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

##### Resumen

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.