Image registration is an inherently ill-posed problem that lacks the constraints needed for a unique mapping between voxels of the two images being registered. As such, one must regularize the registration to achieve physically meaningful transforms. The regularization penalty is usually a function of derivatives of the displacement-vector field, and can be calculated either analytically or numerically. The numerical approach, however, is computationally expensive depending on the image size, and therefore a computationally efficient analytical framework has been developed. Using cubic B-splines as the registration transform, we develop a generalized mathematical framework that supports five distinct regularizers: diffusion, curvature, linear elastic, third-order, and total displacement. We validate our approach by comparing each with its numerical counterpart in terms of accuracy. We also provide benchmarking results showing that the analytic solutions run significantly faster -- up to two orders of magnitude -- than finite differencing based numerical implementations.
We construct a family of monotone and convex $C^1$ integro cubic splines under a strictly convex position of the dataset. Then, we find an optimal spline by considering its approximation properties. Finally, we give some examples to illustrate the convex-preserving properties of these splines.
We prove quartic convergence of cubic spline interpolation for curves into Riemannian manifolds as the grid size of the interpolation grid tends to zero. In contrast to cubic spline interpolation in Euclidean space, where this result is classical, the interpolation operator is no longer linear. Still, concepts from the linear setting may be generalized to the Riemannian case, where we try to use intrinsic Riemannian formulations and avoid charts as much as possible.
Spline interpolation has been used in several applications due to its favorable properties regarding smoothness and accuracy of the interpolant. However, when there exists a discontinuity or a steep gradient in the data, some artifacts can appear due to the Gibbs phenomenon. Also, preservation of data monotonicity is a requirement in some applications, and that property is not automatically verified by the interpolator. In this paper, we study sufficient conditions to obtain monotone cubic splines based on Hermite cubic interpolators and propose different ways to construct them using non-linea…
Isogeometric analysis with $C^1$ cubic Powell-Sabin splines
Faculty of Mathematics and Physics, University of Ljubljana（卢布尔雅那大学数学与物理学院）；Institute of Mathematics, Physics and Mechanics（数学、物理与力学研究所）；Department of Mathematics, University of Rome Tor Vergata（罗马托维加塔大学数学系）
In this paper I uncover and explain---using contour integrals and residues---a connection between cubic splines and a popular compact finite difference formula. The connection is that on a uniform mesh the simplest Padé scheme for generating fourth-order accurate compact finite differences gives \textsl{exactly} the derivatives at the interior nodes needed to guarantee twice-continuous differentiability for cubic splines. %I found this connection surprising, because the two problems being solved are different. I also introduce an apparently new spline-like interpolant that I call a compact cub…
In the classical theory of cubic interpolation splines there exists an algorithm which works with only $O\left( n\right)$ arithmetic operations. Also, the smoothing cubic splines may be computed via the algorithm of Reinsch which reduces their computation to interpolation cubic splines and also performs with $O\left( n\right)$ arithmetic operations. In this paper it is shown that many features of the polynomial cubic spline setting carry over to the larger class of $L$-splines where $L$ is a linear differential operator of order $4$ with constant coefficients. Criteria are given such that the …
For spaces of constant, linear, and quadratic splines of maximal smoothness on the Powell-Sabin 12-split of a triangle, the so-called S-bases were recently introduced. These are simplex spline bases with B-spline-like properties on the 12-split of a single triangle, which are tied together across triangles in a Bézier-like manner. In this paper we give a formal definition of an S-basis in terms of certain basic properties. We proceed to investigate the existence of S-bases for the aforementioned spaces and additionally the cubic case, resulting in an exhaustive list. From their nature as simpl…
Numerical Solution of two dimensional coupled viscous Burgers Equation using the Modified Cubic B Spline Differential Quadrature Method
DDU Gorakhpur University（德杜戈拉克普尔大学）；ISRO Telemetry, Tracking and Command Network (ISTRAC)（印度空间研究组织遥测、跟踪和指挥网络）；ISRO Satellite Center (ISAC)（印度空间研究组织卫星中心）
This is a summary of articles based on higher order B-splines methods and the variation of B-spline methods such as Quadratic B-spline Finite Elements Method, Exponential Cubic B-Spline Method Septic B-spline Technique, Quintic B-spline Galerkin Method, and B-spline Galerkin Method based on the Quadratic B-spline Galerkin method (QBGM) and Cubic B-spline Galerkin method (CBGM). In this paper we study the B-spline methods and variations of B-spline techniques to find a numerical solution to the Burgers' equation. A set of fundamental definitions including Burgers equation, spline functions, and…
A Multi-step Scheme based on Cubic Spline for solving Backward Stochastic Differential Equations
Lehrstuhl für Angewandte Mathematik und Numerische Analysis, Fakultät für Mathematik und Naturwissenschaften, Bergische Universität Wuppertal（伍珀塔尔大学数学与自然科学学院应用数学与数值分析讲席）
A Numerical Algorithm for C2-splines on Symmetric Spaces
Chalmers and University of Gothenburg（查尔姆斯理工大学和哥德堡大学）；Instituto de Ciencias Matemática（数学科学研究所）；Western Norway University of Applied Sciences（挪威西部应用科学大学）
Explicit pointwise error bounds for the interpolation of a smooth function by piecewise exponential splines of order four are given. Estimates known for cubic splines are extended to a natural class of piecewise exponential splines which are appearing in the construction of multivariate polysplines. The error estimates are derived in an inductive way using error estimates for the interpolation of a smooth function by exponential splines of order two.
On the error bounds of the Gauss-type quadrature formulae associated with spaces of parabolic and cubic spline functions with double equidistant knots
Faculty of Mathematics and Informatics, Sofia University ”St. Kliment Ohridski”（索非亚大学圣克利门特奥赫里斯基数学与信息学院）；Faculty of Pharmacy, Medical University of Sofia（索非亚医科大学药学院）
