Topology-preserving geometric deformable models (TGDMs) are used to segment objects that have a known topology. Their accuracy is inherently limited, however, by the resolution of the underlying computational grid. Although this can be overcome by using fine-resolution grids, both the computational cost and the size of the resulting contour increase dramatically. In order to maintain computational efficiency and to keep the contour size manageable, we have developed a new framework, termed QTGDMs, for topology-preserving geometric deformable models on balanced quadtree grids (BQGs). In order to do this, definitions and concepts from digital topology on regular grids were extended to BQGs so that characterization of simple points could be made. Other issues critical to the implementation of geometric deformable models are also addressed and a strategy for adapting a BQG during contour evolution is presented. We demonstrate the performance of the QTGDM method using both mathematical phantoms and real medical images.