With the advent of active and passive 3D sensing techniques, 3D geometric data now play vital roles in many applications spanning a broad range of disciplines. Yet, little attention has been given to the fact that real-world objects and scenes consist of geometric structures of varying scales. For instance, a human face has a handful of discriminative local surface structures that span a wide range of spatial extents, such as the forehead, chin, nose, eyes, mouth, nostrils, earlobes, wrinkles, and dimples, from large to small scales. The relative sizes and the spatial configuration of these local structures collectively define the characteristic geometry of the face. In turn, if extracted properly, they add significant information for accurately describing the geometry of the object or scene. The goal of this research program is to establish a general theoretical and computational foundation for analyzing and exploiting this hidden dimension of 3D geometry -- the geometric scale variability. At the heart of the research program are the investigation and derivation of a formal scale-space representation of surface geometry, novel local and global geometric representations that faithfully encode the scale variability, and novel computational methods for leveraging the extra scale-related information in a number of important applications. These key componential research thrusts will individually and collectively enable one to unveil and harness the hidden characteristic properties of 3D geometry.
This research program also focuses on investigating the use of geometric scale variability in a number of fundamental applications, including 3D matching, registration, and recognition, all of which serve as vital building blocks in many other applications that use 3D geometric data. The research will lead to not only more robust and efficient analysis and processing of 3D geometry, but will also enable novel approaches to handling geometry, for instance stitching together range images just like mosaicing intensity images, and set the foundation for novel use of 3D geometric data, for example, in appearance modeling. Due to the ubiquitous use of geometric data, the results are expected to have a significant impact across a broad range of disciplines, especially in nationally and societally vital domains. For instance, it will enable finer analysis of anomalous geometric structures of human organs, such as those recovered with 3D endoscopy, leading to more accurate medical diagnosis; provide rich discriminative information for sorting and matching a large collection of geometric data as often encountered in digital archaeology; and undoubtedly serve as an integral component of any 3D sensing-based surveillance application for homeland security. The use of geometric scale variability can go far beyond these examples, leading to a new paradigm for exploiting 3D geometric data.
