Solving Minimal Problems Without Matrix Inversion Using FFT-Based Interpolation

TL;DR

This paper introduces a fast, matrix inversion-free method for solving minimal problems in camera geometry using FFT-based interpolation, improving stability and efficiency.

Contribution

It presents a novel sampling-based solver that constructs solutions via sparse hidden-variable resultants and FFT interpolation, avoiding symbolic expansion and matrix inversion.

Findings

Achieves strong numerical stability in diverse minimal problems.

Provides competitive runtime, especially for small-scale problems.

Offers a practical alternative to traditional algebraic solvers.

Abstract

Estimating camera geometry typically involves solving minimal problems formulated as systems of multivariate polynomial equations, which often pose computational challenges when using existing Gr\"obner-basis or resultant-based methods due to matrix inversion needed in the online solver. Here we propose a sampling-based, matrix inversion-free method that constructs the solvers using sparse hidden-variable resultants. The determinant polynomial in the hidden variable is efficiently reconstructed via inverse fast Fourier transform interpolation from sampled evaluations, avoiding symbolic expansion. Solving this polynomial yields the hidden variable, and the remaining unknowns are recovered by identifying rank-1 deficient submatrices and applying Cramer's rule. A greatest common divisor-based criterion ensures robust submatrix identification under noise. Experiments on diverse minimal problems demonstrate that the proposed solver achieves strong numerical stability and competitive runtime, particularly for small-scale problems, providing a practical alternative to traditional Gr\"obner-basis and resultant-based solvers.