Existing shape models with spherical topology are typically designed either in the discrete domain using interpolating polygon meshes or in the continuous domain using smooth but non-interpolating schemes such as subdivision or NURBS. Both polygon models and subdivision methods require a large number of parameters to model smooth surfaces. NURBS need fewer parameters but have a complicated rational expression and non-uniform shifts in their formulation. We present a new method to construct deformable closed surfaces, which includes exact spheres, by combining the best of two worlds: a smooth, interpolating model with a continuously varying tangent plane and well-defined curvature at every point on the surface.Our formulation is considerably simpler than NURBS and requires fewer parameters than polygon meshes. We demonstrate the generality of our method with applications including intuitive user-interactive shape modeling, continuous surface deformation, shape morphing, reconstruction of shapes from parameterized point clouds, and fast iterative shape optimization for image segmentation. Comparisons with discrete methods and non-interpolating approaches highlight the advantages of our framework.
A B S T R A C TIn applications that involve interactive curve and surface modeling, the intuitive manipulation of shapes is crucial. For instance, user interaction is facilitated if a geometrical object can be manipulated through control points that interpolate the shape itself. Additionally, models for shape representation often need to provide local shape control and they need to be able to reproduce common shape primitives such as ellipsoids, spheres, cylinders, or tori. We present a general framework to construct families of compactly-supported interpolators that are piecewise-exponential polynomial. They can be designed to satisfy regularity constraints of any order and they enable one to build parametric deformable shape models by suitable linear combinations of interpolators. They allow to change the resolution of shapes based on the refinability of B-splines. We illustrate their use on examples to construct shape models that involve curves and surfaces with applications to interactive modeling and character design.
EPFLFigure 1: Smooth modeling of shapes with spherical toplogy. The continuous deformation of the sphere into the Gargoyle is shown in the top row where a wood texture has been added to the surface. The shapes in the bottom row consist of a single surface patch and are constructed through the interactive deformation of the sphere. The interpolating structure of the model allows us to intuitively design surfaces that can adopt shapes beyond the classical spherical topology. Our framework is inherently smooth, which facilitates natural texturing. AbstractExisting shape models with spherical topology are typically designed either in the discrete domain using interpolating polygon meshes or in the continuous domain using smooth but noninterpolating schemes such as NURBS. Polygon models and subdivision methods require a large number of parameters to model smooth surfaces. NURBS need fewer parameters but have a complicated rational expression and non-uniform shifts in their formulation. We present a new method to construct deformable closed surfaces, which includes the exact sphere, by combining the best of two worlds: a smooth and interpolating model with a continuously varying tangent plane and well-defined curvature at every point on the surface. Our formulation is simpler than NURBS while it requires fewer parameters than polygon meshes. We demonstrate the generality of our method with applications ranging from intuitive user-interactive shape modeling, continuous surface deformation, reconstruction of shapes from parameterized point clouds, to fast iterative shape optimization for image segmentation.
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