Deformable Medical Image Registration: A Survey <s> 4) Curvature Registration: <s> This paper offers a new fast algorithm for non-rigid Viscous Fluid Registration of medical images that is at least an order of magnitude faster than the previous method by Christensen et al. [4]. The core algorithm in the fluid registration method is based on a linear elastic deformation of the velocity field of the fluid. Using the linearity of this deformation we derive a convolution filter which we use in a scalespace framework. We also demonstrate that the ’demon’-based registration method of Thirion [13] can be seen as an approximation to the fluid registration method and point to possible problems. <s> BIB001 </s> Deformable Medical Image Registration: A Survey <s> 4) Curvature Registration: <s> 1. Introduction 2. The Human Neuroscanning Project 3. The mathematical setting I PARAMETRIC IMAGE REGISTRATION 4. Landmark based registration 5. Principal axes based registration 6. Optimal linear registration 7. Summarizing parametric image registration II NON-PARAMETRIC IMAGE REGISTRATION 8. Non-parametric image registration 9. Elastic registration 10. Fluid registration 11. Diffusion registration 12. Curvature registration 13. Concluding remarks <s> BIB002 </s> Deformable Medical Image Registration: A Survey <s> 4) Curvature Registration: <s> A fully automated, non-rigid image registration algorithm is presented. The deformation field is found by minimizing a suitable measure subject to a curvature based constraint. It is a well-known fact that non-rigid image registration techniques may converge poorly if the initial position is not sufficiently near to the solution. A common approach to address this problem is to perform a time consuming rigid pre-registration step. In this paper we show that the new curvature registration not only produces accurate and smooth solutions but also allows for an automatic rigid alignment. Thus, in contrast to other popular registration schemes, the new method no longer requires a pre-registration step. Furthermore, we present an implementation of the new scheme based on the numerical solution of the underlying Euler-Lagrange equations. The real discrete cosine transform is the backbone of our implementation and leads to a stable and fast {\cal O}(N log N) algorithm, where N denotes the number of voxels. Finally, we report on some numerical test runs. <s> BIB003 </s> Deformable Medical Image Registration: A Survey <s> 4) Curvature Registration: <s> Abstract Image registration is central to many challenges in medical imaging today. It has a vast range of applications. The purpose of this note is twofold. First, we review some of the most promising non-linear registration strategies currently used in medical imaging. We show that all these techniques may be phrased in terms of a variational problem and allow for a unified treatment. Second, we introduce, within the variational framework, a new non-linear registration model based on a curvature type smoother. We show that affine linear transformations belong to the kernel of this regularizer. As a result, the approach becomes more robust against poor initializations of a pre-registration step. Furthermore, we develop a stable and fast implementation of the new scheme based on a real discrete cosine transformation. We demonstrate the advantages of the new technique for synthetic data sets and present an application of the algorithm for registering MR-mammography images. <s> BIB004 </s> Deformable Medical Image Registration: A Survey <s> 4) Curvature Registration: <s> Image registration, i.e., finding an optimal displacement field u which minimizes a distance functional D(u) is known to be an ill-posed problem. In this paper a novel variational image registration method is presented, which matches two images acquired from the same or from different medical imaging modalities. The approach proposed here is also independent of the image dimension. The proposed variational penalty against oscillations in the solutions is the standard H2(?) Sobolev semi-inner product for each component of the displacement. We investigate the associated Euler-Lagrange equation of the energy functional. Furthermore, we approach the solution of the underlying system of biharmonic differential equations with higher order boundary conditions as the steady-state solution of a parabolic partial differential equation (PDE). ::: ::: One of the important aspects of this approach is that the kernel of the Euler-Lagrange equation is spanned by all rigid motions. Hence, the presented approach includes a rigid alignment. Experimental results on both synthetic and real images are presented to illustrate the capabilities of the proposed approach. <s> BIB005 </s> Deformable Medical Image Registration: A Survey <s> 4) Curvature Registration: <s> Labeling of discrete Markov Random Fields (MRFs) has become an attractive approach for solving the problem of non-rigid image registration. Here, regularization plays an important role in order to obtain smooth deformations for the inherent ill-posed problem. Smoothness is achieved by penalizing the derivatives of the displacement field. However, efficient optimization strategies (based on iterative graph-cuts) are only available for second-order MRFs which contain cliques of size up to two. Higher-order cliques require graph modifications and insertion of auxiliary nodes, while pairwise interactions actually allow only regularization based on the first-order derivatives. In this paper, we propose an approximated curvature penalty using second-order derivatives defined on the MRF pairwise potentials. In our experiments, we demonstrate that our approximated term has similar properties as higher-order approaches (invariance to linear transformations), while the computational efficiency of pairwise models is preserved. <s> BIB006 </s> Deformable Medical Image Registration: A Survey <s> 4) Curvature Registration: <s> Non-parametric image registration is still among the most challenging problems in both computer vision and medical imaging. Here, one tries to minimize a joint functional that is comprised of a similarity measure and a regularizer in order to obtain a reasonable displacement field that transforms one image to the other. A common way to solve this problem is to formulate a necessary condition for an optimizer, which in turn leads to a system of partial differential equations (PDEs). In general, the most time consuming part of the registration task is to find a numerical solution for such a system. In this paper, we present a generalized and efficient numerical scheme for solving such PDEs simply by applying 1-dimensional recursive filtering to the right hand side of the system based on the Green's function of the differential operator that corresponds to the chosen regularizer. So in the end we come up with a general linear algorithm. We present the associated Green's function for the diffusive and curvature regularizers and show how one may efficiently implement the whole process by using recursive filter approximation. Finally, we demonstrate the capability of the proposed method on realistic examples. <s> BIB007
In this case, the deformation is modeled by the following equilibrium equation: BIB002 This regularization scheme does not penalize affine linear transformations. As a consequence, unless an initial significant misalignment in space is present, these registration frameworks do not necessarily require an additional affine linear preregistration step. Fischer and Modersitzki used this constraint in BIB003 , BIB004 . To solve (5), the Gâteaux derivatives with respect to the data and regularization terms were calculated and a finite difference scheme was employed to solve the resulting PDE. Neumann boundary conditions were used since they result in a highly structured matrix problem that can be solved efficiently. Despite this fact, the resulting underlying function space penalizes the affine linear displacements as pointed out by Henn in BIB005 . Thus, Henn proposed to include second-order terms as boundary conditions in the energy and applied a semi-implicit time discretization scheme to solve the full curvature registration problem. Glocker et al. BIB006 used an approximation of the curvature penalty in the case of parametric grid-based deformation models. The approximation was derived by simultaneously examining the displacements of two neighboring grid nodes while the third was assumed to be fixed. Beuthien et al. BIB007 , inspired by the approach presented in BIB001 for the viscous fluid registration scenario, proposed another way to solve the curvature based registration problem. Instead of devising a numerical scheme to solve the PDE that results from the equilibrium equa-tion (5), recursive convolutions with an appropriate Green's function were used.