This paper addresses the problem of finding the closest generalized essential matrix from a given \(6\times 6\) matrix, with respect to the Frobenius norm. To the best of our knowledge, this nonlinear constrained optimization problem has not been addressed in the literature yet. Although it can be solved directly, it involves a large number of constraints, and any optimization method to solve it would require much computational effort. We start by deriving a couple of unconstrained formulations of the problem. After that, we convert the original problem into a new one, involving only orthogonal constraints, and propose an efficient algorithm of steepest descent type to find its solution. To test the algorithms, we evaluate the methods with synthetic data and conclude that the proposed steepest descent-type approach is much faster than the direct application of general optimization techniques to the original formulation with 33 constraints and to the unconstrained ones. To further motivate the relevance of our method, we apply it in two pose problems (relative and absolute) using synthetic and real data.

中文翻译：

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关于广义基本矩阵校正：问题的有效解答及其应用

本文解决了从给定\（6 \ times 6 \）中找到最接近的广义基本矩阵的问题关于Frobenius范数的矩阵。据我们所知，该非线性约束优化问题尚未在文献中得到解决。尽管可以直接解决，但是它涉及大量约束，解决该问题的任何优化方法都需要大量的计算工作。我们首先得出问题的两个不受约束的表述。之后，我们将原始问题转换为仅涉及正交约束的新问题，并提出了一种有效的最速下降类型算法来找到其解决方案。要测试算法，我们用综合数据评估了这些方法，并得出结论，提出的最速下降型方法要比将一般优化技术直接应用于具有33个约束条件的原始公式和不受约束条件的方法快得多。为了进一步激发我们方法的相关性，我们使用综合和真实数据将其应用于两个姿势问题（相对和绝对）。