Spatial Distortion Techniques

Daniel Cohen-Or, David Levin, Amira Solomovici

Ionel Daniel Stroe
CS 563

March 24th, 1999 


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Surface metamorphosis is a continuously transformation of a surface from a source surface, through intermediate surfaces into a designated surface. An example of surface metamorphosis is given below (click image).The method is extremely useful in Computer Graphics to generate the surface of an (imaginary) solid object when only models of some approximations of it are available.


The method performs an object-space metamorphosis between two or more surfaces of solid objects with general topology using an extended distance field interpolation. The metamorphosis of the surface is guided by corresponding control points that define a warp function. The warp function is decomposed into a rigid part and an elastic part in order to reduce the distortion of intermediate models. The rigid transformation serves as an approximation to the overall warp transformation, which yields better control over the path of the in-between shape throughout the metamorphosis.

The method presented here allows an intuitive and simple manual intervention of the animator who can control the animation process by prescribing a small number of anchor points. The advantage of this approach is that no correspondence between the geometric primitives in the model needs to be established. Moreover, the input objects do not necessarily need to have the same representation or the same topological genus, and can have a different number of connectivity components.

Although there is no explicit definition of what the in-between objects should look like, the sequence of shapes should look "natural" to the viewer. There is a variety of ideas of what the sequence of in-between objects should satisfy to appear natural. However, it is rather easy to agree that a morph sequence should satisfy the following criteria to yield a pleasing morph

1. The volume and the boundary surface area of the objects should change smoothly and monotonically.
2. The boundary surface of the objects should retain the smoothness of the original objects.
3. Features common to both source and target objects (e.g., head or legs) should be preserved during the process.

In general, these criteria aim at treating the objects as rigidly as possible and at avoiding redundant global and local deformations. A morph sequence satisfying the preceding criteria probably yields a good metamorphosis, but it still remains difficult to obtain an objective comparison between different sequences, resulting either from different algorithms or from the same algorithm with different parameters.

Distance Field Interpolation

The method can be described in terms of n dimensional surface as a reconstruction from n-1 dimensional cross-sections. Consider a solid object V and a finite set of n-1 dimensional cross-sections of the object:

Now, using univariate interpolation (with respect to the parameter t) we interpolate, between the cross-sections, the distance values of points having the same first n-1 coordinates. The resulting interpolant approximates the n Euclidean distance. Once this approximated distance field is available, the surface of the object can be determined by the zero points of the distance field or, in the discretized version, by the boundary between the positive and negative valued lattice points, whereas the volume itself (its interior) is defined as the set of all negative valued points.

The gain of this method over a classical cross-disolving method is presented below:

Warp Transformations

A warp function is a transformation that "aligns" as much as possible two objects before interpolation. The process is illustrated below:

The quality of the morph is very much subject to a proper warp. That is, it works well provided that "corresponding parts" of the two objects are properly aligned by the warp. Otherwise, some parts may unexpectedly disappear and reappear, exactly as can happen in pure DFI (with no warp). The fact that the warp is guided by the user is probably the weak part of the technique. However, a high-fidelity, eye-pleasing, automatic alignment of arbitrary objects is still a major challenge. The necessity of using warp functions is illustrated in the example below:


The results are quite promising from the perspective of the overall quality and generality. Two of them are presented below (click the appropriate images):


[1]    Cohen-Or, D.; Levin, D; Solomovici, A: "Three-Dimensional Distance Field Metamorphosis",
        ACM Transactions on Graphics, 1998


Some of the figures used in this presentation as well as the movies have been obtained from [1].