Training phase

The four-dimensional feature distribution as sampled from a surface in 3D space is described by a histogram. Each feature $S$ is mapped onto exactly one bin $i$ of the histogram $H(i)$,

$$h: S \mapsto i \in \{1, 2, \ldots, d\}\ ;$$ (9)

$d$ is the number of bins in the histogram. The mapping $h(S)$ is defined by quantizing each of the four feature dimensions in five equal intervals. The resulting number of $d=5^4=625$ bins for the complete histogram is both easy to handle and sufficient for classification. The length dimension $\delta$ [cf. Equation (8)] is normalized to the maximal occurring length $\Delta$. An entry $H(i)$ of the histogram is the normalized frequency of features $S$ that are mapped onto bin $i$,

$$H(i)=\frac{{\rm card}\{S\in{\cal{S}}\vert h(S)=i\}}{{\rm card}{\cal{S}}}\ ,$$ (10)

where ${\cal{S}}$ is the set of all sampled features and card denotes the cardinality of a set.

When working with meshed surfaces, it is a good idea to collect for training all samples from multiple meshes of the same surface. In this way, we incorporate variations introduced by the mesh procedure.

The histogram $H(i)$ together with the maximal length $\Delta$ constitute an object model. The additional information of $\Delta$ is necessary for scaling at recognition time. We store a collection of such models in a database, one for each object we want to recognize.