An affine shape constraint for geometric active contours - Emerging Trends in Image Processing, Computer Vision, and Pattern Recognition - page 419

There have been promising results given that the classical snake models can detect object with
smooth contours whereas the level set-based models have the additional ability to detect sim-
ultaneously many objects in the image. However, several work of the prior knowledge-driv-
en models reached good segmentation results by detecting partially occluded shapes present-
ing missing part in low contrast or very noisy image. To define prior knowledge, the authors
use either shape alignment [ 7 ] or registration [ 9 ] between the contour of the target shape and
the shape of reference or a distance between invariant shape descriptors like [ 8 ] . All these
work manage the case of Euclidean transformations (translation, rotation, and scale factor).
But in real situations, the object of interest can submit large transformations (affine or project-
ive cases) and distortions. Hence the class of rigid transformations is not appropriate to model
this type of movement between the shape of reference and the target one. To our knowledge,
there has been only the work of Foulonneau et al. [ 15 ] that treats the case of affine transform-
ations. Although this approach, which is based on the distance between an invariant set of
Legendre descriptors of the target and reference shapes, presented good results, this method
suffers from the instability of the Legendre moments and the order that has to be fixed em-
pirically. Besides, the method requires a heavy execution time to achieve satisfactory results.
In order to extend the work of shape priors presented in Refs. [ 16 , 17 ], we propose in this re-
search a geometric approach to manage the class of affine transformations [ 26 ] that can occur
between the shape of interest and the reference one. Prior knowledge is obtained by aligning
the evolving contour and the template using an alignment method which is based on Fourier
descriptors. For this purpose, we will start by introducing our approach in the case of rigid
transformations. Then we generalize the idea by treating the case of affine ones that often oc-
cur in real situations. We will also propose a geometric method based on a set of complete and
stable set of invariant shape descriptors to manage the case of mutireferences. Combining the
proposed shape prior information with the level set-based model, the improved model can re-
tain all the advantages of the active contour model and have the additional ability of being able
to represent the global shape of an object. The remainder of this paper is organized as follows:
In Section 2 , a description of the outlines of the used shape alignment method in both cases
(Euclidean and affine transformations) will be presented. In Section 3 , we will briefly recall the
principle of the used edge-based active contours, then we will move to present the proposed
shape prior. Experimental results are presented and commented in Section 4 . Finally, we con-
clude the work and highlight some possible perspectives in Section 5 .
2 Shape alignment using fourier descriptors
In this section, we recall the used alignment methods to define the proposed energy in both
cases rigid and affine transformations.
2.1 Euclidean Shapes Alignment
It is well known that the closed contour of a planar object can be given by its parametric rep-
resentation