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.

翻译：估计相机几何通常涉及求解最小问题，这些问题被表述为多元多项式方程组，在使用现有的 Gröbner 基或基于结式的方法时，由于在线求解器所需的矩阵求逆，往往会带来计算挑战。本文提出了一种基于采样的、无需矩阵求逆的方法，该方法利用稀疏隐变量结式构建求解器。通过从采样评估中进行逆快速傅里叶变换插值，高效重构了隐变量中的行列式多项式，避免了符号展开。求解该多项式可得到隐变量，而其余未知量则通过识别秩亏缺子矩阵并应用克莱默法则来恢复。基于最大公约数的准则确保了在噪声下鲁棒的子矩阵识别。在多种最小问题上的实验表明，所提出的求解器具有强大的数值稳定性和有竞争力的运行时间，尤其适用于小规模问题，为传统的基于 Gröbner 基和结式的求解器提供了一种实用的替代方案。