Multiprojective Geometry of Compatible Triples of Fundamental and Essential Matrices

In structure from motion, quadrifocal tensors capture more information than their pairwise counterparts (essential matrices), yet they have often been thought of as impractical and only of theoretical interest. In this work, we challenge such beliefs by providing a new framework to recover $n$ cameras from the corresponding collection of quadrifocal tensors. We form the block quadrifocal tensor and show that it admits a Tucker decomposition whose factor matrices are the stacked camera matrices, and which thus has a multilinear rank of (4,~4,~4,~4) independent of $n$. We develop the first synchronization algorithm for quadrifocal tensors, using Tucker decomposition, alternating direction method of multipliers, and iteratively reweighted least squares. We further establish relationships between the block quadrifocal, trifocal, and bifocal tensors, and introduce an algorithm that jointly synchronizes these three entities. Numerical experiments demonstrate the effectiveness of our methods on modern datasets, indicating the potential and importance of using higher-order information in synchronization.

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.

Estimating a camera pose from 3D-2D correspondences - Perspective-n-Point - is one of the most fundamental problems in computer vision. Existing methods either solve a minimal problem or replace the objective with an easier one. Surprisingly, no scalable certifiable solver for the actual Maximum Likelihood Estimation (MLE) exists. We propose PnP*, the first practical solver that guarantees efficient computation and returns a certifiable globally optimal solution. We prove that the underlying convex relaxation is exact under moderate noise, which to our knowledge is the first stability guarantee for any relaxation for the MLE of PnP. Furthermore, we reduce the relaxation by exploiting chordal sparsity pattern and quotient ring reduction such that the relaxation is tractable and scalable. The stability and scalability results are verified numerically, as PnP* returns a provably correct camera pose at over 10 Hz.