Shape Spaces from Morphing

In computer graphics, models of three-dimensional shapes are nowadays mainly represented as meshes. A mesh contains a set of vertices describing geometric positions (and other attributes such as color, etc.) and topological information describing edges containing vertices and forming faces. Meshes are universal in the sense that they can represent every shape with arbitrary precision (assuming infinite space to store the description). In many applications one deals not only with one single mesh but with many meshes. The most prominent example are geometric animations, which is typically stored a set of meshes describing the shape over time. We like to exploit the idea of a shape space, where shapes are described as the combination of a few base shapes. Here, base shapes are meshes, and all combinations are meshes. We start exploring this idea by looking at the simple case of only two base meshes. The main idea of this work is to use morphing techniques to generate the family of shapes described as the combination of two base shapes. Morphing techniques are used to generate smooth transitions from one object to another. They have become popular and widespread in the special effects industry but have applications in many areas such as medical imaging and scientific visualization. We can say that a morph represents the family of shapes generated by two base shapes, i.e. the space is one dimensional. By adding a third base shape and morphing between an element of the family resulting from the first two base shapes we add another dimension. This process can be repeated to add any number of dimensions. Such spaces of shapes allow to represent each shape in the space with a vector of scalars not longer than the number of base objects spanning the space. Assuming the number of base shapes is relatively small with respect to the amount of information needed to describe a single shape, this is an extremely compact and meaningful way of describing a shape. Why is the representation meaningful? Imagine a set of faces (smiling, frowning, blinking, staring, etc.) comprising the base of a space. If we want to generate a particular expression we simply describe the face in terms of the features we want. The modeling process is intuitive and simple. In addition, if such a face has to be stored or communicated only the small vector is needed. The major aim of the dissertation is to build spaces of polyhedral objects and demonstrate their usefulness in practical applications. However, at the current state of science even morphing between two polyhedral objects is a difficult process. For that reason, a large part of the work is dedicated to generating morph sequences between two meshes. Potential applications discussed in detail include geometric animations and information visualization.