Spatial Perspective Transform Estimation from Fourier Spectrum Analysis of 2D Patterns in 3D Space

TL;DR

This paper introduces a novel Fourier spectrum analysis method for 3D surface imaging, deriving mathematical relationships for perspective transformations to enhance point-cloud data with localized pattern information.

Contribution

It develops a new approach to extract perspective transformation coefficients from Fourier spectra, expanding on affine transformation pairs for improved 3D surface imaging.

Findings

Validated mathematical relationship for perspective transformation in Fourier domain

Demonstrated congruence with known spatial transformation pairs

Enhanced 3D point-cloud sampling with pattern transformation data

Abstract

A novel approach to 3D surface imaging is proposed, allowing for the continuous sampling of 3D surfaces to extract localized perspective transformation coefficients from Fourier spectrum analysis of projected patterns. The mathematical relationship for Spatial-Fourier Transformation Pairs is derived, defining the transformation of spatial transformed planar surfaces in the Discrete Fourier Transform spectrum. The mathematical relationship for the twelve degrees of freedom in perspective transformation is defined and validated, asserting congruity with independent and uniform transform pairs for spatial Euclidean, similarity, affine and perspective transformations. This work expands on previously derived affine Spatial-Fourier Transformation Pairs and characterizes its implications towards 3D surface imaging as a means of augmenting (X,Y,Z)-(R,G,B) point-clouds to include additional information from localized sampling of pattern transformations.