A class of spline functions for landmark‐based image registration

Abstract

A class of spline functions, called Lobachevsky splines, is proposed for landmark‐based image registration. Analytic expressions of Lobachevsky splines and some of their properties are given, reasoning in the context of probability theory. Because these functions have simple analytic expressions and compact support, landmark‐based transformations can be advantageously defined using them. Numerical results point out accuracy and stability of Lobachevsky splines, comparing them with Gaussians and thin plate splines. Moreover, an application to a real‐life case (cervical X‐ray images) shows the effectiveness of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.

Two and Three Dimensional Partition of Unity Interpolation by Product-Type Functions

“…is symmetric and depends on the choice of n and α in (3). As also these splines turn out to be strictly positive definite for any even n ≥ 2, the interpolation matrixà in (4) is positive definite for any distinct node set.…”

“…[10,17,30]). On the other hand, the latter, firstly considered in probability theory [18,26], have successfully been proposed for multivariate scattered data interpolation and integration in [4,5,6] and for landmark-based image registration in [1,3]. We remark that both of these families of univariate functions (depending on a shape parameter) are compactly supported, strictly positive definite, and enjoy noteworthy theoretical and computational properties, such as the spline convergence to the Gaussian function.…”

Local interpolation schemes for landmark-based image registration: A comparison

“…They were introduced by Lobachevsky in 1842 (see, e.g., [39]) and are identical (up to a scaling factor) to zero centered uniform B-splines (see below). The use of Lobachevsky splines for interpolation and integration of multivariate scattered data was considered in [4,5,6] and for image registration in [1,3].…”

Multidimensional Lobachevsky Spline Integration on Scattered Data

“…In particular, we consider the problem of constructing new integration formulas for high-dimensional scattered data by a class of spline functions, called Lobachevsky splines, arisen in probability theory [17,21] and then also proposed in multivariate interpolation on scattered data [4] and landmark-based image registration [1,3,6]. Lobachevsky splines consist in an infinite sequence of univariate spline functions depending on a shape parameter, which are compactly supported, strictly positive definite, and enjoy noteworthy theoretical and computational properties, such as the convergence to the Gaussian function and the convergence of the sequence of their integrals and derivatives to integrals and derivatives of the Gaussian, respectively (see [4]).…”