The goal of this research is the investigation of a computational framework that allows for group-wise joint segmentation, smoothing and registration of images and shapes. With recent advances in sensor technology, images (shapes and other types of data) are being generated in abundance, and there is now a need for algorithms that operate and process images in collections instead of individually. In particular, this requires segmentation, smoothing and registration, three most important image processing operations, to be formulated in new ways that emphasize the relational aspects of their inputs. In addition, image data in computer vision applications are usually sampled from low-dimensional manifolds embedded in high-dimensional features spaces, and an important problem is to construct versatile and expressive computational models that exploit their geometries for solutions. The proposed computational framework addresses these two issues by formulating a variational framework that unifies smoothing, segmentation and registration. Specifically, it uses hypergraphs to model the multiple geometric relations among the inputs, and the three operations are integrated in one single discrete variational framework defined over a hypergraph. The proposed framework provides a foundation for several principled joint segmentation and registration algorithms for images and shapes that can guarantee crucial properties such as compatibility, consistency, unbiasedness and symmetry. Furthermore, it also provides a new and more discriminative numerical signature for 2D and 3D shapes that can be important for many shape-related vision applications such as shape recognition, shape retrieval and image-based medical diagnosis.
