Measuring 3-D shape similarity using progressive transformations

Abstract We present a quantitative approach to the measurement of shape similarity among 3-D (threedimensional) objects. Using voxels, an object is mapped to a representation invariant under translation and rotation. The different objects to be compared are normalized to have the same amount of information (equal number of voxels) and this is termed invariance under volume. When the different objects to be compared are normalized under translation, rotation and volume, a quantity of work (from a physics point of view) is performed that transforms an object O1 into object O2 (the transformation of an object into another is performed moving voxels, as if they were bricks). Voxels to move are selected so as to minimize the work involved. The work done by transforming O1 into O2 is the measure of dissimilarity between them. Dissimilar objects will have a large quantity of work done to transform one into other, while analogous objects will have a small quantity of work done. When two objects are identical, the quantity of work done is zero. Thus, the distance or shape dissimilarity between two objects can be defined as the amount of work needed to convert one into another. Informally, if two objects to be compared consist of bricks, their shape difference could be ascertained by counting how many bricks we have to move and how far to change one object into another.