Topology-preserving thinning in 2-D pseudomanifolds

International audiencePreserving topological properties of objects during thinning procedures is an important issue in the field of image analysis. In the case of 2-D digital images (i.e. images defined on Z^2) such procedures are usually based on the notion of simple point. By opposition to the case of spaces of higher dimensions (i.e. Z^n, n ≥ 3), it was proved in the 80’s that the exclusive use of simple points in Z^2 was indeed sufficient to develop thinning procedures providing an output that is minimal with respect to the topological characteristics of the object. Based on the recently introduced notion of minimal simple set (generalising the notion of simple point), we establish new properties related to topology-preserving thinning in 2-D spaces which extend, in particular, this classical result to more general spaces (the 2-D pseudomanifolds) and objects (the 2-D cubical complexes)

Topological properties of thinning in 2-D pseudomanifolds

International audiencePreserving topological properties of objects during thinning procedures is an important issue in the field of image analysis. In the case of 2-D digital images (i.e. images defined on Z^2) such procedures are usually based on the notion of simple point. In contrast to the situation in Z^n , n>=3, it was proved in the 80s that the exclusive use of simple points in Z^2 was indeed sufficient to develop thinning procedures providing an output that is minimal with respect to the topological characteristics of the object. Based on the recently introduced notion of minimal simple set (generalising the notion of simple point), we establish new properties related to topology-preserving thinning in 2-D spaces which extend, in particular, this classical result to cubical complexes in 2-D pseudomanifolds

Gradient Vector Fields of Discrete Morse Functions and Watershed-cuts

In this paper, we study a class of discrete Morse functions, coming from Discrete Morse Theory, that are equivalent to a class of simplicial stacks, coming from Mathematical Morphology. We show that, as in Discrete Morse Theory, we can see the gradient vector field of a simplicial stack (seen as a discrete Morse function) as the only relevant information we should consider. Last, but not the least, we also show that the Minimum Spanning Forest of the dual graph of a simplicial stack is induced by the gradient vector field of the initial function. This result allows computing a watershed-cut from a gradient vector field

Writing Reusable Digital Geometry Algorithms in a Generic Image Processing Framework

Digital Geometry software should reflect the generality of the underlying mathe- matics: mapping the latter to the former requires genericity. By designing generic solutions, one can effectively reuse digital geometry data structures and algorithms. We propose an image processing framework focused on the Generic Programming paradigm in which an algorithm on the paper can be turned into a single code, written once and usable with various input types. This approach enables users to design and implement new methods at a lower cost, try cross-domain experiments and help generalize resultsComment: Workshop on Applications of Discrete Geometry and Mathematical Morphology, Istanb : France (2010

A graph-based mathematical morphology reader

This survey paper aims at providing a "literary" anthology of mathematical morphology on graphs. It describes in the English language many ideas stemming from a large number of different papers, hence providing a unified view of an active and diverse field of research

Collapses and watersheds in pseudomanifolds

This work is settled in the framework of abstract simplicial complexes. We propose a definition of a watershed and of a collapse for maps defined on pseudomanifolds of arbitrary dimension. Through an equivalence theorem, we establish a deep link between these two notions: any watershed can be obtained by collapse iterated until idempotence, and conversely any collapse iterated until idempotence induces a watershed. We also state an equivalence result which links the notions of a watershed and of a collapse with the one of a minimum spanning forest

Collapses and watersheds in pseudomanifolds of arbitrary dimension

International audienceThis work is settled in the framework of abstract simplicial complexes. We propose a definition of a watershed and of a collapse (i.e., a homotopic retraction) for maps defined on pseudomanifolds of arbitrary dimension. Then, we establish two important results linking watersheds and homotopy. The first one generalizes a property known for distance transforms in a continuous setting to any map on pseudomanifolds: a watershed of any map is a subset of an ultimate collapse of the support of this map. The second result establishes, through an equivalence theorem, a deep link between watershed and collapse of maps: any watershed of any map can be straightforwardly obtained from an ultimate collapse of this map, and conversely any ultimate collapse of the initial map straightforwardly induces a watershed