Graph-theoretic Issues in Remote Sensing and Landscape Ecology

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1 EnviroInfo 2002 (Wien) Environmental Communication in the Information Society - Proceedings of the 16th Conference Graph-theoretic Issues in Remote Sensing and Landscape Ecology Joachim Steinwendner 1 Abstract The goal of remote sensing applications in landscape ecology have evolved from mere landcover/landuse classification to include also structural information and patterns of landscape elements. The importance of landscape patterns to describe landscape units is also mirrored in a number of current EU projects and initiatives (e.g. Natura 2000, SPIN Spatial Indicators for European Nature Conservation, Envip Nature, and others). Spatial properties of landscape units are closely related to the topology of landscape elements. Topology and topology-related applications make up the core of graph theory. This mathematical branch provides data structures and methods based on a firm mathematical background suitable for remote sensing and landscape ecological applications. 1. Introduction By reason of coverage, high spatial resolution and exact allocation in time, remote sensing data is a vital source of information for mapping and modelling for various applications in landscape ecology. Although recent and future sensors offer better spatial resolution (1-4 m), satellite data with mid-resolution (i.e m resolution, e.g. Landsat ETM+, SPOT, ASTER) play still a major role for areas under investigation on a state or continental scale. Automatic methods for the extraction of landscape elements and landscape units from remote sensing data are an important prerequisite for large area landscape analysis. Computer vision methods such as image segmentation are used to accomplish this task (Baatz (et al.), 2000; Felszenszwalb (et al.), 1998; Gagalowicz (et al.), 1986; Schneider (et al.), 1999). The following papers use also graph-based methods for segmentation (Barr (et al.), 1997; Kropatsch, 1995; Urquhart, 1982; Zahn, 1971). 1 Institute of Surveying, Remote Sensing and Land Information, Univ. of Agricultural Sciences Vienna (BOKU), Peter-Jordan-Str. 82, A-1190 Vienna, Austria steinwendner@boku.ac.at,

2 Basic Definitions in Graph theory This section introduces a few graph theory definitions utilized in the following sections. The definitions are mainly taken from (West, 2000). Definition 1: A graph G(V,E) consists of a set of vertices, V(G)={v i }, and a set of edges, E(G)={e i }. Each edge e=v i v j from the set E(G) is defined by two vertices v i and v j defining the edge. Vertices v i and v j are said to be adjacent. A vertex v is said to be incident to the edges that contain v as a start or end vertex. Definition 2: The number of edges incident to a vertex is termed deg(v) or degree of vertex v, respectively. Fig. 1: A graph with vertex set {x 0,x 1,x 2,x 3,x 4,x 5 } and the edge set {x 0 x 1,x 0 x 5,x 0 x 2,x 1 x 2,x 1 x 3,x 1 x 4,x 2 x 3 } Definition 3: A tree is a graph than contains no cycles, i.e. there exists only one path between a randomly chosen pair of vertices in a tree graph. The graph in fig. 1 is thus no tree, because there are two paths from the vertices x1 to x2 (x1x3x2 and x1x0x2). Definition 4: The subgraph of G(V,E) with a minimum number of edges connecting all vertices is called minimum spanning tree of G, mst(g). If B=msb(G) then the following must be valid: V(B)=V(G) and E(B) E(V). Figure 3 shows a graph and its minimum spanning tree. Definition 5: A planar graph G is a graph that can be embedded in a twodimensional surface so that no edge crossing exists. The graph displayed in figure 1 is a planar graph.

