This cumulative thesis presents research from five selected publications in the field of topology preservation in images using irregular graph pyramids. Preserving the structure and the topology of data is a thorough research problem in the field of image analysis and representation, with ample applications.The thesis introduces a monotonically connected topological subspace called Slope region from previous research. The theoretical contribution of this thesis focuses on the characteristics and generalization of the slope regions. It enumerates the salient features of the slope regions for its identification. From the representation and modeling point of view, the core contribution consists of the origination of the inner boundary (or the folded boundary) and the outer boundary. The inner boundary of the slope region helps to incorporate the holes that are geometrically encapsulated by the outer boundary of the slope region but is topologically excluded from the interior of the slope region. This helps to model the slope region as homeomorphic to a disc. The presence and modeling of holes marks as one of the important distinctions between the proposed slope regions and the previously existing topological subspace.The slope regions are generalized into two prototypes depending on the components of the slope region and irrespective of the geometric features like the size and shape of the regions. From the implementing point of view, the thesis proposes an algorithm to build a hierarchy of the slope region over the 4-neighbourhood Region Adjacency Graph (RAG) of an image. Another core contribution of this thesis deals with the dual of the RAG, its significance, and utilization in merging the adjacent slope regions. The slope region hierarchy decomposes the image into slope regions while preserving its topology. The top level of the hierarchy reveals the structure of the image that consists of the critical points and the connections between them. The results from the implementation suggest that an image is a combination of its structure and a few colors. The proposed hierarchical algorithm has the computational complexity of O(log d). Finally, the last goal of this thesis is to exploit the link between the proposed ingeniously designed algorithm and machine learning. This goal is achieved by deriving an objective function that simplifies a few of the many steps in the proposed algorithm and opens the domain of learning an irregular image pyramid.


Batavia, D. (2022). A walk inside slope region hierarchy [Dissertation, Technische Universit├Ąt Wien]. reposiTUm. https://doi.org/10.34726/hss.2022.104840