Many image analysis tasks lead to or make use of graph structures that are related through the analysis process with the planar layout of a digital image. This paper presents a theory that allows to build different types of hierarchies on top of such image graphs. The theory is based on the properties of a pair of dual image graphs that the reduction process should preserve, e.g. the structure of a particular input graph. The reduction process is controlled by decimation parameters, i.e. a selected subset of vertices, called survivors, and a selected subset of the graph's edges, the parent-child connections. It is formally shown that two phases of contractions transform a dual image graph to a dual image graph built by the surviving vertices. Phase one operates on the original (neighborhood) graph and eliminates all non-surviving vertices. Phase two operates on the dual (face) graph and eliminates all degenerated faces that have been created in phase one. The resulting graph preserves the structure of the survivors, it is minimal and unique with respect to the selected decimation parameters. The result is compared with two modified specifications, the one already in use for building stochastic and adaptive irregular pyramids.

A new approach toward image segmentation is proposed. A set of slightly different segmentations are derived from the same input and the final result is based on the consensus among them. The perturbations are introduced by exploiting the probabilistic component of a region adjacency graph (RAG) pyramid based segmentation. From the set of initial segmentations the cooccurrence probability field is obtained in which global information about the delineated regions becomes locally available. The final segmentation is based on this field and is obtained with the same hierarchical, RAG pyramid technique. No user set parameters or context dependent thresholds are required. Keywords: Image Segmentation, Integration of Modules, Low-Level Processing. 3 Current address: Open Solution Center, Samsung Data System, 219-1 Migun-Dong, Seodaemun-Gu, Seoul, Korea. 1 Introduction In image segmentation, given a homogeneity criterion, the image must be partitioned into regions within which the criterio...

We present a new method to determine contraction kernels for the construction of graph pyramids. The new method works with undirected graphs and yields a reduction factor of at least 2:0. This means that with our method the number of vertices in the subgraph induced by any set of contractible edges is reduced to half or less by a single parallel contraction. Our method yields better reduction factors than the stochastic decimation algorithm, in all tests. The lower bound of the reduction factor becomes crucial with large images.

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. This definition tries to encapsulate the intuitive notion of contrast. Two components are merged if there is a low-cost connection between them. Each component's internal difference is represented by the maximum edge weight of its minimum spanning tree. External differences are the smallest weight of edges connecting components. We use this idea for building a minimum spanning tree to find region borders quickly and effortlessly in a bottom-up way, based on local differences in a specific feature.

The region's internal properties (color, texture, ...) help to identify them and their external relations (adjacency, inclusion, ...) 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.

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 (ECKs) combine two or more contraction kernels into one single contraction kernel which generates the same result in one single dual contraction. Decimation parameters of any individual pyramid level can be reconstructed from the ECK of the pyramid's apex if both vertices and edges of this ECK receive labels indicating their annihilation level in the pyramid. This is a labeled spanning tree (LST) of the base graph which allows efficient design and control of different types of dual irregular pyramids. Since the LST determines the pyramid, primitive modifications of the LST transform also pyramids into other pyramids on the same base graph. They open a large variety of possibilities to explore the domain of 'all' pyramids.

: Irregular pyramids are sequences of graphs with decreasing numbers of vertices, edges and faces from level to level. The advantage of describing images with graphs helps to overcome problems which can occur in regular pyramid structures, e.g. shift-variance and rotation-variance. Here we present some ideas and results how a recently developed parallel algorithm to reduce the number of elements (vertices, edges and faces) in a graph can be used for finding connected components in an image. The advantage of using this method to build irregular pyramids is that the degree of vertices in one of two parallel constructed graphs stays bounded, which cannot be provided with other techniques. To achieve this, the conventional region adjacency graph is extended, i.e. additional edges between and around vertices are allowed. 1 Introduction In image processing, pyramids are used for a variety of applications. They provide a multiresolution representation of the image, which can be calculated i...

We present a hierarchical partitioning of images using a pairwise similarity function on a graph-based representation of an image.

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. Lowlevel 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 graphbased 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.

This paper presents a new formalism for irregular pyramids based on combinatorial maps. The combinatorial map formalism allows us to encode a planar graph thanks to two permutations encoding the edges and the vertices of the graph.The combinatorial map formalism encode explicitly the orientation of the planar graph. This last property is useful to describe the partitions of an image which may be considered as a subset of the oriented plane IR 2 . This new constraint allows us to design interesting properties for irregular pyramids. Finally the combinatorial formalism allows us to encode efficiently the graph transformations used in irregular pyramids. 1 This Work was supported by the Austrian Science Foundation under S7002-MAT. 1 Introduction The decomposition of an image into connected components, called the segmentation of an image, is necessary when we want to take decisions from an image or more generally when we want to analyze the different objects that compose this image. ...