Constructive Fitting and Extraction of Geometric Primitives

We propose a constructive method for fitting and extracting geometric primitives. This method formalizes the merging process of geometric primitives, which is often used in computer vision. Constructive fitting starts from small uniform fits of the data, which are called elemental fits, and uses them to construct larger uniform fits. We present formal results that involve the calculation of the fitting cost, the way in which the elemental fits must be selected, and the way in which they must be combined to construct a large fit. The rules used to combine the elemental fits are very similar to the engineering principles used when building rigid mechanical constructions with rods and joins. In fact, we will characterize the quality of a large fit by a rigidity parameter. Because of its bottom-up approach constructive fitting is particularly well suited for the extraction of geometric primitives when there is a need for a flexible system. To illustrate the main aspects of constructive fitting we discuss the following applications: exact Least Median of Squares fitting, linear regression with a minimal number of elemental fits, the design of a flatness estimator to compute the local flatness of an image, the decomposition of a digital arc into digital straight line segments, and the merging of circle segments.