The deep structure of Gaussian scale space images

Abstract

In order to be able to deal with the discrete nature of images in a continuous way, one can use results of the mathematical field of 'distribution theory'. Under almost trivial assumptions, like 'we know nothing', one ends up with convolving the image with a Gaussian filter. In this manner scale is introduced by means of the filter's width. The ensemble of the image and its convolved versions at al scales is called a 'Gaussian scale space image'. The filter's main property is that the scale derivative equals the Laplacean of the spatial variables: it is the Greens function of the so-called Heat, or Diffusion, Equation. The investigation of the image all scales simultaneously is called 'deep structure'. In this thesis I focus on the behaviour of the elementary topological items 'spatial critical points' and 'iso-intensity manifolds'. The spatial critical points are traced over scale. Generically they are annihilated and sometimes created pair wise, involving extrema and saddles. The locations of these so-called 'catastrophe events' are calculated with sub-pixel precision. Regarded in the scale space image, these spatial critical points form one-dimensional manifolds, the so-called critical curves. A second type of critical points is formed by the scale space saddles. They are the only possible critical points in the scale space image. At these points the iso-intensity manifolds exhibit special behaviour: they consist of two touching parts, of which one intersects an extremum that is part of the critical curve containing the scale space saddle. This part of the manifold uniquely assigns an area in scale space to this extremum. The remaining part uniquely assigns it to 'other structure'. Since this can be repeated, automatically an algorithm is obtained that reveals the 'hidden' structure present in the scale space image. This topological structure can be hierarchically presented as a binary tree, enabling one to (de-)select parts of it, sweeping out parts, simplify, etc. This structure can easily be projected to the initial image resulting in an uncommitted 'pre-segmentation': a segmentation of the image based on the topological properties without any user-defined parameters or whatsoever. Investigation of non-generic catastrophes shows that symmetries can easily be dealt with. Furthermore, the appearance of creations is shown to be nothing but (instable) protuberances at critical curves. There is also biological inspiration for using a Gaussian scale space, since the visual system seems to use Gaussian-like filters: we are able of seeing and interpreting multi-scale.