Advances in physiology and medical imaging have been essential tools in prognosis, diagnosis and therapy fostering. These advances have provided the impetus for the development of increasingly sophisticated and tightly coupled imaging algorithms and computational biophysical models that target clinical applications. Typical image analysis tasks are segmentation, feature extraction for statistical inference (e.g., outlier detec- tion, population statistics, prognosis), and image registration (for segmentation and surgical planing). Image analysis is essentially a data-assimilation inverse problem that involve nonlinear partial differential equations (PDEs). In our project in cooperation with the ICES (www.ices.utexas.edu) institute at UT Austin, we focus on image registration as a mean to transfer labels for functional brain regions from a healthy statistical brain (the so-called) atlas to an actual patient brain represented by a 3D MRI scan. Both images are represented by probability maps m for each brain tissue type (in our simple model gray matter, white matter, cerebrospinal fluid). For the atlas, additional information on functional units is given that shall be transfered to the actual patient image. Image registration is a correspondence problem. The basic assumption is that there exists a geometric transformation that relates each point in one image, the so called reference image m R (x), to its corresponding point in another image, the so called template image m T (x). We introduce a pseudo-time variable t ∈ [0, 1] and model this geometric transformation based on a transport equation for the intensity values of m T . The forward model of our problem is: Given a stationary velocity field v(x) and a template image m T (x) compute the transported intensities m 1 (x) := m(x, t = 1) at t = 1 by solving ∂ t m + v · ∇m = 0 m = m T in Ω × (0, 1], in Ω × {0}, with periodic boundary conditions on ∂Ω forward in time. This PDE constrained optimization problem requires highly efficient and accurate solver strategies to be feasible within a runtime that is tolerable in clinical settings. One major contribution to efficiency is the use of semi-Lagrangian methods for the advection problem. The latter requires computing backward trajectories in time from fixed grid points along the given velocity field v. This requires spatial interpolation of velocities and tissue probability maps in every time step of the forward convection solver. The current implementation uses a cubic spline interpolation and a RK2 scheme for the trajectory tracing. |