We review multilevel hierarchies under the special aspect of their potential for segmentation and grouping. The one-to-one correspondence between salient image features and salient model features are a limiting assumption that makes prototypical or generic object recognition impossible. The region's internal properties (color, texture, shape, ...) help to identify them and their external relations (adjacency, inclusion, similarity of properties) are used to build groups of regions having a particular consistent meaning in a more abstract context. Low-level cue image segmentation in a bottom-up way, cannot and should not produce a complete final "good" segmentation. We present a hierarchical partitioning of images using a pairwise similarity function on a graph-based representation of an image. This function measures the difference along the boundary of two components relative to a measure of differences of the components' internal differences. Two components are merged if there is a low-cost connection between them. We use this idea to find region borders quickly and effortlessly in a bottom-up way, based on local differences in a specific feature. The aim of this paper is to build a minimum weight spanning tree (MST) in order to find region borders quickly in a bottom-up 'stimulus-driven' way based on local differences in a specific feature.
It is shown by evaluation against the standard fractal encoding by partitioned IFS that global IFS are suited best because which are often self similar even not always exactly. Global iterated function system (IFS) represent an object by the union of affine contractive transformed copies of the object itself. The objects are decomposed into a minimal set of copies by calculating their touching points (TP) - which have no neighborhood that can be affinely and expansively be mapped to a neighborhood of any other TP - on the object boundary. This boundary is computed by a fractal hull. An affine invariant representation of feature points are mapped to those of the sub objects to calculate their affine transformations. This technique can be generalized to encode assemblies of arbitrary colored objects, using extensions of the IFS-Theory. According to Barnsley's collage theorem not exactly affine self similar objects can be also encoded. Even for worse approximations by an IFS, compression ratios in the range of the standard encoding methods can be reached. For good approximations the compression is even far better.
Burt introduced 1983 'equivalent weighting function': 'Iterative pyramid generation is equivalent to convolving the image g<SUB>0</SUB> with a set of 'equivalent weighting functions' h<SUB>l</SUB>' g<SUB>l</SUB> equals h<SUB>l</SUB> * g<SUB>0</SUB> equals h * g<SUB>l-1</SUB>, l > 1. It allowed him to study the effects of iterated reduction using the single parameter h<SUB>l</SUB> without giving up the efficient iterative computation. A similar concept applies to graph pyramids built by dual graph contraction. This new algorithm reduces the number of vertices and of edges of a pair of dual image graphs while, at he same time, the topological relations among the 'surviving' components are preserved. Repeated application produces a stack of successively smaller graphs: a pari of dual irregular pyramids. The process is controlled by selected decimation parameters which consist of a subset of surviving vertices and associated contraction kernels. These pay a similar role for graph pyramids than the convolution kernels of Gaussian pyramids. Equivalent contraction kernels combine two or more contraction kernels int one single dual contraction. The basic concepts are elaborated and discussed. The new theory opens a large variety of possibilities to explore the domain of 'all' graph pyramids.
Dual graph contraction reduces the number of vertices and of edges of a pair of dual image graphs while at the same time, the topological relations among the `surviving' components are preserved. Repeated application produces a stack of successively smaller graphs: a pair of dual irregular pyramids. The process is controlled by selected decimation parameters which consist of a subset of surviving vertices and associated contraction kernels. Equivalent contraction kernels combine two or more contraction kernels into one single contraction kernel which generates the same result in one single dual contraction. This is the basis to the proof that any segmentation can be represented in one single level of such a pyramid. Experimental results demonstrate the applicability on synthetic and real images respectively.