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A three-dimensional (3D) theoretical morphospace of gomphonemoid and cymbelloid diatoms was skeletonized using concepts from extended Reeb graph analysis and Morse theory. The resultant skeleton tree was matched to a cladogram of the same group of related taxa using adjacency matrices of the trees and ordinated with multidimensional scaling (MDS) of leaf nodes. From this, an unweighted path matrix based on the number of branches between leaf nodes was ordinated to determine degree of matched tree structures. A constrained MDS of the path matrix, weighted by ranked MDS leaf node groups as facets, was used to interpret taxon environmental tolerances and habitat preferences with respect to adaptive value. The methods developed herein provided a way to combine results from morphological and phylogenetic analyses and interpret those results with respect to an aspect of evolutionary process, namely, adaptation.

This paper deals with the definition of abstraction tools for deriving high-level descriptions of complex shape models. Among the wide range of shape descriptors, topological graph-like representations provide a powerful and synthetic sketch of the object, and also capture its inner structure, that is how features connect together to give the overall shape. This aspect makes them useful to describe complex 3D objects in various applications as modelling, morphing, matching and recognition. The approach therein presented deals with the extraction of protrusion-like features of the input object based on a multi-scale curvature evaluation of its surface; then, a skeleton is constructed by taking the tips of the protrusions as seed points for expanding topological rings on it.

This article presents a 3D shape matching method for 3D mesh models applied to content-based search in database of 3D objects. The approach is based on the multiresolution Reeb graph (MRG) proposed by Hilaga et al.¹ MRG provides a rich representation of shapes able in particular to embed the object topology. In our framework, we consider 3D mesh models of various geometrical complexity, of different resolution, and when available with color texture map. The original approach, mainly based on the 3D object topology, is not accurate enough to obtain satisfying matching. Therefore we propose to reinforce the topological consistency conditions of the matching and to merge within the graph geometrical and visual information to improve matching and calculation of shape similarity between models. Besides, all these new attributes can be freely weighted to fit the user requirements for object retrieval. We obtain a flexible multiresolutional and multicriteria representation that we called augmented multiresolution Reeb graph (aMRG). The approach has been tested and compared with other methods. It reveals very performant for the retrieval and the classification of similar 3D shapes.

To analyze given object shapes, it is necessary first to model the shapes and then to analyze the models. This paper proposes a method of modeling and analyzing two-dimensional (2D) and three-dimensional (3D) shapes based on singularities. First, a function is defined on an object. The object is then modeled by the distribution of the singularities of the function. Finally, the extracted singular points are analyzed by a Reeb graph together with wavelets for multiresolution analysis. The applications of the method include analysis of botanical leaf shapes and human facial expressions.

We discuss in the present paper the following natural question: is the space of all Morse functions with fixed number of minima and maxima on a closed surface linearly connected? We give an algorithm for reduction of any Morse function on a closed orientable surface to some canonical form. We apply this result to the new representation for the inversion of 2-sphere in Euclidean 3-space, in terms of Reeb graph of the height function.

With the recent advances in computed tomography and magnetic resonance devices, cross-sectional images are now commonly used for diagnosis. However, how contours between cross-sections should be connected is often ambiguous. In this paper, we propose an algorithm that enumerates all possible cases of the correspondence of contours. This is useful for achieving fully automatic interpolation of contours, although our current implementation still requires some degree of manual interaction.

A new adaptive homotopy modeling is described for 3D skeleton-based key frame animation. Using the homotopy model that constructs object surfaces from their cross-sectional contours and Reeb graphs, an animator can develop a motion sequence by editing the graphs and contours of the objects. The Reeb graph is a skeletal abstraction of an object providing topological information while the contour provides surface information. It is converted to a set of articulated Reeb graphs in order to satisfy the height function restriction when they are transformed. A user interface which allows interactive and predictable editing of the Reeb graphs and contours, denning of key frames, reconstructing the objects and then animating the objects is also presented.