  Traditionally, shape transformation using implicit functions is performed in
two distinct steps: 1) creating two implicit functions, and 2) interpolating
between these two functions. We present a new shape transformation method that
combines these two tasks into a single step. We create a transformation between
two N-dimensional objects by casting this as a scattered data interpolation
problem in N + 1 dimensions. For the case of 2D shapes, we place all of our
data constraints within two planes, one for each shape. These planes are placed
parallel to one another in 3D. Zero-valued constraints specify the locations of
shape boundaries and positive-valued constraints are placed along the normal
direction in towards the center of the shape. We then invoke a variational
interpolation technique (the 3D generalization of thin-plate interpolation),
and this yields a single implicit function in 3D. Intermediate shapes are
simply the zero-valued contours of 2D slices through this 3D function. Shape
transformation between 3D shapes can be performed similarly by solving a 4D
interpolation problem. To our knowledge, ours is the first shape transformation
method to unify the tasks of implicit function creation and interpolation. The
transformations produced by this method appear smooth and natural, even between
objects of differing topologies. If desired, one or more additional shapes may
be introduced that influence the intermediate shapes in a sequence. Our method
can also reconstruct surfaces from multiple slices that are not restricted to
being parallel to one another.
