The Visual Computer (1997) 13:465±478  Springer-Verlag 1997 465 Warp-guided object-space morphing Eyal Carmel 1 , Daniel Cohen-Or 2 1 Department of Mathematics and Computer Science, Ben-Gurion University, Beer Sheva 84015, Israel 2 Computer Science Department, School of Mathematical Sciences, Tel-Aviv University, Ramat-Aviv 69978, Israel, e-mail: daniel@math.tau.ac.ll We present an algorithm that builds a cor- respondence between two arbitrary genus-0 objects and generates a sequence of inbe- tween objects. A warp function deforms the source object and aligns it with the tar- get object. An iterative polygon-evolution algorithm blurs the details of the warped source and target objects into two convex objects with similar shapes that are project- ed to two identical circles. Merging the topologies of the projected objects and re- constructing the original objects results in two objects with identical topologies. A two-part transformation produces the morph sequence. The rigid part moves and rotates the objects to their relative po- sitions. The elastic part establishes the po- sition of each of the vertices forming the inbetween object. Key words: Warp ± Morphing ± Blending ± Metamorphosis Introduction Metamorphosis is defined as a change from one form, shape, or substance into another by any means (Lerios 1990). However, for a computer graphics application, this definition is not suffi- cient to fully describe the requirements of the metamorphosis process. Morphing a source object to a target object involves the creation of inbe- tween objects that gradually change from resem- bling the source object to a blend of the source and target to resemble the target object. Although there is no explicit definition of what the inbe- tween objects should look like, the sequence of shapes should look ªnaturalº to the viewer. There is a variety of ideas of what the sequence of inbetween objects should satisfy to appear natural (Goldstein and Gotsman 1995; Sun et al. 1995). However, it is rather easy to agree that a morph se- quence should satisfy the following criteria to yield a pleasing morph. l The volume and circumference of the objects should change monotonically. l The boundary of the objects should retain the smoothness of the original objects. l Features common to both source and target ob- jects (e.g., head or legs), should be preserved during the process. Generally speaking, these criteria aim at treating the objects as rigidly as possible and at avoiding redundant global and local deformations. A morph sequence satisfying the criteria given probably yields a good metamorphosis, but it still remains difficult to compare different sequences resulting either from different algorithms or from the same algorithm with different parameters. This compar- ison is a subjective impression of the viewer, but there is clearly a consensus that the morph se- quence in Fig. 1a, which violates the criteria, is not acceptable, in contrast to that in Fig. 1b. Morphing algorithms are classified as image-space algorithms (Beier and Neely 1992; Wolberg 1976) and their 3D voxel-space counterparts (He et. al. 1994; Lerios et. al. 1995) and object-space algo- rithms. Object-space methods are commonly ap- plied to objects represented by polygons (Gold- stein and Gotsman 1995; Sederberg and Green- wood 1992; Shapira and Rappoport 1995) or poly- hedra (Sun et al. 1995) [see also the classification by Costa et. al. (1995)]. Polygons and polyhedra are the most popular representation in computer Correspondence to: D. Cohen-Or