| Abstract: |
In this paper we consider the problem of updating principal components of a point set in $\mathbb{R}^d$ when points are added or deleted from the point set. A recent result of \cite{Pebay-08} implies an efficient solution for that problem when points are added to a discrete point set. Here, we extend that result for deletions in the discrete case, and for both additions and deletions for continuous point sets in $\mathbb{R}^2$ and $\mathbb{R}^3$. In both cases, discrete and continuous, no additional data structures or storage are needed for computing the new principal components. An important application of the above results is the dynamical computation of bounding boxes based on principal component analysis. PCA bounding boxes are very often used in many fields, among others in computer graphics, for example, for ray tracing, fast rendering, collision detection, or video compression algorithms. Since some version of PCA bounding boxes have guaranties on their size (volume), they are also of interest in applications where the guaranty of the approximation quality is required. We have designed and implemented algorithms for computing dynamically PCA bounding boxes in $\mathbb{R}^3$. |