Coding shape inside the shape

The shape of an object lies at the interface between vision and cognition, yet the field of statistical shape analysis is far from developing a general mathematical model to represent shapes that would allow computational descriptions to express some simple tasks that are carried out robustly and e↵ortlessly by humans. In this thesis, novel perspectives on shape characterization are presented where the shape information is encoded inside the shape. The representation is free from the dimensions of the shape, hence the model is readily extendable to any shape embedding dimensions (i.e 2D, 3D, 4D). A very desirable property is that the representation possesses the possibility to fuse shape information with other types of information available inside the shape domain, an example would be reflectance information from an optical camera. Three novel fields are proposed within the scope of the thesis, namely ‘Scalable Fluctuating Distance Fields’, ‘Screened Poisson Hyperfields’, ‘Local Convexity Encoding Fields’, which are smooth fields that are obtained by encoding desired shape information. ‘Scalable Fluctuating Distance Fields’, that encode parts explicitly, is presented as an interactive tool for tumor protrusion segmentation and as an underlying representation for tumor follow-up analysis. Secondly, ‘Screened Poisson Hyper-Fields’, provide a rich characterization of the shape that encodes global, local, interior and boundary interactions. Low-dimensional embeddings of the hyper-fields are employed to address problems of shape partitioning, 2D shape classification and 3D non-rigid shape retrieval. Moreover, the embeddings are used to translate the shape matching problem into an image matching problem, utilizing existing arsenal of image matching tools that could not be utilized in shape matching before. Finally, the ‘Local Convexity Encoding Fields’ is formed by encoding information related to local symmetry and local convexity-concavity properties. The representation performance of the shape fields is presented both qualitatively and quantitatively. The descriptors obtained using the regional encoding perspective outperform existing state-of-the-art shape retrieval methods over public benchmark databases, which is highly motivating for further study of regional-volumetric shape representations.