Lidija Čomić

Morse-Smale decompositions for modeling terrain knowledge

In this paper, we describe, analyze and compare techniques for extracting spatial knowledge from a terrain model. Specifically, we investigate techniques for extracting a morphological representation from a terrain model based on an approximation of a Morse-Smale complex. A Morse-Smale complex defines a decomposition of a topographic surface into regions with vertices at the critical points and bounded by integral lines which connect passes to pits and peaks. This provides a terrain representation which encompasses the knowledge on the salient characteristics of the terrain. We classify the various techniques for computing a Morse-Smale complexe based on the underlying terrain model, a Regular Square Grid (RSG) or a Triangulated Irregular Network (TIN), and based on the algorithmic approach they apply. Finally, we discuss hierarchical terrain representations based on a Morse-Smale decomposition.

Modeling and Manipulating Cell Complexes in Two, Three and Higher Dimensions

Cell complexes have been used in geometric and solid modeling as a discretization of the boundary of 3D shapes. Also, operators for manipulating 3D shapes have been proposed. Here, we review first the work on data structures for encoding cell complexes in two, three and arbitrary dimensions, and we develop a taxonomy for such data structures. We review and analyze basic modeling operators for manipulating complexes representing both manifold and non-manifold shapes. These operators either preserve the topology of the cell complex, or they modify it in a controlled way. We conclude with a discussion of some open issues and directions for future research.

Cancellation of critical points in 2D and 3D Morse and Morse-Smale complexes

Morse theory studies the relationship between the topology of a manifold M and the critical points of a scalar function f defined on M. The Morse-Smale complex associated with f induces a subdivision of M into regions of uniform gradient flow, and represents the topology of M in a compact way. Function f can be simplified by cancelling its critical points in pairs, thus simplifying the topological representation of M, provided by the Morse-Smale complex. Here, we investigate the effect of the cancellation of critical points of f in Morse-Smale complexes in two and three dimensions by showing how the change of connectivity of a Morse-Smale complex induced by a cancellation can be interpreted and understood in a more intuitive and straightforward way as a change of connectivity in the corresponding ascending and descending Morse complexes. We consider a discrete counterpart of the Morse-Smale complex, called a quasi-Morse complex, and we present a compact graph-based representation of such complex and of its associated discrete Morse complexes, showing also how such representation is affected by a cancellation.

Morphological Modeling of Terrains and Volume Data

This book describes the mathematical background behind discrete approaches to morphological analysis of scalar fields, with a focus on Morse theory and on the discrete theories due to Banchoff and Forman. The algorithms and data structures presented are used for terrain modeling and analysis, molecular shape analysis, and for analysis or visualization of sensor and simulation 3D data sets. It covers a variety of application domains including geography, geology, environmental sciences, medicine and biology. The authors classify the different approaches to morphological analysis which are all based on the construction of Morse or Morse-Smale decompositions. They describe algorithms for computing such decompositions for both 2D and 3D scalar fields, including those based on the discrete watershed transform. Also addressed are recent developments in the research on morphological shape analysis, such as simplification operators for Morse and Morse-Smale complexes and their multi-resolution representation. Designed for professionals and researchers involved with modeling and algorithm analysis, Morphological Modeling of Terrains and Volume Data is a valuable resource. Advanced-level students of computer science, mathematics and geography will also find the content very helpful.

Simplification Operators on a Dimension-Independent Graph-Based Representation of Morse Complexes

Ascending and descending Morse complexes are defined by the critical points and integral lines of a scalar field f defined on a manifold M. They induce a subdivision of M into regions of uniform gradient flow, thus providing a compact description of the topology of M and of the behavior of f over M. We represent the ascending and descending Morse complexes of f as a graph, that we call the Morse incidence graph (MIG). We have defined a simplification operator on the graph-based representation, which is atomic and dimension-independent, and we compare this operator with a previous approach to the simplification of 3D Morse complexes based on the cancellation operator. We have developed a simplification algorithm based on a simplification operator, which operates on the MIG, and we show results from this implementation as well as comparisons with the cancellation operator in 3D.

Fuzzy prediction based on regression models: Evidence from the Belgrade Stock Exchange - prime market and graphic industry

In the conventional regression model, deviations between the observed values and the estimated values are supposed to be due to measurement errors. Here, taking a different perspective, these deviations are regarded as the fuzziness of the systems parameters. Thus, these deviations are reflected in a linear function with fuzzy parameters. Using linear programming algorithm, this fuzzy linear regression model might be very convenient and useful for finding a fuzzy structure in an evaluation system. In this paper, the details of the fuzzy linear regression concept and its applications in an uncertain environment are shown and discussed on data of the Belgrade Stock Exchange. In addition, the prediction of stock market prices is performed using fuzzy linear trend. Having in mind the characteristics of trading methods, fuzzy linear trend is used for prediction of stock market prices based on historical data, which are not precisely given within a trading day. Results of the research indicate the significance of fuzzy prediction based on regression models, i.e. fuzzy linear trend.

Simplifying morphological representations of 2D and 3D scalar fields

We describe a dual graph-based representation for the ascending and descending Morse complexes of a scalar field, and a compact and dimension-independent data structure based on it, which assumes a discrete representation of the field as a simplicial mesh. We present atomic dimension-independent simplification operators on the graph-based representation. Based on such operators, we have developed a simplification algorithm, which allows generalization of the ascending and descending Morse complexes at different levels of resolution. We show here the results of our implementation, discussing the computation times and the size of the resulting simplified graphs, also in comparison with the size of the original full-resolution graph.

Modeling and Simplifying Morse Complexes in Arbitrary Dimensions

Ascending and descending Morse complexes, defined by a scalar function f over a manifold domain M, decompose M into regions of influence of the critical points of f, thus representing themorphology of the scalar function f over M in a compact way. Here, we introduce two simplification operators on Morse complexes which work in arbitrary dimensions and we discuss their interpretation as n-dimensional Euler operators. We consider a dual representation of the two Morse complexes in terms of an incidence graph and we describe how our simplification operators affect the graph representation. This provides the basis for defining a multi-scale graph-based model of Morse complexes in arbitrary dimensions.

A topological 4-coordinate system for the face centered cubic grid

Abstract Non traditional 3D grids, such as face centered cubic and body centered cubic, have some advantages over the traditional cubic grid. They offer, combined with the use of topological coordinates that address not only the voxels, but also the lower dimensional cells of the grid, a way to avoid some topological paradoxes of digital geometry. In this paper, a symmetric topological 4-coordinate system is presented for the face centered cubic grid. Incidence (boundary and co-boundary) and adjacency relations between cells can easily be deduced from cell coordinates through simple integer operations. An application of the proposed coordinate system to graphics/visualization area is shown.

A Combinatorial 4-Coordinate System for the Diamond Grid

A new combinatorial coordinate system for cells in the diamond grid is presented, and some of its properties are detailed. Four dependent coordinates are used to address the voxels (triakis truncated tetrahedra), their faces (hexagons and triangles), their edges and the points at their corners. The incidence (boundary and co-boundary) and adjacency relations of the cells can easily be captured by these coordinate values. Therefore, the new coordinate system can effectively by applied in morphological and topological operations.