Spatial Free Form Deformation with Scattered

The problem of deforming a given spatial shape is treated. There are many examples of applications in visual computing: tting surfaces to sampled data points in space, correction of distortions in tomographic imaging, modeling of free form geometric shapes, and animating metamorphoses of geometric objects. Our solution warps the space surrounding the given shape with the eeect of deforming the embedded shape, too, with a function derived with scattered data interpolation methods from the displacements of a nite set of control points that can be placed arbitrarily and adaptively. We present algorithms implementing this idea on parametric surfaces and rasterized volume data. 1 The Problem Suppose we are given a cloud of points in space obtained by digitizing the surface of a solid, for instance with a laser scanner. One approach to t a surface to the points is to start oo with a crudely approximating surface, for example a rectangle or a sphere, and deform this surface until it ts the given points. In the physical world, the approximating surface might be an elastic skin. Deformation may be achieved by assigning some or all sample points to points on the elastic skin. Then, by pulling the points on the skin towards their corresponding sample points, it hopefully deforms to a natural approximation of the sampled points. A further example is the correction of image distortions caused for instance by the lense of a camera. This problem exists in three dimensions as well, in relation with rasterized volume data acquired by tomographic scanners (CT, MRI, and others). A typical problem is to match data sets of the same patient obtained at diierent times with diierent imaging techniques, or to map a patient's individual image data to a standard anatomical data set. Generally, this requires a true 3D deformation rather than just warping individual slices. The deformation can be deened by assigning corresponding anatomical \landmarks" in the two volume sets involved (Thirion and Gourdon 23]). Then one of the volumes can be deformed along the resulting displacement vectors, like a block of rubber, so that it matches the second volume. Apart from matching, free form deformation of an object can also be quite useful for purposes of geometric modeling. Starting with a simple elastic object, a complex goal object might be modeled by moving selected control points. In computer animation, the process of deformation might be of interest for designing animated …