Title: 图像坐标导数的表达式 Expression for Image Coordinate Derivate
文章目录
多视图几何读书笔记
[2] [笔记] 仿射变换性质的代数证明
[3] 多视图几何中向量叉乘的矩阵转换恒等式 Matrix Transformation for Cross Product in MVG
[5] 图像坐标导数的表达式 Expression for Image Coordinate Derivate ← \leftarrow ← 本篇
简单推导一下多视图几何中图像坐标导数的表达式[1].
I. 图像坐标 Image Coordinates
给定二维射影平面 P 2 \mathbb{P}^2 P2 内的一组点集 { x i } \{\mathbf{x}_i\} {xi} 以及另一二维射影平面 P 2 \mathbb{P}^2 P2 内的一组点集 { x i ′ } \{\mathbf{x}'_i\} {xi′}. 这两组点集就代表两张不同图片上的像素点.
如果图像之间存在对应关系
x
i
↔
x
i
′
\mathbf{x}_i \leftrightarrow \mathbf{x}'_i
xi↔xi′. 则满足如下关系的
3
×
3
3\times 3
3×3 矩阵称为二维单应矩阵 (2D Homography),
H
x
i
=
x
i
′
for each
i
(I-1)
H \mathbf{x}_i = \mathbf{x}'_i \quad \text{for each}\; i \tag{I-1}
Hxi=xi′for eachi(I-1)
假设二维射影平面上存在一点
x
\mathbf{x}
x 的其次坐标为
x
=
(
x
y
w
)
\mathbf{x}=\begin{pmatrix}x \\ y \\ w\end{pmatrix}
x=
xyw
, 非齐次坐标为
x
~
=
(
x
~
y
~
)
=
(
x
/
w
y
/
w
)
\tilde{\mathbf{x}}= \begin{pmatrix}\tilde{x} \\ \tilde{y} \end{pmatrix} =\begin{pmatrix} {x}/{w} \\ {y}/{w} \end{pmatrix}
x~=(x~y~)=(x/wy/w).
另一二维射影平面上的对应点 x ′ \mathbf{x}' x′ 的其次坐标为 x ′ = ( x ′ y ′ w ′ ) \mathbf{x}'=\begin{pmatrix}x' \\ y' \\ w'\end{pmatrix} x′= x′y′w′ , 非齐次坐标为 x ~ ′ = ( x ~ ′ y ~ ′ ) = ( x ′ / w ′ y ′ / w ′ ) \tilde{\mathbf{x}}'= \begin{pmatrix}\tilde{x}' \\ \tilde{y} ' \end{pmatrix} = \begin{pmatrix}{x'}/{w'} \\ {y'}/{w'} \end{pmatrix} x~′=(x~′y~′)=(x′/w′y′/w′).
