A polynomial formula for the perspective four points problem (2501.13058v1)

Abstract: We present a fast and accurate solution to the perspective n-points problem, by way of a new approach to the n=4 case. Our solution hinges on a novel separation of variables: given four 3D points and four corresponding 2D points on the camera canvas, we start by finding another set of 3D points, sitting on the rays connecting the camera to the 2D canvas points, so that the six pair-wise distances between these 3D points are as close as possible to the six distances between the original 3D points. This step reduces the perspective problem to an absolute orientation problem (which has a solution via explicit formula). To solve the first problem we set coordinates which are as orientation-free as possible: on the 3D points side our coordinates are the squared distances between the points. On the 2D canvas-points side our coordinates are the dot products of the points after rotating one of them to sit on the optical axis. We then derive the solution with the help of a computer algebra system.

• The paper introduces a novel algebraic method that explicitly solves the four-point perspective problem without iterative optimization.
• It employs canonical mappings and algebraic transformations to reduce complex configurations into solvable polynomial equations.
• The proposed solution minimizes reprojection errors and is optimized for real-time applications in computer vision and robotics.

A Polynomial Formula for the Perspective Four Points Problem

The paper "A polynomial formula for the perspective four points problem," authored by David Lehavi and Brian Osserman, presents a novel approach to efficiently and accurately solving the perspective $n$-points problem, particularly focusing on the $n=4$ case. The perspective $n$-points problem (PnP) is a classical issue in computer vision, where given $n$ correspondences between 3D points and 2D image projections, the task is to determine the six degrees of freedom (DoF) of the camera's pose. The challenge becomes apparent, especially when $n=4$, as this case is typically overdetermined and traditionally solved through optimization-based methods.

The authors introduce a sound methodology that circumvents the usual iterative optimization processes. Key to their innovation is a technique grounded in algebraic geometry, which involves a clever separation of variables and transforms the original problem into one capable of explicit algebraic manipulation. They achieve this via a novel canonical mapping from the original configuration space to a lower-dimensional vector space, reducing the complexity involved in solving for the intermediate variables, or $z_i$, which denote the $z$-depths of the points along the camera rays.

This paper leverages two mathematical mappings: one for the quadruples of 3D points, represented by squared distances, and another for the lines, represented by dot products. These representations simplify the problem into algebraic forms, leading to a correspondence variety in a reduced space where the problem can be tackled by solving polynomial equations.

The resulting polynomial equations, particularly the quadrics $Q_i$, pertinent to each $z_i^2$, are central to solving the P4P problem. The authors derive these quadrics using a combination of symbolic computation performed in the Singular computer algebra system, supplemented by logical human intervention. Additionally, they harness the symmetry from an $S_3$ permutation action among the indices of their variables to derive linear conditions from pairs of quadrics, hence reducing the computational load and ensuring robustness in numerical settings.

This explicit algebraic solution is computationally efficient, expeditiously producing solutions that generally exhibit lower reprojection errors compared to traditional minimization methods like EPnP and SQPnP. The authors underscore its potential architectural optimization, making it highly compatible with SIMD implementation, which enhances its practical utility in real-time applications.

The implications of this research extend into various practical realms of computer vision, robotics, and augmented reality, where real-world environments necessitate robust and rapid pose estimation algorithms. Theoretically, this work enriches the paper of algebraic geometry's applications in computational problems, encouraging further exploration of deterministic solutions over heuristic optimizations.

Speculating on future developments, this research could inspire algorithms for larger $n$-thing problems, incorporating more sophisticated algebraic manipulations or hybrid approaches that blend their explicit method with optimization techniques for even greater accuracy and efficiency. Furthermore, the integration of such approaches into hardware accelerators promises advancements in on-device computation for mobile and embedded systems, a critical requirement for next-generation AI applications.