Image hierarchy in gaussian scale space

We investigate the topological structure of an image and the hierarchical relationship between local and global structures provided by spatial gradients at different levels of scale in the Gaussian scale space. The gradient field curves link stationary points of an image, including a local minimum at infinity, and construct the topological structure of the image. The evolution of the topological structure with respect to scale is analyzed using pseudograph representation. The hierarchical relationships among the structures at different scales are expressed as trajectories of the stationary points in the scale space, which we call the stationary curves. Each top point of the local extremum curve generically has a specific gradient field curve, which we call the antidirectional figure-flow curve. The antidirectional figure-flow curve connects the top-point and another local extremum to which the toppoint is subordinate. A point at infinity can also be connected to the top points of local minimum curves. These hierarchical relationships among the stationary points are expressed as a tree. This tree expresses a hierarchical structure of dominant parts. We clarify the graphical grammar for the construction of this tree in the Gaussian scale space. Furthermore, we show a combinatorial structure of singular points in the Gaussian scale space using conformal mapping from Euclidean space to the spherical surface. We define segment edges as a zero-crossing set in the Gaussian scale space using the singular points. An image in the Gaussian scale space is the convolution of the image and the Gaussian kernel. The Gaussian kernel of an appropriate variance is a typical presmoothing operator for segmentation. The variance is heuristically selected using statistics of images such as the noise distribution in images. The variance of the kernel is determined using the singular-point configuration in the Gaussian scale space, since singular points in the Gaussian scale space allow the extraction of the dominant parts of an image. This scale-selection strategy derives the hierarchical structure of the segments. Unsupervised segmentation methods, however, have difficulty in distinguishing valid segments associated with the objects from invalid random segments due to noise. By showing that the number of invalid segments monotonically decreases with increasing scale, we characterize the valid and invalid segments in the Gaussian scale space. This property allows us to identify the valid segments from coarse to fine and does us to prevent undersegmentation and oversegmentation. Finally, we develop principal component analysis (PCA) of a point cloud on the basis of the scale-space representation of its probability density function. We explain the geometric features of a point cloud in the Gaussian scale space and observe reduced dimensionality with respect to the loss of information. Furthermore, we introduce a hierarchical clustering of the point cloud and analyze the statistical significance of the clusters and their subspaces. Moreover, we present a mathematical framework of the scale-based PCA, which derives a statistically reasonable criterion for choosing the number of components to retain or reduce the dimensionality of a point cloud. Finally, we also develop a segmentation algorithm using configurations of singular points in the Gaussian scale space

The deep structure of Gaussian scale space images

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