存在单应矩阵
x
′
=
H
x
⇒
(
x
′
y
′
w
′
)
=
[
h
11
h
12
h
13
h
21
h
22
h
23
h
31
h
32
h
33
]
(
x
y
w
)
(I-2)
\mathbf{x}' = H \mathbf{x}\quad \Rightarrow \\ \begin{pmatrix} x' \\ y'\\w' \end{pmatrix} = \begin{bmatrix}h_{11} & h_{12} &h_{13} \\ h_{21} & h_{22} &h_{23} \\ h_{31} & h_{32} &h_{33} \end{bmatrix} \begin{pmatrix} x \\ y \\w \end{pmatrix} \tag{I-2}
x′=Hx⇒
x′y′w′
=
h11h21h31h12h22h32h13h23h33
xyw
(I-2)
将
H
H
H 的第
i
i
i 行行向量记为
h
i
T
=
[
h
i
1
,
h
i
2
,
h
i
3
]
\mathbf{h}^{i\rm{T}} = \begin{bmatrix} h_{i1}, h_{i2}, h_{i3}\end{bmatrix}
hiT=[hi1,hi2,hi3], 则
(
x
′
y
′
w
′
)
=
[
h
1
T
x
h
2
T
x
h
3
T
x
]
(I-3)
\begin{pmatrix} x' \\ y'\\w' \end{pmatrix} = \begin{bmatrix} \mathbf{h}^{1\rm{T}} \mathbf{x} \\ \mathbf{h}^{2\rm{T}} \mathbf{x}\\ \mathbf{h}^{3\rm{T}} \mathbf{x} \end{bmatrix} \tag{I-3}
x′y′w′
=
h1Txh2Txh3Tx
(I-3)
那么非齐次坐标的表达式为
x
~
′
=
(
x
~
′
y
~
′
)
=
(
x
′
/
w
′
y
′
/
w
′
)
=
[
h
1
T
x
h
3
T
x
h
2
T
x
h
3
T
x
]
(I-4)
\tilde{\mathbf{x}}'= \begin{pmatrix}\tilde{x}' \\ \tilde{y} ' \end{pmatrix} = \begin{pmatrix}{x'}/{w'} \\ {y'}/{w'} \end{pmatrix} =\left[ \begin{aligned} \frac{\mathbf{h}^{1\rm{T}} \mathbf{x}}{\mathbf{h}^{3\rm{T}} \mathbf{x}} \\ \frac{\mathbf{h}^{2\rm{T}} \mathbf{x}}{\mathbf{h}^{3\rm{T}} \mathbf{x}} \end{aligned}\right] \tag{I-4}
x~′=(x~′y~′)=(x′/w′y′/w′)=
h3Txh1Txh3Txh2Tx
(I-4)
另外
H
H
H 写成 9 维向量形式
h
=
[
h
1
h
2
h
3
]
=
[
h
11
h
12
h
13
h
21
⋮
h
33
]
(I-5)
\mathbf{h} = \begin{bmatrix} \mathbf{h}^{1} \\ \mathbf{h}^{2} \\ \mathbf{h}^{3} \end{bmatrix} = \begin{bmatrix} h_{11} \\\ h_{12} \\ h_{13} \\ h_{21} \\ \vdots \\ h_{33} \end{bmatrix} \tag{I-5}
h=
h1h2h3
=
h11 h12h13h21⋮h33
(I-5)
II. 关于 x \mathbf{x} x 的导数 Derivative wrt x \mathbf{x} x
由向量的导数的分子布局 (参考之前博文 Derivative of Vector with Respect to Vector (Numerator-Layout Notation - in common use) ) 可知
∂
x
~
′
∂
x
=
[
∂
x
~
′
/
∂
x
∂
y
~
′
/
∂
x
]
(II-1)
\frac{\partial \tilde{\mathbf{x}}'}{\partial\mathbf{x}} = \begin{bmatrix} {\partial \tilde{{x}}'}/{\partial\mathbf{x}} \\ {\partial \tilde{{y}}'}/{\partial\mathbf{x}} \end{bmatrix} \tag{II-1}
∂x∂x~′=[∂x~′/∂x∂y~′/∂x](II-1)
上式分为两部分分别计算.