3 548 Definition 6: A planar graph G divides a surface into faces. A new graph can be derived in such a way that each face is represented by a vertex. An edge is included into the edge set if two faces are adjacent. The so-produced graph G d is called dual graph. In figure 2, the gray graph is the dual graph of the black graph and vice versa. Definition 7: A parametrized graph G p contains a vertex set V(G p )={v i (X i )} and an edge set E(G p )={e i (Y i )}, where X i F V (the feature space of vertices) and Y i F E (the feature space of edges). 3. Graph representation of remote sensing data and landscapes In order to use graph theoretic methods for remote sensing data, it is necessary to transform the data into graphs. A straight-forward way is that a vertex represents a pixel. An edge is added to the edge set if pixels are neighboured. In this work, fourneighbourhood of pixels only is considered in order to obtain planar graphs. This restriction does not imply a limitation on the degree of vertices. Fig. 2: Part of a satellite image and its pixel graph (black) and its pixel border graph (gray) In figure 2 the black graph (pixel graph) is the result of this construction. The gray graph is the dual graph of the pixel graph (called pixel border graph).

4 Graphs and Segmentation Image segmentation in general is the process of partitioning an image into segments (i.e. sets of adjacent pixels) having a meaning in the real world. Graph-based segmentation means thus the process of partitioning a graph into connected components (i.e. sets of adjacent vertices) having a meaning in the real world. This definition offers several advantages: - It covers also image segmentation if a graph is constructed as described in the previous section. - One is not restricted with respect to the regular grid of digital images so that vertices can have arbitrarily many neighbors. - When using parameterised graphs vertices as well as edges are attributed with properties. In terms of image segmentation it is straight forward to develop hybrid segmentation methods considering homogeneity of pixels and discontinuities in the image Graph representation of segmentation results A labeled graph is a simple representation of a graph partitioning where vertices grouped according to some criterion are labeled equally. The set of vertices with the same label is called a component C i. Their must be a path in G between any randomly chosen pair of vertices of C i traveling only through vertices of C i. For the whole set of components, the following must be valid V(G)= V(C i ) and i,j i j: V(C i ) V(C j )=. Based on this graph partitioning a new graph G S can be construced where a vertex represents a component and edges are added to E(G S ) if there is at least one edge between vertices of two differently labeled components. Note that this construction is not specific about how many edges are added to the edge set. In terms of landscape ecology a component corresponds to a landscape element as mentioned in (Grillmayer, 2002). The newly created graph G S is consequently termed landscape element graph. So far, only graph-based representations of segmentation results are discussed. In the next section the process of graph-based segmentation is introduced Graph-based segmentation methods There are two groups of graph-based segmentation methods. The one group consists of conventional segmentation methods such as region growing, split-and-merge, watershed methods adapted to be applied on a graph structure the other group of segmentation methods is based on graph-theoretic methods.

5 550 For further discussion let us consider a parameterized pixel graph. The vertex feature space is an n-dimensional space representing the spectral signatures of a pixel. The edge feature space represents the spectral distance between pixels. Conventional segmentation methods can be improved exploiting the properties of the graph structure. In case of region growing, two immediate advantages with respect to computing time are obvious: Firstly, spectral distance computation is performed only once such that pixel comparison is done by simply looking into the edge attribute. Secondly, edges attributed with high spectral distance can be deleted before starting the region growing process reducing thereby the number of pixel comparisons. Fig. 3: Graph G (edge set consists of black and gray edges) and its minimum spanning tree (gray edges only) Some segmentation methods based on graph theory use the minimum spanning tree of a graph. The minimum criteria is again the spectral distance of pixels represented by vertices. In (Zahn, 1971) all edges of the minimum spanning tree are deleted that feature a spectral distance below a certified threshold leading eventually to a set of components (see section 4.1). This approach, however, is problematic with textured objects. Felszenzwalb (Felszenszwalb(et al.), 1998) overcomes this problem by applying a pairwise component comparison taking into account the intercomponent difference and intracomponent differences. In (Kropatsch (et al.), 1996), a graph-based segmentation is described building an image pyramid using dual graph contraction. The step from one level into the next level in the pyramid results from merging of vertices of the pixel graph as well as the pixel border graph using a contraction rule or function, resp. This function can be chosen arbitrarily guiding the segmentation process to match the outcome of any

