Sampling of Planar Curves: Theory and Fast Algorithms

TL;DR

This paper presents a novel continuous domain framework for recovering planar curves from limited samples by modeling them as zero level sets of trigonometric polynomials, with theoretical guarantees and a fast iterative algorithm.

Contribution

It introduces a new low-rank feature map approach for curve recovery, providing theoretical guarantees and an efficient algorithm for noisy and unknown bandwidth scenarios.

Findings

The number of samples needed depends on the curve's complexity.

The proposed algorithm effectively recovers curves from noisy samples.

Preliminary results show utility in image segmentation.

Abstract

We introduce a continuous domain framework for the recovery of a planar curve from a few samples. We model the curve as the zero level set of a trigonometric polynomial. We show that the exponential feature maps of the points on the curve lie on a low-dimensional subspace. We show that the null-space vector of the feature matrix can be used to uniquely identify the curve, given a sufficient number of samples. The worst-case theoretical guarantees show that the number of samples required for unique recovery depends on the bandwidth of the underlying trigonometric polynomial, which is a measure of the complexity of the curve. We introduce an iterative algorithm that relies on the low-rank property of the feature maps to recover the curves when the samples are noisy or when the true bandwidth of the curve is unknown. We also demonstrate the preliminary utility of the proposed curve representation in the context of image segmentation.