在密集范围图像中的表面法线的估计

问题描述

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我正在尝试实施，但我不清楚一些要点：

1. 在等式（9）像素位置x处的深度值（Z轴）


2. 如何估计渐变值▽D使用感兴趣点周围的8个邻近点？
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解决方案

1. 如上所述， 意味着对于其中 x = [xy]的 中的 x ， D（ x ）是像素位置的深度

li>

假设我们在 D（ x ）中的深度值5周围有以下邻域 x ：

  8 1 6 
 3 5 7 
 4 9 2 
  

然后使用泰勒展开

d x 。 grad / strong> + 错误 = D（ x + dx ） - D（ x

我们为邻域点获得八个方程。

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

我们可以以矩阵形式表示 Ag + e = b 为

  [1 0; -1 0; 0; 0 -1; 11; 1 -1; -1 1; -1 -1] g + e = [2; -2; 4; -4; -3; 
  

然后最小化平方误差
|| Ag - b | | 。用于最小化此错误的 g 的解析解法形式为

g =（A A） A b p>

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

1. In equation (9), is D(x) corresponding to the depth value (Z-axis) at pixel location x?
2. How to estimate the value of the gradient ▽D using 8 neighboring points around the point of interest?

解决方案

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

2. Estimating the optimal Gradient in a least-square sense

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

Then, using the Taylor expansion

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

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

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]

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

g = (AA)Ab

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