Reliability / precision uncertainity in shape fitting problems

The precision/uncertainty duality has been long known in the context of Hough transform, where a shape in an image cannot be fit precisely using the Hough transform without compromising the certainty or reliability of the fitting. This paper mathematically shows that such duality also exists while using the least squares based method. This paper also proposes a method to quantify the reliability of a fit. Further, based on the proposed measure of reliability, an optimization scheme to strike a balance between the precision and reliability is suggested. Though the mathematical formulations deal with only straight line, considering it as the simplest and basic geometric primitive, it is argued that such duality exists for any shape fitting and applies to any shape fitting method.