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Reconstructing Gridded Data from Higher Autocorrelations

W. Riley Casper, Bobby Orozco

TL;DR

This work studies the problem of reconstructing a rational-valued function $f$ on a finite $r$-dimensional grid $G$ from its higher-order autocorrelations $M_n(f;x_1, dots,x_{n-1})$, with reconstruction defined up to a translation. The authors develop an explicit two-stage algorithm based on the discrete Fourier transform and field automorphisms of cyclotomic fields: first reconstruct $\\widehat{f}$ up to a local phase using second- and third-order moments, then fix local phases via a remodulation framework and group-theoretic alignment to obtain $\\widehat{f}$ up to a global translation. They prove a sharp bound: if $|G|$ is odd, $f$ is determined from autocorrelations up to order $2r+2$, and if $|G|$ is even, up to order $3r+3$; they also provide counterexamples showing these bounds are tight. The approach blends harmonic analysis on finite Abelian groups with Galois theory and constructive algebraic methods, yielding a practical reconstruction procedure and insight into when higher-order correlations suffice for unique recovery. The results have implications for imaging, crystallography, and pattern recognition, informing which autocorrelation orders are theoretically sufficient and how to algorithmically recover the underlying data.

Abstract

The higher-order autocorrelations of integer-valued or rational-valued gridded data sets appear naturally in X-ray crystallography, and have applications in computer vision systems, correlation tomography, correlation spectroscopy, and pattern recognition. In this paper, we consider the problem of reconstructing a gridded data set from its higher-order autocorrelations. We describe an explicit reconstruction algorithm, and prove that the autocorrelations up to order 3r + 3 are always sufficient to determine the data up to translation, where r is the dimension of the grid. We also provide examples of rational-valued gridded data sets which are not determined by their autocorrelations up to order 3r + 2.

Reconstructing Gridded Data from Higher Autocorrelations

TL;DR

This work studies the problem of reconstructing a rational-valued function on a finite -dimensional grid from its higher-order autocorrelations , with reconstruction defined up to a translation. The authors develop an explicit two-stage algorithm based on the discrete Fourier transform and field automorphisms of cyclotomic fields: first reconstruct up to a local phase using second- and third-order moments, then fix local phases via a remodulation framework and group-theoretic alignment to obtain up to a global translation. They prove a sharp bound: if is odd, is determined from autocorrelations up to order , and if is even, up to order ; they also provide counterexamples showing these bounds are tight. The approach blends harmonic analysis on finite Abelian groups with Galois theory and constructive algebraic methods, yielding a practical reconstruction procedure and insight into when higher-order correlations suffice for unique recovery. The results have implications for imaging, crystallography, and pattern recognition, informing which autocorrelation orders are theoretically sufficient and how to algorithmically recover the underlying data.

Abstract

The higher-order autocorrelations of integer-valued or rational-valued gridded data sets appear naturally in X-ray crystallography, and have applications in computer vision systems, correlation tomography, correlation spectroscopy, and pattern recognition. In this paper, we consider the problem of reconstructing a gridded data set from its higher-order autocorrelations. We describe an explicit reconstruction algorithm, and prove that the autocorrelations up to order 3r + 3 are always sufficient to determine the data up to translation, where r is the dimension of the grid. We also provide examples of rational-valued gridded data sets which are not determined by their autocorrelations up to order 3r + 2.

Paper Structure

This paper contains 13 sections, 10 theorems, 136 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Mathematical tools
  3. Basic algorithm
  4. When things go wrong
  5. Group notation
  6. Discrete Fourier transforms
  7. Discrete Fourier transforms and autocorrelations
  8. Discrete Fourier transforms and field automorphisms
  9. Reconstruction Algorithm
  10. Reconstruction up to local phase
  11. Reconstructing the local phases
  12. Sharpness of the bound
  13. Examples demonstrating sharpness

Key Result

Proposition 2.2

Let $f$ be a function on a finite Abelian group $G$. The $n$'th autocorrelation of $f$ can be expressed in terms of the discrete Fourier transform as where the sum on the right is taken over all $n$-tuples $(y_1,\dots,y_n)$ which sum to $0$.

Figures (1)

  • Figure 1: A basic pixel image of a crab. This is can be encoded as data on a $13\times 13$ grid with $1$'s representing red values and $2$'s representing black values and zeros everywhere else.

Theorems & Definitions (30)

  • Example 1.1
  • Example 1.2
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • proof
  • Theorem 2.4
  • proof
  • Example 2.5
  • ...and 20 more