用代数方法高效求解4发6收定位问题，精度提升千倍。
An implicitization-based solution to the minimal 4s/6r ToA problem using Cayley--Menger determinants
- 结合凯利-梅格纳行列式与隐式化技术构建新参数化
- 在无噪声数据上精度超现有方法1000倍，速度更快
- 适合需要高精度初始值的声学定位系统
本文提出一种高效的代数求解器，用于解决4发6收时间到达（ToA）自定位问题，即根据收发端之间的距离测量值确定所有节点的相对位置。该方法通过结合凯利-梅格纳行列式与隐式化技术进行新参数化。算法分三步：首先构建148×211的Macaulay矩阵；其次通过PLU分解得到63×63矩阵对；最后利用广义特征分解获得最多38个实数解，并经过验证筛选。在合成无噪声数据上的实验表明，该求解器数值精度比现有方法高出约三个数量级，平均运行时间比最快替代方案快1.3倍。真实声学数据集上的实验进一步验证，将其嵌入RANSAC框架后，可为捆绑调整提供可靠初始猜测。
原文摘要 · Abstract (English)
The paper introduces an efficient algebraic solver for the 4-sender/6-receiver (4s/6r) Time-of-Arrival (ToA) self-localization problem, which involves determining the relative positions of all receivers and senders given their pairwise distance measurements. The problem is addressed through a new parametrization combining Cayley--Menger determinants with an implicitization technique. The proposed algorithm proceeds in three steps. First, a 148 x 211 Macaulay matrix is constructed from the coefficients of the original polynomial system. Second, PLU decomposition of this matrix yields a 63 x 63 matrix pair. Finally, up to 38 real solutions are obtained via generalized eigendecomposition followed by a validation step. Experiments on synthetic noise-free data demonstrate that the proposed solver outperforms existing methods by approximately three orders of magnitude in numerical accuracy while achieving an average runtime of 1.3x faster than the fastest alternative. Experiments on a real-world acoustic dataset confirm that, when integrated within a RANSAC framework, the solver provides a reliable initial guess for bundle adjustment refinement.
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