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

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.