The world of both natural and man-made objects provides us with an abundance of geometric forms obtained through evolution or design. The shapes of humans, animals, vehicles, furniture, clothes, and organs has been forged by a myriad of forces or considerations related to their functionality, structure, aesthetics, tradition, or materials. Computational disciplines that deal with models of 3D geometry, including computer graphics, computer vision, and robotics have studied shape representations and used them to develop various notions of shape similarity, as needed by particular applications, such as shape search. So far, though, relatively little effort has been spent on trying to quantify differences between the shapes of 3D forms, beyond various notions of distance or "dissimilarity". This makes it difficult to navigate the shape space around a given shape -- as, for example, when one wants to refine a search in specific ways (e.g., "shoes like these, but with higher heels"). It is especially interesting to try to develop low-dimensional parametrizations of shape differences and variability that reflect the underlying semantics of the shapes.
This talk will present some mathematical and algorithmic efforts in this direction, trying to address the many challenges present due to the vast diversity of ways that shapes can be compared in both their geometry and their structure. For example, shape differences can be continuous (e.g., changes in size) or discrete (e.g., addition/removal of parts). In fact, beyond comparing just two shapes, one may want to understand variability across an entire shape collection, or to compare two collections. It can be important to separate different kinds of variability, for some may be nuisance and others may matter (e.g., normal human organ variability vs. various pathologies). Furthermore, variability of form is interesting not only across shapes but even within a single shape, to capture articulations of deformations, often related to shape function. Finally, the correlation between geometry and language when it comes to describing variability is an interesting object of study on its own. The talk, through small vignettes, will aim to throw a bit of light on these topics.
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