# Reconstruction of convex bodies from moments

arXiv: https://arxiv.org/abs/1605.06362 · HTML: https://mlsanity.com/paper/1605.06362/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 May 2016.

## Built on by (most cited first, 3 of 3)

- [ShipHullGAN: A generic parametric modeller for ship hull design using deep convolutional generative model](https://mlsanity.com/paper/2305.00210/) (2023)
- [VariGrad: A Novel Feature Vector Architecture for Geometric Deep Learning on Unregistered Data](https://mlsanity.com/paper/2307.03553/) (2023)
- [Physics-Informed Geometric Operators to Support Surrogate, Dimension Reduction and Generative Models for Engineering Design](https://mlsanity.com/paper/2407.07611/) (2024)

## Abstract (arXiv)

We investigate how much information about a convex body can be retrieved from a finite number of its geometric moments. We give a sufficient condition for a convex body to be uniquely determined by a finite number of its geometric moments, and we show that among all convex bodies, those which are uniquely determined by a finite number of moments form a dense set. Further, we derive a stability result for convex bodies based on geometric moments. It turns out that the stability result is improved considerably by using another set of moments, namely Legendre moments. We present a reconstruction algorithm that approximates a convex body using a finite number of its Legendre moments. The consistency of the algorithm is established using the stability result for Legendre moments. When only noisy measurements of Legendre moments are available, the consistency of the algorithm is established under certain assumptions on the variance of the noise variables.