Trivariate Functions in Solid Modeling, Medical Imaging and Computer Graphics

Over the past decade, volume graphics has been recognized as an important field of research. Most early research in volume graphics was focused on the problem of rendering of volumetric models. Only recently, volume modeling had been suggested. Volume modeling encompass the tools, data representations and algorithms that are needed to describe and manipulate the inner parts of a model, as well as its outer shell. Volume modeling can applied in wide range of fields such as material sciences, medical imaging, the movie-making industry, etc. As with surface-based geometric modeling, volume modeling seeks to supply the designer with a set of robust, fast and predictable manipulation tools for designing volumetric models. Several alternative representations can be used to model volumetric data. The most common is that of voxels. This representation is well understood and more suitable for volume rendering. However, for modeling purposes, voxels are probably not the best of choices mainly due to their bulkiness. For example, even a moderatesize CT scan may contain 256 × 256 × 256 scalars with a 16 bit-per-voxel resolution. Such a model requires more than 128 MByte of storage and even with today’s powerful computers, interaction with models as large as this can become cumbersome and slow. Moreover, since our aim is to develop tools for volumetric modeling, we should consider the state-of-the-art in surface-based modeling. In computer aided geometric design (CAGD), tensor product B-spline functions are, perhaps, the most commonly used surface representations. Therefore, we consider their extension into three-dimensional parametric space as a data representation for volume modeling. This thesis focuses on tools and algorithms for modeling with trivariate tensor product B-spline functions. These functions are used to solve several geometric modeling and computer graphics problems. We present diverse and novel applications 1 T ec hn io n C om pu te r S ci en ce D ep ar tm en t P h. D . T he si s P H D -2 00 605 20 06 such as placement of deformable objects inside virtual scenes, incision simulation of surface models using trivariate functions and silhouette extraction from volume data where trivariate tensor product B-spline functions are effectively used. In an attempt to make trivariate tensor product B-spline functions a better deformation tool, we also supply bounds on the deformation error of polygonal models inside free-form deformation. Lastly, in order to reduce the increased computational burden of trivariate functions, relative to the B-spline surfaces, we propose fast evaluation techniques that employ programmable graphics hardware. These evaluation techniques are used for real-time model deformation and for adding arbitrary surface details in realtime applications. 2 T ec hn io n C om pu te r S ci en ce D ep ar tm en t P h. D . T he si s P H D -2 00 605 20 06