Computing Topology Preservation of RBF Transformations for Landmark-Based Image Registration

Abstract

Abstract. In image registration, a proper transformation should be topology preserving. Especially for landmark-based image registration, if the displacement of one landmark is larger enough than those of neighbourhood landmarks, topology violation will be occurred. This paper aim to analyse the topology preservation of some Radial Basis Functions (RBFs) which are used to model deformations in image registration. Matérn functions are quite common in the statistic literature (see, e.g. [9,12]). In this paper, we use them to solve the landmark-based image registration problem. We present the topology preservation properties of RBFs in one landmark and four landmarks model respectively. Numerical results of three kinds of Matérn transformations are compared with results of Gaussian, Wendland's, and Wu's functions.

Topology analysis of global and local RBF transformations for image registration

“…Nevertheless, ψ 3,1 is better than τ 2,7/2 and τ 2,5 because the optimal locality parameter is smaller. Some numerical results about topology preservation in this case can be found in [5,10]. We observe that if locality parameter c is much smaller than the optimal one, the deformed image will deeply be misrepresented above all around the shifted point.…”

“…To preserve topology, the necessary condition is the continuity of the function Φ and the positivity of the determinant of Jacobian matrix generated by F at each point. In this section, we briefly recall some results contained in [5,10]. We consider the situation of one landmark matching where the source landmark p is shifted by ∆ x along the x-axis direction and by ∆ y along the y-axis direction to the target landmark q.…”

“…From (10) we deduce that the optimal parameter c for the Gneiting function τ 2,5 is obtained by finding the solution of the equation ( 15), provided that…”

“…In this paper, we evaluate the topology preservation of Gneiting and Matérn functions. Although Matérn functions are well known in the statistics literature [13,25,34], as far as we know, excepting [10], there is no work focuses on applying them to image registration. We study and evaluate the properties of such functions in topology preservation based on a small number of landmarks.…”

On the topology preservation of Gneiting’s functions in image registration

“…Locality parameter is the optimal support size of different CSRBFs under topology preservation condition. In [19], we instead evaluated the performances of topology preservation for a GSRBF family such as Matérn functions in landmark-based image registration.…”