1. 第一部分
∂ x ~ ′ ∂ x = ∂ ( h 1 T x h 3 T x ) / ∂ x = ( ∂ h 1 T x / ∂ x ) ( h 3 T x ) − ( h 1 T x ) ( ∂ h 3 T x / ∂ x ) ( h 3 T x ) ( h 3 T x ) = 1 w ′ h 1 T ( h 3 T x ) − ( h 1 T x ) h 3 T h 3 T x = 1 w ′ ( h 1 T − h 1 T x h 3 T x h 3 T ) = 1 w ′ ( h 1 T − x ~ ′ h 3 T ) (II-1-1) \begin{aligned} \frac{\partial \tilde{{x}}'}{\partial\mathbf{x}} &= \partial \left(\frac{\mathbf{h}^{1\rm{T}} \mathbf{x}}{\mathbf{h}^{3\rm{T}} \mathbf{x}}\right)/\partial \mathbf{x}\\ &= \frac{\left({\partial {\mathbf{h}^{1\rm{T}} \mathbf{x}}}/{\partial \mathbf{x}}\right) \left({\mathbf{h}^{3\rm{T}} \mathbf{x}}\right) - \left({\mathbf{h}^{1\rm{T}} \mathbf{x}}\right) \left({\partial {\mathbf{h}^{3\rm{T}} \mathbf{x}}}/{\partial \mathbf{x}}\right) }{\left({\mathbf{h}^{3\rm{T}} \mathbf{x}}\right) \left({\mathbf{h}^{3\rm{T}} \mathbf{x}}\right)}\\ &=\frac{1}{w'} \frac{ \mathbf{h}^{1\rm{T}} \left({\mathbf{h}^{3\rm{T}} \mathbf{x}}\right) - \left({\mathbf{h}^{1\rm{T}} \mathbf{x}}\right) {\mathbf{h}^{3\rm{T}}} } {{\mathbf{h}^{3\rm{T}} \mathbf{x}}}\\ & = \frac{1}{w'} \left(\mathbf{h}^{1\rm{T}} - \frac{\mathbf{h}^{1\rm{T}}\mathbf{x}}{\mathbf{h}^{3\rm{T}}\mathbf{x}}\mathbf{h}^{3\rm{T}} \right)\\ & = \frac{1}{w'} \left(\mathbf{h}^{1\rm{T}} - \tilde{x}' \mathbf{h}^{3\rm{T}} \right) \end{aligned} \tag{II-1-1} ∂x∂x~′=∂(h3Txh1Tx)/∂x=(h3Tx)(h3Tx)(∂h1Tx/∂x)(h3Tx)−(h1Tx)(∂h3Tx/∂x)=w′1h3Txh1T(h3Tx)−(h1Tx)h3T=w′1(h1T−h3Txh1Txh3T)=w′1(h1T−x~′h3T)(II-1-1)
2. 第二部分
∂ y ~ ′ ∂ x = ∂ ( h 2 T x h 3 T x ) / ∂ x = ( ∂ h 2 T x / ∂ x ) ( h 3 T x ) − ( h 2 T x ) ( ∂ h 3 T x / ∂ x ) ( h 3 T x ) ( h 3 T x ) = 1 w ′ h 2 T ( h 3 T x ) − ( h 2 T x ) h 3 T h 3 T x = 1 w ′ ( h 2 T − h 2 T x h 3 T x h 3 T ) = 1 w ′ ( h 2 T − y ~ ′ h 3 T ) (II-2-1) \begin{aligned} \frac{\partial \tilde{{y}}'}{\partial\mathbf{x}} &= \partial \left(\frac{\mathbf{h}^{2\rm{T}} \mathbf{x}}{\mathbf{h}^{3\rm{T}} \mathbf{x}}\right)/\partial \mathbf{x}\\ &= \frac{\left({\partial {\mathbf{h}^{2\rm{T}} \mathbf{x}}}/{\partial \mathbf{x}}\right) \left({\mathbf{h}^{3\rm{T}} \mathbf{x}}\right) - \left({\mathbf{h}^{2\rm{T}} \mathbf{x}}\right) \left({\partial {\mathbf{h}^{3\rm{T}} \mathbf{x}}}/{\partial \mathbf{x}}\right) }{\left({\mathbf{h}^{3\rm{T}} \mathbf{x}}\right) \left({\mathbf{h}^{3\rm{T}} \mathbf{x}}\right)}\\ &=\frac{1}{w'} \frac{ \mathbf{h}^{2\rm{T}} \left({\mathbf{h}^{3\rm{T}} \mathbf{x}}\right) - \left({\mathbf{h}^{2\rm{T}} \mathbf{x}}\right) {\mathbf{h}^{3\rm{T}}} } {{\mathbf{h}^{3\rm{T}} \mathbf{x}}}\\ & = \frac{1}{w'} \left(\mathbf{h}^{2\rm{T}} - \frac{\mathbf{h}^{2\rm{T}}\mathbf{x}}{\mathbf{h}^{3\rm{T}}\mathbf{x}}\mathbf{h}^{3\rm{T}} \right)\\ & = \frac{1}{w'} \left(\mathbf{h}^{2\rm{T}} - \tilde{y}' \mathbf{h}^{3\rm{T}} \right) \end{aligned} \tag{II-2-1} ∂x∂y~′=∂(h3Txh2Tx)/∂x=(h3Tx)(h3Tx)(∂h2Tx/∂x)(h3Tx)−(h2Tx)(∂h3Tx/∂x)=w′1h3Txh2T(h3Tx)−(h2Tx)h3T=w′1(h2T−h3Txh2Txh3T)=w′1(h2T−y~′h3T)(II-2-1)
3. 两部分合并
所以有
∂
x
~
′
∂
x
=
[
∂
x
~
′
/
∂
x
∂
y
~
′
/
∂
x
]
=
1
w
′
[
h
1
T
−
x
~
′
h
3
T
h
2
T
−
y
~
′
h
3
T
]
(II-3-1)
\frac{\partial \tilde{\mathbf{x}}'}{\partial\mathbf{x}} = \begin{bmatrix} {\partial \tilde{{x}}'}/{\partial\mathbf{x}} \\ {\partial \tilde{{y}}'}/{\partial\mathbf{x}} \end{bmatrix} = \frac{1}{w'} \begin{bmatrix} \mathbf{h}^{1\rm{T}} - \tilde{x}' \mathbf{h}^{3\rm{T}} \\ \mathbf{h}^{2\rm{T}} - \tilde{y}' \mathbf{h}^{3\rm{T}} \end{bmatrix} \tag{II-3-1}
∂x∂x~′=[∂x~′/∂x∂y~′/∂x]=w′1[h1T−x~′h3Th2T−y~′h3T](II-3-1)
III. 关于 H H H 的导数 Derivative wrt H H H
由向量的导数的分子布局, 可知
∂
x
~
′
∂
h
=
[
∂
x
~
′
/
∂
h
∂
y
~
′
/
∂
h
]
(III-1)
\frac{\partial \tilde{\mathbf{x}}'}{\partial\mathbf{h}} = \begin{bmatrix} {\partial \tilde{{x}}'}/{\partial\mathbf{h}} \\ {\partial \tilde{{y}}'}/{\partial\mathbf{h}} \end{bmatrix} \tag{III-1}
∂h∂x~′=[∂x~′/∂h∂y~′/∂h](III-1)
上式分为两部分分别计算.
