9. Nonparametric Image Registration
Society for Industrial and Applied Mathematics eBooks · 2009
All ingredients for the nonparametric image registration (NPIR) have been prepared and are ready for use. Chapter 3 addressed solutions to the forward problem: computing the transformed template image for a given transformation y. Chapter 7 provided a general framework for distance measures , quantifying the similarity of the transformed template and reference images. Moreover, examples for commonly used distance measures are provided. Chapter 8 explained the necessity of regularization, introduced a general framework for L2-norm based regularization, and provided examples for commonly used regularizers. Using the optimization framework discussed in Section 2.1, the registration problem becomes one of minimizing the joint functional J [y]=D [T [y],R ]+S [y− yref ], 9.1 where yref enables a bias towards a particular solution, for example yref(x) = x. The obvious difference between the parametric case y(x) = y(w, x), as discussed in Chapter 6, and the nonparametric case is that the transformation is no longer parameterized. Numerical schemes are required to solve both the parametric and the nonparametric problem, and a discretization of the domain is a first step. The regularization for the parametric case can be either neglected (for low-dimensional transformation spaces and at one's own risk; see Example 8.3 as a warning) or transferred to the coefficient space cf. Section 6.5. In contrast, the discretization for NPIR depends on the chosen regularizer. For example, the elastic regularizer is discretized on staggered grids and the curvature regularizer is discretized on cell-centered grids. To bypass this ambiguity and to enable a unified treatment of various grids, a cell-centered grid interpolation P is introduced in Section 9.1.1. Let xc = xch denote a certain discretization of the domain and let yc ≈ y(xc). The discrete version of (9.1) reads J (yc) =D (T (P⋅yc),R (P⋅xc) )+ S (yc−yRef) ; 9.2 the discretizations of and have been discussed in Chapters 7 and 8, respectively.