Why are algebraic distances usually not satisfactory ?

The big advantage of use of algebraic distances is the gain in computational efficiency, because closed-form solutions can usually be obtained. In general, however, the results are not satisfactory. There are at least two major reasons.

• The function to minimize is usually not invariant under Euclidean transformations. For example, the function with normalization F=1 is not invariant with respect to translations. This is a feature we dislike, because we usually do not know in practice where is the best coordinate system to represent the data.
• A point may contribute differently to the parameter estimation depending on its position on the conic. If a priori all points are corrupted by the same amount of noise, it is desirable for them to contribute the same way. (The problem with data points corrupted by different noise will be addressed in section 8.)

  
Figure 1: Normalized conic

To understand the second point, consider a conic in the normalized system (see fig:normal_conic):

The algebraic distance of a point to the conic Q is given by [3]:

where is the distance from the point to the center O of the conic, and is the distance from the conic to its center along the ray from the center to the point . It is thus clear that a point at the high curvature sections contributes less to the conic fitting than a point having the same amount of noise but at the low curvature sections. This is because a point at the high curvature sections has a large and its is small, while a point at the low curvature sections has a small and its is higher with respect to the same amount of noise in the data points. Concretely, methods based on algebraic distances tend to fit better a conic to the points at low curvature sections than to those at high curvature sections. This problem is usually termed as high curvature bias.