Extending the quadratic taxonomy of regularizers for nonparametric registration

Quadratic regularizers are used in nonparametric registration to ensure that the registration problem is well posed and to yield solutions that exhibit certain types of smoothness. Examples of popular quadratic regularizers include the diffusion, elastic, fluid, and curvature regularizers.1 Two important features of these regularizers are whether they account for coupling of the spatial components of the deformation (elastic/fluid do; diffusion/curvature do not) and whether they are robust to initial affine misregistrations (curvature is; diffusion/ elastic/fluid are not). In this article, we show how to extend this list of quadratic regularizers to include a second-order regularizer that exhibits the best of both features: it accounts for coupling of the spatial components of the deformation and contains affine transformations in its kernel. We then show how this extended taxonomy of quadratic regularizers is related to other families of regularizers, including Cachier and Ayache's differential quadratic forms2 and Arigovindan's family of rotationally invariant regularizers.3, 4 Next, we describe two computationally efficient paradigms for performing nonparametric registration with the proposed regularizer, based on Fourier methods5 and on successive Gaussian convolution.6, 7 Finally, we illustrate the performance of the quadratic regularizers on the task of registering serial 3-D CT exams of patients with lung nodules.