1 Image Analysis and Computer Graphics, Department of Informatics and Mathematical Modeling, Technical University of Denmark2 Department of Informatics and Mathematical Modeling, Technical University of Denmark3 DTU Data Analysis, Department of Informatics and Mathematical Modeling, Technical University of Denmark4 Department of Applied Mathematics and Computer Science, Technical University of Denmark
The work presented here consists of contributions in three areas. An efficient algorithm for calculating the entire regularization path of the support vector domain description (SVDD) is presented. The ability to calculate the entire path with a complexity in the same order as solving the original quadratic problems gives inspiration to utilize the extra information available from the entire path. A method for hierarchical support vector clustering, based on information from the entire regularization path, and multiple Gaussian kernels is described. Bayesian methods are applied in the attempt to draw direct statistical conclusions from the SVDD analysis. In the context of image registration, different assumptions on the warp fields, namely diffeomorphism and a linear elastic potential in the form of regularization are discussed. A new warp representation which allows statistical analysis on an unrestricted linear parameter space, where all derivatives are defined, is introduced. Furthermore, it is shown that L2-norm the parameter space introduces a reasonable metric in the actual space of modelled diffeomorphisms. A new parametrization of 3D deformation fields, using potentials and Helmholtz decomposition is also presented. The representation can be considered a natural parametrization for both elastic and fluid image registration due to the decoupling of the parameters. The determinant gradient field is shown to be the first-order small-deformation approximation to the determinant of the Jacobian matrix. Spline approximations of functions and in particular image registration warp fields are discussed. It is shown how spline bases may be learned from the optimization process, i.e. image registration optimization, and how this may contribute with a reasonable prior, or regularization in the method. A new formula, based on the multivariate divided difference, for explicit calculation of the simplex splines is presented. The formula additionally admits easy calculation of derivatives, both spatial, and with respect to the position of the knots. It is demonstrated that conditions may be set on the knot movements, which ensures that the splines form a partition of unity, even if the knots are not Delaunay. A subdivision scheme is also presented, which requires no recalculation of the configurations of the splines. The use of the splines for image registration is demonstrated, and the inherent smoothing or averaging cost, of selecting warp parameterizations at a specific kernel resolution, has been analyzed. A refinement measure has been derived, which is shown to be efficient for guiding the local mesh layout. With the combination of the refinement measure and the local flexibility of the multivariate B-splines, the warp field is automatically refined in areas where it results in the minimization of the registration cost function.