Medial Spheres for Shape Representation - CIM - McGill University

the shape. Blinn [15] proposed using a truncated Gaussian density field (cf. Figure 1–2).
Simultaneously, Omura proposed ‘metaballs’, where the density is given by a polynomial
function. Wyvill et al. [123] introduced ‘soft objects’, where the density function is a
polynomial expression that is cheaper to evaluate than that of metaballs. To facilitate
modeling shapes that are locally flat, Bloomenthal [16] proposed convolving a piecewise-
planar 2D skeleton with a density function.
These ‘blobby’ representations are generated by artists using modeling tools. They
are particularly appropriate for modeling amorphous objects, such as raindrops, mud and
dough. Objects of high artistic value for computer graphics can be designed using a small
number of primitives. The smoothness properties of these representations provide the ob-
jects with either an organic or a plasticine, cartoon-like appearance. These representations
are an important addition to an artist’s toolbox for shape generation. As with other im-
plicit representations, efficient rendering of such objects is challenging. For additional
information, the reader is referred to a survey in [88].
1.2 Preliminaries
Section 1.2.1 will formally introduce the medial surface transform and its properties.
Its discrete approximation is the ‘union of medial spheres’ shape representation, which is
the topic of this thesis. Section 1.2.2 provides an overview of basic space partitions that
will be referred to throughout the thesis.
1.2.1 Medial Surface Transform
In this thesis, we will propose a shape representation for a shape that is a solid. As a
formal definition of a solid, we follow [2, ch. 5], and say that a subset X of R n is a solid
if the closure of the interior of X is X. In this definition, for example, the union of a cube
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