视觉测量中的变换关系

1.平移变换（Translation）

[left{ egin{array}{l}
{ m{x'}} = x + {t_x}\
y' = y + {t_y}
end{array} ight.]

写成矩阵为:

[left[ {egin{array}{*{20}{c}}
{{ m{x}}'}\
{y'}
end{array}} ight] = left[ {egin{array}{*{20}{c}}
1&0&{{t_x}}\
0&1&{{t_y}}
end{array}} ight]left[ {egin{array}{*{20}{c}}
x\
y\
1
end{array}} ight]]

齐次坐标形式为:

[left[ {egin{array}{*{20}{c}}
{{ m{x}}'}\
egin{array}{l}
y'\
1
end{array}
end{array}} ight] = left[ {egin{array}{*{20}{c}}
1&0&{{t_x}}\
egin{array}{l}
0\
0
end{array}&egin{array}{l}
1\
0
end{array}&egin{array}{l}
{{ m{t}}_y}\
1
end{array}
end{array}} ight]left[ {egin{array}{*{20}{c}}
x\
y\
1
end{array}} ight]]变量为2个自由度。

2.旋转变换（Euclidean变换）

用单位向量表示为：

[left{ egin{array}{l}
overrightarrow { m{x}} { m{' = }}overrightarrow { m{x}} cos b + overrightarrow y sin b\
overrightarrow y ' = - overrightarrow x sin b + overrightarrow y cos b
end{array} ight.]

P在坐标系中关系表示为:

$$left[ {egin{array}{*{20}{c}}
{{ m{OA}}}\
{OB}
end{array}} ight] = left[ {egin{array}{*{20}{c}}
{cos b}&{ - sin b}\
{sin b}&{cos b}
end{array}} ight]left[ {egin{array}{*{20}{c}}
{{ m{OA'}}}\
{{ m{OB'}}}
end{array}} ight]$$

将xy坐标系和x’y’坐标系建立起了联系。加入上面的平移变换写成齐次形式为：

[left[ {egin{array}{*{20}{c}}
{{ m{x}}'}\
egin{array}{l}
y'\
1
end{array}
end{array}} ight] = left[ {egin{array}{*{20}{c}}
{cos heta }&{{ m{ - }}sin heta }&{{t_x}}\
egin{array}{l}
sin heta \
0
end{array}&egin{array}{l}
cos heta \
0
end{array}&egin{array}{l}
{{ m{t}}_y}\
1
end{array}
end{array}} ight]left[ {egin{array}{*{20}{c}}
x\
y\
1
end{array}} ight]]平移2个自由度加旋转一个自由度总共3个自由度。

3.相似变换（Similarity transform）

[left[ {egin{array}{*{20}{c}}
{{ m{x}}'}\
egin{array}{l}
y'\
1
end{array}
end{array}} ight] = left[ {egin{array}{*{20}{c}}
{alpha cos heta }&{{ m{ - }}alpha sin heta }&{{t_x}}\
egin{array}{l}
alpha sin heta \
0
end{array}&egin{array}{l}
alpha cos heta \
0
end{array}&egin{array}{l}
{{ m{t}}_y}\
1
end{array}
end{array}} ight]left[ {egin{array}{*{20}{c}}
x\
y\
1
end{array}} ight]]基于以上的变换又多了一个比例系数为4个自由度。

4.仿射变换（Affine transform）

[left[ {egin{array}{*{20}{c}}
{{ m{x}}'}\
egin{array}{l}
y'\
1
end{array}
end{array}} ight] = left[ {egin{array}{*{20}{c}}
{ m{a}}&b&c\
egin{array}{l}
d\
0
end{array}&egin{array}{l}
e\
0
end{array}&egin{array}{l}
f\
1
end{array}
end{array}} ight]left[ {egin{array}{*{20}{c}}
x\
y\
1
end{array}} ight]]

为6个自由度.

5.透视变换

 

我们人眼通过窗外看外面的景物，景物投射到玻璃上面的现状就是最典型的例子。我们看的角度不同，在玻璃上投射的情况就不同。物体的基本形状已经改变。

[left[ {egin{array}{*{20}{c}}
{{ m{x}}'}\
egin{array}{l}
y'\
w'
end{array}
end{array}} ight] = left[ {egin{array}{*{20}{c}}
{ m{a}}&b&c\
egin{array}{l}
d\
{ m{g}}
end{array}&egin{array}{l}
e\
h
end{array}&egin{array}{l}
f\
1
end{array}
end{array}} ight]left[ {egin{array}{*{20}{c}}
x\
y\
1
end{array}} ight]]为8个自由度。