Fast and Stable Polynomial Equation Solving and Its Application to Computer Vision

Abstract

This paper presents several new results on techniques for solving systems of polynomial equations in computer vision. Gröbner basis techniques for equation solving have been applied successfully to several geometric computer vision problems. However, in many cases these methods are plagued by numerical problems. In this paper we derive a generalization of the Gröbner basis method for polynomial equation solving, which improves overall numerical stability. We show how the action matrix can be computed in the general setting of an arbitrary linear basis for C[x]/I . In particular, two improvements on the stability of the computations are made by studying how the linear basis for C[x]/I should be selected. The first of these strategies utilizes QR factorization with column pivoting and the second is based on singular value decomposition (SVD). Moreover, it is shown how to improve stability further by an adaptive scheme for truncation of the Gröbner basis. These new techniques are studied on some of the latest reported uses of Gröbner basis methods in computer vision and we demonstrate dramatically improved numerical stability making it possible to solve a larger class of problems than previously possible.

Robust Fitting for Multiple View Geometry

“…In our current matlab implementation, we do not use a specialized polynomial equation solver. Replacing this component with a state-of-the-art method such as [15] would speed up the computations significantly as each minimal problem typically can be solved in a couple of milliseconds. Another easily accomplished speed-up would be to do the computations in parallel.…”

“…The polynomial equations were solved using a generic polynomial solver based on [15]. Experiments were run on the well-known Dinosaur sequence using the step loss function, i.e.…”

Optimal Geometric Fitting under the Truncated L2-Norm

“…With 3 active constraints we get 3 equations of type (26) and the embedding h(θ) = θ [5] we have implemented a minimal solver in MATLAB that runs in about 0.6 ms on a standard computer (Intel I5).…”

Towards a minimal solution for the relative pose between axial cameras

“…However, as the number of variables grows, the more difficult it becomes to generate a stable polynomial solver. Solving polynomial systems can be achieved with the action matrix method described in [2]. It falls outside the scope of this paper to provide a full account of this method, however, some of its requirements should be discussed.…”