Prediction of lower bounds of the number of sampling points for approximating shapes of planar contours

As a result of rapid advancements of technology, high dimensional data can be easily found almost everywhere. One type of such data is the shape of a contour, where a contour may be viewed as the 2D outline of the image of an object. This type of data arises in medical imaging as well as in computer vision and can be modeled as data on a manifold. It can thus be studied using statistical shape analysis. Practically speaking, each observed contour, while theoretically infinite dimensional, must be discretized for computations. As such, the coordinates for each contour are obtained at k sampling times, resulting in the contour being represented as a k-dimensional complex vector. While choosing large values of k will result in closer approximations to the original contour, this will also result in higher computational costs in the subsequent analysis. The goal of this study is to determine reasonable values for k so as to keep the computational cost low, while maintaining accuracy. To do this, we consider three methods for selecting sample points and determine lower bounds for k for obtaining a desired level of approximation error using two different criteria. Because this process is computationally inefficient to perform on a large scale, we then develop models for predicting the lower bounds for k based on simple characteristics of the contours.