Local Analysis of Scale Space

Scale space theory is a framework which permits one to analyze an image at diierent resolutions. The crucial feature is to let the resolution of an image vary continuously rather than discrete. This opens up the toolbox of mathematical analysis for image analysis. When extrema and saddle points of an image are tracked with increasing scale we know that they will have disappeared at suuciently large scale. In this report we investigate what happens when extrema and saddles disappear. It turns out that extrema and saddles also appear for two dimensional images when scale is increased. Two dimensional images are more complicated than one dimensional signals, and some results generalize from two to higher dimensions. We consider two dimensional images as a compromise between generality and simplicity. One generic event is a saddle and an extremum merging and annihilating. Prior to annihilation they approach each other from opposite directions. The other generic event is the creation of a saddle-extremum pair. It will be shown that these are the only generic events. Studying scale space events for images in an image sequence or for a symmetric image, one discovers that more complicated interactions between extrema and saddles take place when scale is increased. The situations which arise when extrema and saddles disappear and appear can be classiied by their codimension. The codimension is a measure of complexity. Generic events are the least complex and have codimension one. We also analyse events of codimension two. The mathematical tool which will be used for computing the local structure of toppoints is diierential geometry of curves. A short version of this report was published in 8].