Topological Analysis and Characterization of Discrete Scalar Fields

Discrete Geometry for Computer Imagery

2008

Self Cite

Abstract. 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.

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Cancellation of Critical Points in 2D and 3D Morse and Morse-Smale Complexes

Čomić

Discrete Geometry for Computer Imagery

2008

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“…In this Section, we describe a dimension independent region-based method, introduced in [4], for extracting an approximation of a descending Morse complex related to maxima, called a Smale-like decomposition, starting from a scalar field f defined on the vertices of a triangulated manifold domain Σ.…”

Section: The Smale-like Discrete Decompositionmentioning

confidence: 99%

Smale-Like Decomposition and Forman Theory for Discrete Scalar Fields

Čomí

Mesmoudi

Discrete Geometry for Computer Imagery

2011

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Abstract. Forman theory, which is a discrete alternative for cell complexes to the well-known Morse theory, is currently finding several applications in areas where the data to be handled are discrete, such as image processing and computer graphics. Here, we show that a discrete scalar field f , defined on the vertices of a triangulated multidimensional domain Σ, and its gradient vector field Grad f through the Smale-like decomposition of f [6], are both the restriction of a Forman function F and its gradient field Grad F that extends f over all the simplexes of Σ. We present an algorithm that gives an explicit construction of such an extension. Hence, the scalar field f inherits the properties of Forman gradient vector fields and functions from field Grad F and function F .

“…Region growing algorithms [9,12,20] mimic, in the discrete case, the definition of 2-cell for a Morse complex, which is the set of the integral lines that originate or converge to a critical point. We discuss here how the algorithms extract the unstable Morse complex.…”

Section: Region-growing Algorithmsmentioning

confidence: 99%

“…The algorithm in [12], denoted here as DIS, is based on the elevation values at the TIN vertices, while the one in [11], denoted as GRD, is based on the gradient associated with the triangles. The algorithms sort the vertices according to their elevation and process them in decreasing elevation order.…”

Section: Region-growing Algorithmsmentioning

confidence: 99%

Computing Morse Decompositions for Triangulated Terrains: An Analysis and an Experimental Evaluation

Vitali

Image Analysis and Processing – ICIAP 2011

Magillo

2011

Self Cite

Abstract. We consider the problem of extracting the morphology of a terrain discretized as a triangle mesh. We discuss first how to transpose Morse theory to the discrete case in order to describe the morphology of triangulated terrains. We review algorithms for computing Morse decompositions, that we have adapted and implemented for triangulated terrains. We compare the the Morse decompositions produced by them, by considering two different metrics.