1. 第一部分
由向量的导数的分子布局, 可知
∂
x
~
′
∂
h
=
[
∂
x
~
′
/
∂
h
1
T
∂
x
~
′
/
∂
h
2
T
∂
x
~
′
/
∂
h
3
T
]
=
[
∂
(
h
1
T
x
h
3
T
x
)
/
∂
h
1
T
∂
(
h
1
T
x
h
3
T
x
)
/
∂
h
2
T
∂
(
h
1
T
x
h
3
T
x
)
/
∂
h
3
T
]
=
[
x
T
h
3
T
x
0
T
−
(
h
1
T
x
)
x
T
(
h
3
T
x
)
(
h
3
T
x
)
]
=
1
w
′
[
x
T
0
T
−
h
1
T
x
h
3
T
x
x
T
]
=
1
w
′
[
x
T
0
T
−
x
~
′
x
T
]
(III-1-1)
\begin{aligned} \frac{\partial \tilde{{x}}'}{\partial\mathbf{h}} &= \begin{bmatrix} {\partial \tilde{{x}}'}/{\partial\mathbf{h}^{1\rm{T}}} & {\partial \tilde{{x}}'}/{\partial\mathbf{h}^{2\rm{T}}} & {\partial \tilde{{x}}'}/{\partial\mathbf{h}^{3\rm{T}}} \end{bmatrix}\\ & = \begin{bmatrix} \partial\left(\frac{\mathbf{h}^{1\rm{T}} \mathbf{x}}{\mathbf{h}^{3\rm{T}} \mathbf{x}}\right)/{\partial\mathbf{h}^{1\rm{T}}} &\partial\left(\frac{\mathbf{h}^{1\rm{T}} \mathbf{x}}{\mathbf{h}^{3\rm{T}} \mathbf{x}}\right)/{\partial\mathbf{h}^{2\rm{T}}} &\partial\left(\frac{\mathbf{h}^{1\rm{T}} \mathbf{x}}{\mathbf{h}^{3\rm{T}} \mathbf{x}}\right)/{\partial\mathbf{h}^{3\rm{T}}} \end{bmatrix}\\ & = \begin{bmatrix} \frac{\mathbf{x}^{\rm{T}}}{\mathbf{h}^{3\rm{T}} \mathbf{x}} & \mathbf{0}^{\rm{T}} & -\frac{\left(\mathbf{h}^{1\rm{T}} \mathbf{x}\right) \mathbf{x}^{\rm T}}{\left(\mathbf{h}^{3\rm{T}} \mathbf{x}\right)\left(\mathbf{h}^{3\rm{T}} \mathbf{x}\right)} \end{bmatrix}\\ & = \frac{1}{w'} \begin{bmatrix} {\mathbf{x}^{\rm{T}}} & \mathbf{0}^{\rm{T}} & -\frac{\mathbf{h}^{1\rm{T}} \mathbf{x} }{\mathbf{h}^{3\rm{T}} \mathbf{x}} \mathbf{x}^{\rm{T}} \end{bmatrix}\\ & = \frac{1}{w'} \begin{bmatrix} {\mathbf{x}^{\rm{T}}} & \mathbf{0}^{\rm{T}} & - \tilde{x}' \mathbf{x}^{\rm{T}} \end{bmatrix} \end{aligned} \tag{III-1-1}
∂h∂x~′=[∂x~′/∂h1T∂x~′/∂h2T∂x~′/∂h3T]=[∂(h3Txh1Tx)/∂h1T∂(h3Txh1Tx)/∂h2T∂(h3Txh1Tx)/∂h3T]=[h3TxxT0T−(h3Tx)(h3Tx)(h1Tx)xT]=w′1[xT0T−h3Txh1TxxT]=w′1[xT0T−x~′xT](III-1-1)
2. 第二部分
∂ y ~ ′ ∂ h = [ ∂ y ~ ′ / ∂ h 1 T ∂ y ~ ′ / ∂ h 2 T ∂ y ~ ′ / ∂ h 3 T ] = [ ∂ ( h 2 T x h 3 T x ) / ∂ h 1 T ∂ ( h 2 T x h 3 T x ) / ∂ h 2 T ∂ ( h 2 T x h 3 T x ) / ∂ h 3 T ] = [ 0 T x T h 3 T x − ( h 2 T x ) x T ( h 3 T x ) ( h 3 T x ) ] = 1 w ′ [ 0 T x T − h 2 T x h 3 T x x T ] = 1 w ′ [ 0 T x T − y ~ ′ x T ] (III-2-1) \begin{aligned} \frac{\partial \tilde{{y}}'}{\partial\mathbf{h}} &= \begin{bmatrix} {\partial \tilde{{y}}'}/{\partial\mathbf{h}^{1\rm{T}}} & {\partial \tilde{{y}}'}/{\partial\mathbf{h}^{2\rm{T}}} & {\partial \tilde{{y}}'}/{\partial\mathbf{h}^{3\rm{T}}} \end{bmatrix}\\ & = \begin{bmatrix} \partial\left(\frac{\mathbf{h}^{2\rm{T}} \mathbf{x}}{\mathbf{h}^{3\rm{T}} \mathbf{x}}\right)/{\partial\mathbf{h}^{1\rm{T}}} &\partial\left(\frac{\mathbf{h}^{2\rm{T}} \mathbf{x}}{\mathbf{h}^{3\rm{T}} \mathbf{x}}\right)/{\partial\mathbf{h}^{2\rm{T}}} &\partial\left(\frac{\mathbf{h}^{2\rm{T}} \mathbf{x}}{\mathbf{h}^{3\rm{T}} \mathbf{x}}\right)/{\partial\mathbf{h}^{3\rm{T}}} \end{bmatrix}\\ & = \begin{bmatrix} \mathbf{0}^{\rm{T}} &\frac{\mathbf{x}^{\rm{T}}}{\mathbf{h}^{3\rm{T}} \mathbf{x}} & -\frac{\left(\mathbf{h}^{2\rm{T}} \mathbf{x}\right) \mathbf{x}^{\rm T}}{\left(\mathbf{h}^{3\rm{T}} \mathbf{x}\right)\left(\mathbf{h}^{3\rm{T}} \mathbf{x}\right)} \end{bmatrix}\\ & = \frac{1}{w'} \begin{bmatrix} \mathbf{0}^{\rm{T}} & {\mathbf{x}^{\rm{T}}} & -\frac{\mathbf{h}^{2\rm{T}} \mathbf{x} }{\mathbf{h}^{3\rm{T}} \mathbf{x}} \mathbf{x}^{\rm{T}} \end{bmatrix}\\ & = \frac{1}{w'} \begin{bmatrix} \mathbf{0}^{\rm{T}} & {\mathbf{x}^{\rm{T}}} & - \tilde{y}' \mathbf{x}^{\rm{T}} \end{bmatrix} \end{aligned} \tag{III-2-1} ∂h∂y~′=[∂y~′/∂h1T∂y~′/∂h2T∂y~′/∂h3T]=[∂(h3Txh2Tx)/∂h1T∂(h3Txh2Tx)/∂h2T∂(h3Txh2Tx)/∂h3T]=[0Th3TxxT−(h3Tx)(h3Tx)(h2Tx)xT]=w′1[0TxT−h3Txh2TxxT]=w′1[0TxT−y~′xT](III-2-1)
3. 合并
∂ x ~ ′ ∂ h = [ ∂ x ~ ′ / ∂ h ∂ y ~ ′ / ∂ h ] = 1 w ′ [ x T 0 T − x ~ ′ x T 0 T x T − y ~ ′ x T ] (III-3-1) \frac{\partial \tilde{\mathbf{x}}'}{\partial\mathbf{h}} = \begin{bmatrix} {\partial \tilde{{x}}'}/{\partial\mathbf{h}} \\ {\partial \tilde{{y}}'}/{\partial\mathbf{h}} \end{bmatrix} = \frac{1}{w'} \begin{bmatrix} {\mathbf{x}^{\rm{T}}} & \mathbf{0}^{\rm{T}} & - \tilde{x}' \mathbf{x}^{\rm{T}} \\ \mathbf{0}^{\rm{T}} & {\mathbf{x}^{\rm{T}}} & - \tilde{y}' \mathbf{x}^{\rm{T}} \end{bmatrix} \tag{III-3-1} ∂h∂x~′=[∂x~′/∂h∂y~′/∂h]=w′1[xT0T0TxT−x~′xT−y~′xT](III-3-1)
参考文献
[1] Richard Hartley, Andrew Zisserman, Multiple View Geometry in Computer Vision, 2nd edition, Cambridge University Press, 2004
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本文链接：https://blog.youkuaiyun.com/woyaomaishu2/article/details/144831870
本文作者：wzf@robotics_notes
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