# Revealing tissue architecture through the hypercomplex Fourier analysis of spatial transcriptomics data

## TL;DR

This paper introduces a new method using quaternions to analyze spatial transcriptomics data, enabling advanced image analysis and visualization techniques.

## Contribution

The novel contribution is applying quaternions and Fourier analysis to spatial transcriptomics data for enhanced analysis and visualization.

## Key findings

- The quaternion-based model allows multidimensional ST data analysis using Fourier techniques.
- Transformations in transcriptomic state can be interpreted as three-dimensional rotations.
- The model supports visualization of transcriptomic uncertainty in spatial data.

## Abstract

We present an approach for analyzing spatial transcriptomics (ST) data using a quaternion-domain discrete Fourier transform. Quaternions are four-dimensional hypercomplex numbers that have been primarily employed to represent rotations in computer graphics with biomedical applications focused on biomolecule structure and orientation.

According to our proposed model, the quaternion associated with each location in an ST dataset represents a vector in R3 whose length captures sequencing depth and whose direction captures three transcriptomic features (individual genes, gene sets, or latent variables). This representation has several important benefits: (i) it enables the use of powerful Fourier-based image analysis techniques on a multidimensional representation of ST data, (ii) it implies that transformations in transcriptomic state can be viewed as three-dimensional rotations with a corresponding representation as rotation quaternions, and (iii) it supports an ST visualization that captures transcriptomic uncertainty. We demonstrate the features of this model through the analysis of Visium HD data and discuss how a similar model can be applied to single-cell RNA-sequencing data.

An implementation of this model, support for the hypercomplex Fourier analysis of ST data, and example vignettes are included in the QSC R package (https://hrfrost.host.dartmouth.edu/QSC/).