Geometric matching problems are among the most intensely studied fields in Computational Geometry. A geometric matching problem can be formulated as follows: given are two geometric objects P and Q. These objects are taken from a class of geometric objects G and P is called the "pattern" and Q is called the "model". A geometric matching instance is defined for a distance measure dist and a transformation class T. The task is to find the transformations t of T that minimizes dist(t(P),Q).
In this thesis, geometric hybrid registration problems are studied. Registration problems are closely related to geometric matching problems. The term geometric registration problem describes the task of mapping points from one space ("pattern space") to their corresponding points in a deformed copy of that space called "model space".
This research is motivated by a real world application: navigated surgery. Here, the goal is to register an operation theatre space (pattern space) to the internal coordinate system (model space) of a medical navigation system. The purpose of a medical navigation system is to support surgeons by visualizing the used surgical instruments at their correct position in a 3D-model of a patient. The models are generated beforehand based on CT or MRT scans.
Hybrid registration is a novel strategy to compute solutions for this alignment problem. Geometric hybrid registrations reduce the spatial synchronization problem to a series of (at least two) geometric matching problems that are solved interdependently. Usually, a computationally involved point-to-surface matching is combined with a comparably simpler but underdefined point-to-point matching. The point-to-surface matching is computed for a sufficiently large and suitably distributed set of points (called surface points) measured in the pattern space to a geometric surface in the model space. For the point-to-point matching, a small set of (one to three) characteristic points are measured in the pattern space and are defined in the model space. In the context of the intended application, these points are called anatomic landmarks - anatomically exposed spots within the field of interest.