The authors develop integrated techniques that unify physics based modeling with geometric subdivision methodology and present a scheme for dynamic manipulation of the smooth limit surface generated by the (modified) butterfly scheme using physics based "force" tools. This procedure based surface model obtained through butterfly subdivision does not have a closed form analytic formulation (unlike other well known spline based models), and hence poses challenging problems to incorporate mass and damping distributions, internal deformation energy, forces, and other physical quantities required to develop a physics based model. Our primary contributions to computer graphics and geometric modeling include: (1) a new hierarchical formulation for locally parameterizing the butterfly subdivision surface over its initial control polyhedron, (2) formulation of dynamic butterfly subdivision surface as a set of novel finite elements, and (3) approximation of this new type of finite elements by a collection of existing finite elements subject to implicit geometric constraints. Our new physics based model can be sculpted directly by applying synthesized forces and its equilibrium is characterized by the minimum of a deformation energy subject to the imposed constraints. We demonstrate that this novel dynamic framework not only provides a direct and natural means of manipulating geometric shapes, but also facilitates hierarchical shape and nonrigid motion estimation from large range and volumetric data sets using very few degrees of freedom (control vertices that define the initial polyhedron).

A new dynamic subdivision surface model is proposed for shape recovery from 3D data sets. The model inherits the attractive properties of the Catmull-Clark subdivision scheme and is set in a physics-based modeling paradigm. Unlike other existing methods, our model does not require a parameterized input mesh to recover shapes of arbitrary topology, allows direct manipulation of the limit surface via application of forces and provides a fast, robust, and hierarchical approach to recover complex shapes from 3D data with very few degrees of freedom (control vertices). We provide an analytic formulation and introduce the physical quantities required to develop the dynamic subdivision surface model which can be deformed by applying forces synthesized from the data. Our experiments demonstrate that this new dynamic model has a promising future in shape recovery from volume and range data sets.

Recursive subdivision schemes have been extensively used in computer graphics, computer-aided geometric design, and scientific visualization for modeling smooth surfaces of arbitrary topology. Recursive subdivision generates a visually pleasing smooth surface in the limit from an initial user-specified polygonal mesh through the repeated application of a fixed set of subdivision rules. We present a new dynamic surface model based on the Catmull-Clark subdivision scheme, a popular technique for modeling complicated objects of arbitrary genus. Our new dynamic surface model inherits the attractive properties of the Catmull-Clark subdivision scheme, as well as those of the physics-based models. This new model provides a direct and intuitive means of manipulating geometric shapes, and an efficient hierarchical approach for recovering complex shapes from large range and volume data sets using very few degrees of freedom (control vertices). We provide an analytic formulation and introduce the "physical" quantities required to develop the dynamic subdivision surface model which can be interactively deformed by applying synthesized forces. The governing dynamic differential equation is derived using Lagrangian mechanics and the finite element method. Our experiments demonstrate that this new dynamic model has a promising future in computer graphics, geometric shape design, and scientific visualization.

Recursive subdivision schemes have been extensively used in computer graphics and scientific visualization for modeling smooth surfaces of arbitrary topology. Recursive subdivision generates a visually pleasing smooth surface in the limit from an initial user-specified polygonal mesh through the repeated application of a fixed set of subdivision rules. In this paper, we present a new dynamic surface model based on the Catmull-Clark (1978) subdivision scheme, which is a very popular method to model complicated objects of arbitrary genus because of many of its nice properties. Our new dynamic surface model inherits the attractive properties of the Catmull-Clark subdivision scheme as well as that of the physics-based modeling paradigm. This new model provides a direct and intuitive means of manipulating geometric shapes, a fast, robust and hierarchical approach for recovering complex geometric shapes from range and volume data using very few degrees of freedom (control vertices). We provide an analytic formulation and introduce the physical quantities required to develop the dynamic subdivision surface model which can be interactively deformed by applying synthesized forces in real time. The governing dynamic differential equation is derived using Lagrangian mechanics and a finite element discretization. Our experiments demonstrate that this new dynamic model has a promising future in computer graphics, geometric shape design and scientific visualization.

It is well known that the spatial frequency spectrum of membrane and thin plate splines exhibit self-affine characteristics and, hence, behave as fractals. This behavior was exploited in generating the constrained fractal surfaces, which were generated by using a Gibbs sampler algorithm in the work of Szeliski and Terzopoulos (1989). The algorithm involves locally perturbing a constrained spline surface with white noise until the spline surface reaches an equilibrium state. We introduce a fast generalized Gibbs sampler that combines two novel techniques, namely, a preconditioning technique in a wavelet basis for constraining the splines and a perturbation scheme in which, unlike the traditional Gibbs sampler, all sites (surface nodes) that do not share a common neighbor are updated simultaneously. In addition, we demonstrate the capability to generate arbitrary order fractal surfaces without resorting to blending techniques. Using this fast Gibbs sampler algorithm, we demonstrate the synthesis of realistic terrain models from sparse elevation data.

It is well known that the spatial frequency spectra of membrane and thin-plate splines exhibit self-affine characteristics and hence behave as fractals. This behavior was exploited in generating the constrained fractal surfaces in the work of Szeliski and Terzopoulos (1989), which were generated by using a Gibbs sampler algorithm. The algorithm involves locally perturbing a constrained spline surface with white noise until the spline surface reaches an equilibrium state. In this paper, we introduce a very fast generalized Gibbs sampler that combines two novel techniques, namely a preconditioning technique in a wavelet basis for constraining the splines and a perturbation scheme in which, unlike the traditional Gibbs sampler, all sites (surface nodes) that do not share a common neighbor are updated simultaneously. In addition, we demonstrate the capability to generate arbitrary-order fractal surfaces without resorting to blending techniques. Using this fast Gibbs sampler algorithm, we demonstrate the synthesis of realistic terrain models from sparse elevation data.