and consistently by using the least-squares method. As an application, the proposed GPLSIM can be employed to assess the lack of fit of a postulated generalized linear model (GLM) based on the comparison of the goodness of fit of the GPLSIM and postulated GLM to construct a likelihood ratio test. An extensive simulation study is conducted to examine the finite-sample performance of the likelihood ratio test. The practicality of the proposed methodology is illustrated with a real-life data set from a study of nesting horseshoe crabs.

TCvBsISM: Texture Classification via B-Splines-Based Image Statistical Modeling

This paper presents an image statistical modeling-based texture classification (TC) approach via the Bayesian-driven B-splines probability density estimation of the image textural surface appearance (ITSA), termed TCvBsISM. It approximates the probability density functions (pdfs) of the marginal distribution and joint distribution, involving the global organization and the locally structural layout of local homogeneous patches in the texture surface, respectively, of both the image raw pixel space and the filter response space, by the linear combination of B-spline basis functions (BsBFs) for ITSA feature characterization. The corresponding linear weighting coefficients (LWCs) are determined by an entropy-based optimization criterion with a prior smooth constraint over the LWCs. By leveraging the B-spline-based pdf modeling, distinctive ITSA structural features of texture images are characterized by the LWCs of the pre-defined BsBFs, which are then embedded in an integrated statistical feature dictionary learning, texture pattern representation, and discrimination model to perform TC. Extensive confirmative and comparative experiments on three different texture databases and one natural environmental scene database demonstrate that the proposed TCvBsISM is very promising, especially when the images of different texture patterns appear to be quite similar with the limited training samples. The effects of various parameters on TCvBsISM, such as the choice of the filter bank, the size of the image statistical feature dictionary, as well as the number of BsBFs, are also discussed.

Relativistic ground state of diatomic molecules from the numerical solution of the Dirac equation on parallel computers

The parallel implementation of a variational method for the numerical solution of the time-independent Dirac equation is presented. This is utilized in the calculation of diatomic molecule relativistic wave functions and ground state energy, in the Born-Oppenheimer approximation. Following [1], a functional equation predicting the same spectrum as the Dirac equation is discretized over a set of basis functions. We find that the combination of B-spline basis functions and prolate spheroidal coordinate systems facilitate the vectorization of the technique using domain decomposition. This procedure results in a non-linear eigenvalue problem which is solved by iteration using Brent's method. We validate our methodology by comparing with existing results and show that it yields very accurate ones. We also analyze the parallelization and demonstrate that our implementation has a good scaling performance. Finally, the utilization of this method for laser-matter interaction and other physical applications will be discussed.