6 551 segmentation method. For this reason this segmentation algorithm is called universal in (Kropatsch(et al.), 1996). The reviewed segmentation algorithms can be applied to the pixel graph but also to the resulting landscape element graph. However, the step from landscape element graph to the landscape units is more complex since the homogeneity criteria (or discontinuity criteria) are based on a edge and vertex feature set containing more and diverse information than the pixel graph (see also (Grillmayer, 2002)). In order to cope with this complexity an approach can be taken similar to (Schneider(et al.), 1999). This approach is termed knowledge-based breadth-firstsearch and is applied to the landscape element graph. This method starts from a vertex and groups adjacent vertices if these and the component built so far are similar according to the knowledge base. The knowledge base contains a collection of landscape unit prototypes and their properties. Schneider (Schneider, 2002) discusses a general concept on knowledge based image analysis with respect to remote sensing and landscape ecology 5. Conclusion In this work a graph structure is introduced that serves as a basis for low level as well as high level image understanding problems related to landscape ecology. The advantages of using graph theory is twofold: it provides a universal concept for using computer vision methods for landscape ecological problems and it is based on a firm mathematical background. The concept also follows the idea of object-oriented programming, including encapsulation, inheritance, polymorphism, etc. The objects are vertices and edges with their properties, some of the typical functions applied to this objects are contraction, deletion, insertion. The hierarchical concept of graph pyramids suggests inheritance of properties and functions. A good example for an object-oriented programming library realizing combinatoric (graphs) and geometric (points, lines, polygons) data types is the LEDA library (Mehlhorn (et al.), 2000). Acknowledgement This work has been funded by the Austrian Science Foundation (FWF), project Hierarchies of plane graphs for the acquisition, analysis und visualization of geographical information, grant number Grant-Nr. P14662-INF. This project is conducted with the collaboration of the Institute of Pattern Recognition and Image Processing and the Department of Algorithms and Data Structures of the Institute of Computer Graphics, both at the Technical University Vienna.

7 552 Bibliography Baatz, M. and Schäpe, A., (2000): Multiresolution Segmentation: an optimization approach for high quality multi-scale image segmentation, AGIT, Salzburg. Barr, S. and Barnsley, M., (1997): A region-based, graph-theoretic data model for the inference of second-order thematic information from remotely-sensed images. International Journal of Geographical Information Science, 11(6): pp Felszenszwalb, P.F. and Huttenlocher, D.P., (1998): Efficiently computing a good segmentation, DARPA Image Understanding Workshop. Gagalowicz, A. and Monga, O., (1986): A New Approach for Image Segmentation, 8th International Conference on Pattern Recognition, pp Grillmayer, R., (2002): Landscape Structure Model, this publication. Kropatsch, W., (1995): Building Irregulars Pyramids by Dual Graph Contraction. IEEE Proc. Vision, Image und Signal Processing, 142(6): pp Kropatsch, W. and Benyacoub, S., (1996): A revision of pyramid segmentation. In: W.G. Kropatsch (Editor), 13th International Conference on Pattern Recognition, pp Mehlhorn, K. and Näher, S., (2000): LEDA: A platform for combinatorial and geometric computing. Cambridge University Press. Schneider, W., (2002): Knowledge Based Methods of Analysis of Remotely Sensed Images for Landscape Ecology, this publication. Schneider, W. and Steinwendner, J., (1999): Landcover mapping by interrelated segmentation and classification of satellite images. Fusion of Sensor Data, Knowledge Sources and Algorithms for Extraction and Classification of Topographic Objects, International Archives of Photogrammetry and Remote Sensing, 32(7-4-3W6): pp Urquhart, R., (1982): Graph theoretical clustering based on limited neighborhood sets. Pattern Recognition, 15(3): pp West, D., (2000): Introduction to Graph Theory. Prentice Hall. Zahn, C.T., (1971): Graph-theoretic methods for detecting and describing gestalt clusters. IEEE Trans. Comp., 20: pp