A technique is presented for deforming solid geometric models in a free-form manner. The technique can be used with any solid modeling system, such as CSG or B-rep. It can deform surface primitives of any type or degree: planes, quadrics, parametric surface patches, or implicitly defined surfaces, for example. The deformation can be applied either globally or locally. Local deformations can be imposed with any desired degree of derivative continuity. It is also possible to deform a solid model in such a way that its volume is preserved.The scheme is based on trivariate Bernstein polynomials, and provides the designer with an intuitive appreciation for its effects.

Free-form deformation of solid geometric models

The author addresses the problem of parametric representation and estimation of complex planar curves in 2-D surfaces in 3-D, and nonplanar space curves in 3-D. Curves and surfaces can be defined either parametrically or implicitly, with the latter representation used here. A planar curve is the set of zeros of a smooth function of two variables x-y, a surface is the set of zeros of a smooth function of three variables x-y-z, and a space curve is the intersection of two surfaces, which are the set of zeros of two linearly independent smooth functions of three variables x-y-z For example, the surface of a complex object in 3-D can be represented as a subset of a single implicit surface, with similar results for planar and space curves. It is shown how this unified representation can be used for object recognition, object position estimation, and segmentation of objects into meaningful subobjects, that is, the detection of 'interest regions' that are more complex than high curvature regions and, hence, more useful as features for object recognition. >

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Estimation of planar curves, surfaces, and nonplanar space curves defined by implicit equations with applications to edge and range image segmentation

Constraints on deformable models: recovering 3D shape and nongrid motion

New hierarchical solid modeling operations are developed, which simulate twisting, bending, tapering, or similar transformations of geometric objects The chief result is that the normal vector of an arbitrarily deformed smooth surface can be calculated directly from the surface normal vector of the undeformed surface and a transformation matrix Deformations are easily combined in a hierarchical structure, creating complex objects from simpler ones The position vectors and normal vectors in the simpler objects are used to calculate the position and normal vectors in the more complex forms; each level in the deformation hierarchy requires an additional matrix multiply for the normal vector calculation Deformations are important and highly intuitive operations which ease the control and rendering of large families of three-dimensional geometric shapes

Global and local deformations of solid primitives

The authors present a physically based approach to fitting complex three-dimensional shapes using a novel class of dynamic models that can deform both locally and globally. They formulate the deformable superquadrics which incorporate the global shape parameters of a conventional superellipsoid with the local degrees of freedom of a spline. The model's six global deformational degrees of freedom capture gross shape features from visual data and provide salient part descriptors for efficient indexing into a database of stored models. The local deformation parameters reconstruct the details of complex shapes that the global abstraction misses. The equations of motion which govern the behavior of deformable superquadrics make them responsive to externally applied forces. The authors fit models to visual data by transforming the data into forces and simulating the equations of motion through time to adjust the translational, rotational, and deformational degrees of freedom of the models. Model fitting experiments involving 2D monocular image data and 3D range data are presented. >

Dynamic 3D models with local and global deformations: deformable superquadrics

A new and powerful family of parametric shapes extends the basic quadric surfaces and solids, yielding a variety of useful forms.

/pdf/superquadrics-and-angle-preserving-transformations-52yahad6yj.pdf

Barr

Papers

Superquadrics and Angle-Preserving Transformations