密集范围图像中表面法线的估计 [英] Estimation of Surface Normal in a Dense Range Image

问题描述

我正在尝试实施 Hinterstoisser 等人 (2011) 提出的表面法线估计) 但我不清楚某些点:

I am trying to implement the surface normal estimation proposed by Hinterstoisser et al (2011) but I'm not clear with some points:

1. 在等式(9)中，D(x) 是否对应于像素位置 x 处的深度值(Z 轴)?
2. 如何使用兴趣点周围的 8 个相邻点来估计梯度 ▽D 的值?

推荐答案

1. 如前所述，D 是一个密集范围图像，这意味着对于任何像素位置 x在 D 中 x = [xy]^T，D(x>) 是像素位置x(或简单地D(x, y))处的深度.

1. As mentioned, D is a dense range image meaning that for any pixel location x in D where x = [x y]^T, D(x) is the depth at pixel location x (or simply D(x, y)).

在最小二乘意义上估计最佳梯度

Estimating the optimal Gradient in a least-square sense

假设我们在 D(x) 中的深度值 5 周围有以下邻域，对于某些 x:

Suppose we have the following neighborhood around the depth value 5 in D(x) for some x:

8   1   6
3   5   7
4   9   2

然后，使用泰勒展开

dx^T.grad(x) + 错误 = D(x + dx) - D(x)

dx^T.grad(x) + error = D(x + dx) - D(x)

我们得到邻域点的八个方程

we get eight equations for the neighborhood points

[1   0]g + e = 7 - 5
[-1  0]g + e = 3 - 5
[0   1]g + e = 9 - 5
[0  -1]g + e = 1 - 5
[1   1]g + e = 2 - 5
[1  -1]g + e = 6 - 5
[-1  1]g + e = 4 - 5
[-1 -1]g + e = 8 - 5

我们可以用矩阵形式表示 Ag + e = b 为

that we can represent in matrix form Ag + e = b as

[1  0;-1  0;0  1;0 -1;1 1;1 -1;-1 1;-1 -1]g + e= [2;-2;4;-4;-3;1;-1;3]

然后最小化平方误差||Ag - b||₂².最小化这个误差的 g^{^} 的解析解是形式

Then minimize the squared error ||Ag - b||₂². The analytical solution for g^{^} that minimizes this error is of the form

g^{^} = (A^TA)^-1A^Tb

这篇关于密集范围图像中表面法线的估计的文章就介绍到这了，希望我们推荐的答案对大家有所帮助，也希望大家多多支持IT屋！