Gilles Bertrand

Topological gray-scale watershed transformation

We propose an original approach to the watershed problem, based on topology. We introduce a 1D topology for grayscale images, and more generally for weighted graphs. This topology allows us to precisely define a topological grayscale transformation that generalizes the action of a watershed transformation. Furthermore, we propose an efficient algorithm to compute this topological grayscale transformation,a nd we give an example of application to image segmentation.

A 3D 12-subiteration thinning algorithm based on P -simple points

In this paper, we propose a new methodology to conceive a thinning scheme based on the parallel deletion of P-simple points. This scheme needs neither a preliminary labelling nor an extended neighborhood, in the opposite of the already proposed thinning algorithms based on P-simple points. Moreover, from an existent thinning algorithm A, we construct another thinning algorithm A, such that A deletes at least all the points removed by A, while preserving the same end points. In fact, we propose a 12-subiteration thinning algorithm which deletes at least the points removed by the one proposed by Palagyi and Kuba (Graphical Models Image Process. 61 (1999) 199).

New Notions for Discrete Topology

Some new notions based on orders and discrete topology are introduced. We investigate the notions of unipolar and free points, we propose some discrete definitions for homotopy and a generalization of the notion of simple point.

A New 3D 6-Subiteration Thinning Algorithm Based on P-Simple Points

In a recent study [1], we proposed a new methodology to build thinning algorithms based on the deletion of P-simple points. This methodology may permit to conceive a thinning algorithm A from an existent thinning algorithm A, such that A deletes at least all the points removed by A, while preserving the same end points.In this paper, by applying this methodology, we propose a new 6-subiteration thinning algorithm which deletes at least all the points removed by the 6-subiteration thinning algorithm proposed by Palagyi and Kuba [2].

Segmentation of microscopic images by flooding simulation: a catchment-basins merging algorithm

This work addresses a catchment basins merging algorithm developed to automate the segmentation of microscopic images, which is directly derived from the traditional non- hierarchical watershed algorithm. The proposed merging algorithm, based on digital topology concepts, employs regional criteria to merge the non-significant minima. It can be classified as a region growing method by flooding simulation, working at the scale of the main structures. The shape of the structures is absolutely irrelevant to the merging process. As a characteristic of the flooding simulation methods, the gray level image is viewed as a relief where each gray level is assigned a height. In the proposed method the relief is always flooded from all its local minima which are progressively detected and merged as the flooding takes place. The catchment basins merging process is guided by two parameters: a depth criterion and an area criterion. This solution suppresses the characteristic over-segmentation of the traditional watershed enabling the direct segmentation of the original image without the need of a previous pre-processing step. Due to the automatic detection of all local minima there is not need of the explicit marker extraction step often required by other flooding simulation methods. It is shown that this solution produces excellent segmentation results allowing the characterization of several materials from their microscopic images.

A Model for Digital Topology

In the framework known as digital topology, two different adjacency relations are used for structuring the discrete space Zn.In this paper, we propose a model for digital topology based on the notion of order and discrete topology. We validate our model by considering the two fundamental notions of surface and simple point. At last, we give the different possible configurations that may appear in 2-and 3- dimensional surfaces in Z4 which correspond to our model.

Discrete Surfaces and Frontier Orders

Many applications require the extraction of an object boundary from a discrete image. In most cases, the result of such a process is expected to be, topologically, a surface, and this property might be required in subsequent operations. However, only through careful design can such a guarantee be provided. In the present article we will focus on partially ordered sets and the notion of n-surfaces introduced by Evako et al. to deal with this issue. Partially ordered sets are topological spaces that can represent the topology of a wide range of discrete spaces, including abstract simplicial complexes and regular grids. It will be proved in this article that (in the framework of simplicial complexes) any n-surface is an n-pseudomanifold, and that any n-dimensional combinatorial manifold is an n-surface. Moreover, given a subset of an n-surface (an object), we show how to build a partially ordered set called frontier order, which represents the boundary of this object. Similarly to the continuous case, where the boundary of an n-manifold, if not empty, is an (n−1)-manifold, we prove that the frontier order associated to an object is a union of disjoint (n−1)-surfaces. Thanks to this property, we show how topologically consistent Marching Cubes-like algorithms can be designed using the framework of partially ordered sets.

Gray-scale image processing using topological operators

In recent work, we introduced topological notions for grayscale images based on a cross-section topology. In particular, the notion of destructible point, which corresponds to the classical notion of simple point, allows to build operators that simplify a grayscale image while preserving its topology. In this paper, we introduce new notions and operators in the framework of the cross-section topology. In particular, the notion of (lambda) -destructible point allows us to selectively modify the topology, based on a local contrast parameter (lambda) . By combining homotopic and non-homotopic operators, we introduce new methods for filtering, thinning, segmenting and enhancing grayscale images.

An attribute-based image segmentation method

This work addresses a new image segmentation method founded on Digital Topology and Mathematical Morphology grounds. The ABA (attribute based absorptions) transform can be viewed as a region-growing method by flooding simulation working at the scale of the main structures of the image. In this method, the gray level image is treated as a relief flooded from all its local minima, which are progressively detected and merged as the flooding takes place. Each local minimum is exclusively associated to one catchment basin (CB). The CBs merging process is guided by their geometric parameters as depth, area and/or volume. This solution enables the direct segmentation of the original image without the need of a preprocessing step or the explicit marker extraction step, often required by other flooding simulation methods. Some examples of image segmentation, employing the ABA transform, are illustrated for uranium oxide samples. It is shown that the ABA transform presents very good segmentation results even in presence of noisy images. Moreover, its use is often easier and faster when compared to similar image segmentation methods.

A New 3D 12-Subiteration Thinning Algorithm Based on P-Simple Points

In this paper, we propose a new methodology based on P -simple points, in order to build a thinning algorithm. From an existent thinning algorithm A, we construct another thinning algorithm A � , such that Adeletes at least all the points removed by A, while preserving the same end points. In fact, we propose an algorithm which deletes at least the points removed by a recent 12-subiteration thinning algorithm proposed by Palagyi and Kuba (26).