Non-Rigid Image Registration using a Hierarchical Partition of Unity Finite Element Method

Abstract

We use a Hierarchical Partition of Unity Finite Element Method (H-PUFEM) to represent and analyse the non-rigid deformation fields involved in multidimensional image registration. We make use of the Ritz-Galerkin direct variational method to solve non-rigid image registration problems with various deformation constraints. In this method, we directly seek a set of parameters that minimizes the objective function. We thereby avoid the loss of information that may occur when an Euler-Lagrange formulation is used. Experiments are conducted to demonstrate the advantages of our approach when registering synthetic images having little of or no localizing features. As a special case, conformal mapping problems can be accurately solved in this manner. We also illustrate our approach with an application to Cardiac Magnetic Resonance temporal sequences.

A Novel Consistency Regularizer for Meshless Nonrigid Image Registration

“…Secondly, we introduce a new functional to penalize for inconsistencies between local deformation fields (Section 3). This regularizer greatly simplifies the registration process compared to the classic regularizer based on Sobolev norm [1], or to the non-conformity measure proposed in [6]. Additionally, our regularizer is not biased towards certain lower-order deformations.…”

“…The proposed functional can be minimized hierarchically at varying scales. Unlike previous methods, where a coarse-scale deformation field is only used as an intermediate result [1,6], our method is able to regularize the deformation field by combining image evidence at different scales simultaneously. Finally, these contributions are supported by a number of experiments performed on both synthetic and real images (Section 4).…”

“…Recently, many meshless models (i.e., no explicit control-point connections) have been proposed in different areas [4,5,3,6,7]. In this paper, we will focus ourselves on the problem of nonrigid image registration while also drawing from the results in other areas.…”

“…For example, polynomial-based moving least-squares (MLS) method has been used for heart-motion analysis [7], and prostate image registration [9]. Most notably, Makram-Ebeid et al [6] proposed a nonrigid image registration method based on a partition-of-unity finite-element method (PUFE). However, these approaches inherit heavily from meshless methods developed for mechanical engineering, and essentially solve mechanical PDEs using image evidence as boundary conditions.…”

Prior-Based Piecewise-Smooth Segmentation by Template Competitive Deformation Using Partitions of Unity

“…A detailed description of the minimization scheme is out of the scope of this paper and can be found in [28].…”