Uses of Elliptic Approximations in Computer Vision

One of the problems in Computer Vision is recovery of object shapes from noisy images. Associated with this problem is the question of what is a shape and how is it to be represented. Since answers to these questions have to be ultimately tailored to the uses one has in mind, one has to bring into consideration potential applications and with it, the question of practical algorithms for implementation of the theory. Here we are concerned with mainly two-dimensional shapes. Mathematically, an object is simply an open subset in the image domain, characterized in some way. In the real world, what is an object and what is just noise or clutter depends of course on what one is looking for. For example, Figure 1a shows a noiseless synthetic image. It may be reasonable to assume that the objects in the figure are the four squares in the four corners and the two ellipses in the middle. Figure 1b shows a noisy version obtained from the image in Figure 1a by adding Gaussian noise. The signal-to-noise ratio (i.e. the ratio between the standard deviation of the image with noise removed and the standard deviation of the noise) is 1:4. The problem is to recover the original objects.