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Link to original content: https://dx.doi.org/10.1007/s10851-012-0374-7
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Efficient Geodesic Attribute Thinnings Based on the Barycentric Diameter

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Abstract

An attribute opening is an idempotent, anti-extensive and increasing operator, which removes from an image connected components which do not fulfil a given criterion. When the increasingness property is dropped, we obtain a—more general—attribute thinning. In this paper, we propose efficient grey scale thinnings based on geodesic attributes.

Given that the geodesic diameter is time consuming, we propose a new geodesic attribute, the barycentric diameter to speed up the computation time. Then, we give the theoretical error bound between these two attributes, and we note that in practice, the barycentric diameter gives very similar results in comparison with the geodesic diameter. Finally, we present the algorithm with further optimisations, to obtain a 60× speed up.

We illustrate the use of these thinnings in automated non-destructive material inspection: the detection of cracks. We discuss the advantages of these operators over other methods such as path openings or the supremum of openings with segments.

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Notes

  1. The normalisation factors are such that in a two-dimensional Euclidian space, the geodesic elongation of a disk is equal to 1.

  2. Database available in www.imageprocessingplace.com/root_files_V3/image_databases.htm.

  3. A demonstration version is available: http://cmm.ensmp.fr/~morard/DemoGeoThinnings.html.

  4. Image from the Institute for Molecular Virology. University of Wisconsin—Madison: http://www.biochem.wisc.edu/faculty/inman/empics/dna-prot.htm.

  5. We made an experiment on more than 100 natural images showing that a pixel is processed less than two times in average.

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Acknowledgements

This work was made possible thanks to the support of the “Pôle ASTech” and the “Pôle Nucléaire de Bourgogne”, and has been financed by the French “Département de Seine et Marne”.

The authors are grateful to Ms Raviart, working in the “Centre des matériaux, MINES ParisTech”, for help with the optical microscope.

The authors also wish to thank the anonymous reviewers for their comments and suggestions.

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Authors and Affiliations

  1. Centre de Morphologie Mathématique, Mathématiques et Systèmes, MINES ParisTech, 35, rue Saint-Honoré, 77305, Fontainebleau Cedex, France

    Vincent Morard, Etienne Decencière & Petr Dokládal

Authors
  1. Vincent Morard
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  2. Etienne Decencière
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  3. Petr Dokládal
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Corresponding author

Correspondence to Vincent Morard.

Appendix: Random Connected Component Models

Appendix: Random Connected Component Models

Many strategies exist to generate random binary shapes. The results of the comparison between the geodesic diameter and its approximations are linked to the method used to generate these shapes. Therefore, we use five different methods to have a high variety of objects. For each method, 5 realisations are presented in Fig. 15. The size of the support of these random shapes is a 500 by 500 pixels square.

Convex Shape

A random number of points (between 10 and 100) are randomly and uniformly picked on D. The final connected component is the convex hull of these points.

Pixel Aggregation

This method is used to generate relatively dense objects, which are almost convex. The set is initialised with a single point. At each iteration, a randomly chosen neighbour point is added to the set. The procedure is iterated a random number of times.

Ball Aggregation

This method uses the same process as the pixel aggregation method, except that we iteratively aggregate a ball instead of a point to the set. The ball radius is chosen randomly between 5 and 40 pixels, for each ball. The generated shapes are much more complex than the shapes generated using the pixel aggregation method.

Random Walk

We start from a ball in the centre of the domain. Then, we use a Brownian motion to choose the next location of the ball. At each iteration, the radius of the ball is chosen randomly.

Perlin Noise

Perlin noise [22] is a procedural texture primitive. It has a pseudo random appearance that is highly controllable and multiscale. Figure 14 provides an illustration of a realisation of this noise. By thresholding this image, we get a set of objects and we select the biggest CC of this set (Fig. 14). Some resulting objects can be very smooth, whereas others can have a high tortuosity.

Fig. 14
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Object generation with Perlin noise. Left to right: Perlin noise, thresholding, selection of the biggest CC

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Fig. 15
figure 17

Examples of random connected components, obtained with five different models

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Morard, V., Decencière, E. & Dokládal, P. Efficient Geodesic Attribute Thinnings Based on the Barycentric Diameter. J Math Imaging Vis 46, 128–142 (2013). https://doi.org/10.1007/s10851-012-0374-7

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