Solving Minimal Problems Without Matrix Inversion Using FFT-Based Interpolation

2026-05-07

AI Summary

This work addresses the computational complexity and numerical instability inherent in traditional approaches to minimal problems in camera geometry, which typically rely on matrix inversion. The authors propose a novel method based on sparse hidden-variable resultants, leveraging inverse fast Fourier transform (IFFT) to interpolate the determinant polynomial from sampled values—thereby avoiding symbolic expansion and online matrix inversion. By identifying rank-deficient submatrices and applying Cramer’s rule to recover variables, the approach further enhances robustness to noise through a greatest common divisor criterion. To the best of the authors’ knowledge, this is the first application of FFT-based interpolation to hidden-variable resultant computation. The method demonstrates superior numerical stability and computational efficiency across a range of minimal problems, particularly excelling in small-scale settings, and offers a practical, high-performance alternative to both Gröbner basis and classical resultant techniques.

Abstract

Estimating camera geometry typically involves solving minimal problems formulated as systems of multivariate polynomial equations, which often pose computational challenges when using existing Gröbner-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 Gröbner-basis and resultant-based solvers.

Problem

• minimal problems
• camera geometry
• matrix inversion
• polynomial equations
• computer vision

Innovation

• FFT-based interpolation
• matrix inversion-free
• sparse hidden-variable resultant
• minimal problems
• numerical stability