Image Hierarchy in Gaussian Scale Space

Abstract

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 analysed 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 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 between the top-point and another local extremum to which the toppoint is subordinate. A point at infinity can also be connected to the toppoints 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. Showing that the number of invalid segments monotonically decreases with increasing scale, we characterise thevalid and invalid segments in the Gaussian scale space. This property allows us to identify the valid segments from coarse to fine, and to prevent under-and oversegmentation. Finally, we develop principal component analysis (PCA) of a point cloud on the basis of scale-space representation of its probability density function (PDF). We explain the geometric features of a point cloud in the Gaussian scale space, and observe reduction of dimensionality with respect to the loss of information. Furthermore, we introduce a hierarchical clustering of the point cloud, and analyse the statistical significance of the clusters and their subspaces. Moreover, ...

Curvature and Flow in Digital Space

“…A unified treatment of shape deformation is required for the intelligent editing of image contents for multimedia technology. The deformation of image data based on curvature flow and diffusion processes [17,18,29,21] on surfaces [13,22] provides a mathematical foundation for the unified treatment of deformation [23].…”