Reconstructing curves from lengths of projections onto lines

TL;DR

This paper explores how to reconstruct a curve from the lengths of its projections onto lines, revealing that the projection data uniquely determines a measure on projective space related to the curve.

Contribution

It introduces a novel approach linking projection lengths to a measure on projective space via the cosine transform, enabling unique reconstruction of the measure and classifying curves by this measure.

Findings

Projection lengths determine a measure on projective space.

The length data is the cosine transform of this measure.

The measure uniquely characterizes the curve class.

Abstract

In this paper, we address the problem of reconstructing a curve from the lengths of its projections onto lines. We first note that the curve itself is not uniquely determined from these measurements. However, we find that a curve determines a measure on projective space which, as a function on Borel subsets of projective space, returns the length of curve parallel to elements of the set. We show that the projected length data can be expressed as the cosine transform of this measure on projective space. The cosine transform is a well studied integral transform on the sphere which is known to be injective. We conclude that the measured length data uniquely determines the associated measure on projective space. We then characterize the class of curves that produce a common measure by starting with the case of piecewise linear curves and then passing to limits to obtain results for more general curves.