We are generally interested in finding a class of objects with smooth
boundaries that are continuously deformable.
Simple edge detectors are of limited or no use
under such circumstances as the output of edge detectors do not
necessarily correspond to object boundaries except for very high
quality images. Hence we are concerned with a whole boundary method
so that a global structure can be imposed on the problem.
A variety of deformable contour methods have been proposed. We
consider two such approaches for our purposes here.
First are the so-called energy minimizing snakes (due to Kass et al),
which are attracted to image features such as edges and lines,
while internal spline forces impose smoothness. The weights of the image
force, external force and the smoothness terms can be adjusted for different
behaviors. The solution of this optimization problem is often either
found by variational methods , or by using dynamic programming.
While we choose to focus on a different deformable boundary
finding method (see below) the approach proposed in this paper
can also be used with the 'snakes' approach.
The second approach due to Staib et al and it uses a Fourier parametrization which we have adopted in this work as our primary approach. It expresses a curve in terms of an orthonormal basis. The motivation for such a basis is that it makes the estimation problem easier as the cross correlation matrix becomes diagonal. However, for most practical situations we constrain ourselves to a limited number of harmonics. Thus,
where, is the contour vector consisting of the x and y coordinates
and , , and are the corresponding Fourier
The curve is thus represented by
The estimation of these parameters to find the boundary is posed as an optimization problem, where, the objective function measures the strength of the boundary given the set of parameters from the image. We shall come to this issue in the next section. This approach has several advantages. It is invariant to the starting point, scale and 2-D rotation and translation thus making the whole process view independent. Also, it has the flexibility of incorporating shape priors. A further advantage of this parametrization is that it simplifies the optimization process. For most objects, especially those that appear in the biomedical imaging domain, only a few parameters are needed as opposed to the larger number of points on the contour that needs to be adjusted using snakes.