Skeletonization of Edges Extracted by Natural Images: A Novel Approach for Shape Representation

Skeletonization of Edges Extracted by Natural Images: A Novel Approach for Shape Representation

Donatella Giuliani (University of Bologna, Italy)
DOI: 10.4018/978-1-4666-9435-4.ch008
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Abstract

This chapter presents a method to compute the skeletal curve of shapes extracted by images derived by the real world. This skeletonization approach has been proved effective when applied to recognize biological forms, regardless of their complexity. The coloured and grayscale images have been pre-processed and transformed in binary images, recurring to segmentation. Generally the resulting binary images contain bi-dimensional bounded shapes, not-simply connected. For edge extraction it has been performed a parametric active contour procedure with a generalized external force field. The force field has been evaluated through an anisotropic diffusion equation. It has been noticed that the field divergence satisfies an anisotropic diffusion equation as well. Moreover, the curves of positive divergence can be considered as propagating fronts that converge to a steady state, the skeleton of the extracted object. This methodology has also been tested on shapes with boundary perturbations and disconnections.
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Background

In the traditional parametric models, a bi-dimensional active contour, or snake, is expressed explicitly by a parametric equation . The evolving curve is defined within a given image I(x,y) and subjected to modifications under the action of forces, until it fits well into the final contour (Kass,Witkin, & Terzopoulos,1998). The final shape of the contour to be extracted will be such as to minimize an energy functional associated with it, so given:

(1)

The first term is the internal energy that expresses a priori knowledge of the model in relation to the degree of flexibility of the active contour:

(2) the term controls the contour tension, while regularises its rigidity. The second term represents the external energy, derived from the image whose local minima correspond to the features to be extracted. By using a variational approach. (Courant, & Hilbert,1953), the contour that minimizes the total energy must satisfy the Euler-Lagrange equation:
(3) where ∇ is the gradient operator. Equation (3) can be viewed as a force balance equation: where the internal forces
restrain stretching and bending, the external forces push the curve towards the features of interest. By introducing a time-variable parametric equation , a deformable model is able to create a geometrical shape that evolves over time, so the solution of the static problem (3) will be made dynamic. To this end, indicating with and the density of mass and the damping coefficient respectively, the equation (3) will be transformed:

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