基于R的运动简单正交结构——确定度量约束

我想根据Tomasi和Kanade[1992]从运动程序构建一个简单的结构。文章内容如下：

这种方法看起来优雅而简单，但是，我在计算上述参考文献等式16中概述的度量约束时遇到困难

我正在使用R，并概述了我迄今为止的工作：

给定一组图像

我想追踪三个柜门的角落和一张图片（图片上的黑点）。首先，我们将点读入矩阵w，其中

最后，我们要将w分解为描述三维点的旋转矩阵R和形状矩阵S。我将尽可能多地保留细节，但可以从Tomasi和Kanade[1992]的论文中收集数学的完整描述

我提供w如下：

w.vector=c(0.2076,0.1369,0.1918,0.1862,0.1741,0.1434,0.176,0.1723,0.2047,0.233,0.3593,0.3668,0.3744,0.3593,0.3876,0.3574,0.3639,0.3062,0.3295,0.3267,0.3128,0.2811,0.2979,0.2876,0.2782,0.2876,0.3838,0.3819,0.3819,0.3649,0.3913,0.3555,0.3593,0.2997,0.3202,0.3137,0.31,0.2718,0.2895,0.2867,0.825,0.7703,0.742,0.7251,0.7232,0.7138,0.7345,0.6911,0.1937,0.1248,0.1723,0.1741,0.1657,0.1313,0.162,0.1657,0.8834,0.8118,0.7552,0.727,0.7364,0.7232,0.7288,0.6892,0.4309,0.3798,0.4021,0.3965,0.3844,0.3546,0.3695,0.3583,0.314,0.3065,0.3989,0.3876,0.3857,0.3781,0.3989,0.3593,0.5184,0.4849,0.5147,0.5193,0.5109,0.4812,0.4979,0.4849,0.3536,0.3517,0.4121,0.3951,0.3951,0.3781,0.397,0.348,0.5175,0.484,0.5091,0.5147,0.5128,0.4784,0.4905,0.4821,0.7722,0.7326,0.7326,0.7232,0.7232,0.7119,0.7402,0.7006,0.4281,0.3779,0.3918,0.3863,0.3825,0.3472,0.3611,0.3537,0.8043,0.7628,0.7458,0.7288,0.727,0.7213,0.7364,0.6949,0.5789,0.5491,0.5761,0.5817,0.5733,0.5444,0.5537,0.5379,0.3649,0.3536,0.4177,0.3951,0.3857,0.3819,0.397,0.3461,0.697,0.671,0.6821,0.6821,0.6719,0.6412,0.6468,0.6235,0.3744,0.3649,0.4159,0.3819,0.3781,0.3612,0.3763,0.314,0.7008,0.6691,0.6794,0.6812,0.6747,0.6393,0.6412,0.6235,0.7571,0.7345,0.7439,0.7496,0.7402,0.742,0.7647,0.7213,0.5817,0.5463,0.5696,0.5779,0.5761,0.5398,0.551,0.5398,0.7665,0.7326,0.7439,0.7345,0.7288,0.727,0.7515,0.7062,0.8301,0.818,0.8571,0.8878,0.8766,0.8561,0.858,0.8394,0.4121,0.3876,0.4347,0.397,0.38,0.3631,0.3668,0.2971,0.912,0.8962,0.9185,0.939,0.9259,0.898,0.8887,0.8571,0.3989,0.3781,0.4215,0.3725,0.3612,0.3461,0.3423,0.2782,0.9092,0.8952,0.9176,0.9399,0.925,0.8971,0.8887,0.8571,0.4743,0.4536,0.4894,0.4517,0.446,0.4328,0.4385,0.3706,0.8273,0.8171,0.8571,0.8878,0.8766,0.8543,0.8561,0.8394,0.4743,0.4554,0.4969,0.4668,0.4536,0.4404,0.4536,0.3857)

w=matrix(w.vector,ncol=16,nrow=16,byrow=FALSE)

然后根据方程式2创建注册测量矩阵

wm

，如下所示

借

通过使用奇异值分解，我们可以将

wm

分解为'2FxP'矩阵

o1

对角'PxP'矩阵

e

和'PxP'矩阵

o2

svdwm <- svd(wm)

o1 <- svdwm$u
e <-  diag(svdwm$d)
o2 <- t(svdwm$v) ## dont forget the transpose!

现在我们可以解方程（14）中的

rhat

和

shat

借

rhat可以写成

可以写成

可以写成

我们的方程是：




所以第一个方程可以写成：





这相当于


简而言之，我们现在定义：


（我知道间距非常小，但是是的，这是一个向量…）

对于所有不同帧f中的所有方程，我们可以写出一个大方程：


（为难看的公式感到抱歉…）
现在你只需要用Cholesky分解或其他方法解-矩阵

svdwm <- svd(wm)

o1 <- svdwm$u
e <-  diag(svdwm$d)
o2 <- t(svdwm$v) ## dont forget the transpose!

o1p <- svdwm$u[,1:3]
ep <-  diag(svdwm$d[1:3])
o2p <- t(svdwm$v)[1:3,] ## dont forget the transpose!

rhat <- o1p%*%ep^(1/2)
shat <- ep^(1/2) %*% o2p