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.

strategy for selecting a basis which improves the conditioning of a crucial elimination step, (ii) use this technique to devise a Gröbner basis with improved precision and (iii) show how solving for the eigenvalues instead of eigenvectors can be used to improve precision further while retaining the same speed.We study these methods on some of the latest reported uses of Gröbner basis methods and demonstrate dramatically improved numerical precision using these new techniques making it possible to solve a larger class of problems than previously.

Correspondence problems are of great importance in computer vision. They appear as subtasks in many applications such as object recognition, merging partial 3D reconstructions and image alignment. Automatically matching features from appearance only is difficult and errors are frequent. Thus, it is necessary to use geometric consistency to remove incorrect correspondences. Typically heuristic methods like RANSAC or EM-like algorithms are used, but they risk getting trapped in local optima and are in no way guaranteed to find the best solution.This paper illustrates how pairwise constraints in combination with graph methods can be used to efficiently find optimal correspondences. These ideas are implemented on two basic geometric problems, 3D-3D registration and 2D-3D registration. The developed scheme can handle large rates of outliers and cope with multiple hypotheses. Despite the combinatorial explosion, the resulting algorithm which has been extensively evaluated on real data, yields competitive running times compared to state of the art.

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