On Weighting and Choosing Constraints for Optimally Reconstructing the Geometry of Image Triplets
On Weighting and Choosing Constraints for Optimally Reconstructing the Geometry of Image Triplets
复制标题
关于优化重建图像三元组几何的加权和选择约束
DOI：
10.1007/3-540-45053-x_43
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发表时间：
2000
期刊：
影响因子：
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通讯作者：
W. Förstner
中科院分区：
文献类型：
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作者：
W. Förstner
根据点对应关系最优地重建图像三元组的几何形状需要在观察到的坐标和未知参数之间对所使用的约束进行适当的加权或选择。通过分析 ML 估计过程，本文解决了一系列尚未解决的问题：（1）四个线性独立三线性的最小集合（Shashua 1995，Hartley 1995）实际上只对图像三元组的几何形状施加了三个约束。使用的约束数量（三个与四个）之间看似矛盾，可以使用正规方程自然地解释。 (2)这种估计的直接应用表明具有等级3的4×4矩阵的伪逆，其中包含作为最佳权重矩阵的同源图像点的协方差矩阵。 (3) 除了使用该奇异权重矩阵之外，还可以选择三个线性相关约束。这针对向前和横向运动的两种经典情况进行了讨论，并阐明了 Faugeras 1995 对三线性约束之间依赖关系的代数分析。
具有 800 个图像的图像序列和三焦点张量的欧几里得参数化的结果证明了该方法的可行性。
Optimally reconstructing the geometry of image triplets from point correspondences requires a proper weighting or selection of the used constraints between observed coordinates and unknown parameters. By analysing the ML-estimation process the paper solves a set of yet unsolved problems: (1) The minimal set of four linearily independent trilinearities (Shashua 1995, Hartley 1995) actually imposes only three constraints onto the geometry of the image triplet. The seeming contradiction between the number of used constraints, three vs. four, can be explained naturally using the normal equations. (2) Direct application of such an estimation suggests a pseudoinverse of a 4 × 4-matix having rank 3 which contains the covariance matrix of the homologeous image points to be the optimal weight matrix. (3) Insteadof using this singluar weight matrix one could select three linearily dependent constraints. This is discussed for the two classical cases of forward andlateral motion, and clarifies the algebraic analyis of dependencies between trilinear constraints by Faugeras 1995.
Results of an image sequence with 800 images and an Euclidean parametrization of the trifocal tensor demonstrate the feasibility of the approach.