仿射变换矩阵

图像的几何变换分为三类：刚性变换、仿射变换和透视变换。

尺度变换，比例缩放变换

[ x ′ y ′ 1 ] = [ f x 0 0 0 f y 0 0 0 1 ] [ x y 1 ]

= ⎣ ⎡​x′y′1​⎦ ⎤​=⎣ ⎡​fx​00​0fy​0​001​⎦ ⎤​⎣ ⎡​xy1​⎦ ⎤​

[ x ′ y ′ 1 ] = [ x y 1 ] [ f x 0 0 0 f y 0 0 0 1 ]

= [x′​y′​1​]=[x​y​1​]⎣ ⎡​fx​00​0fy​0​001​⎦ ⎤​

$x^{\prime }=f_{x}x$
$y^{\prime }=f_{y}y$

平移变换

[ x ′ y ′ 1 ] = [ 1 0 Δ x 0 1 Δ y 0 0 1 ] [ x y 1 ]

= ⎣ ⎡​x′y′1​⎦ ⎤​=⎣ ⎡​100​010​ΔxΔy1​⎦ ⎤​⎣ ⎡​xy1​⎦ ⎤​

[ x ′ y ′ 1 ] = [ x y 1 ] [ 1 0 0 0 1 0 Δ x Δ y 1 ]

= [x′​y′​1​]=[x​y​1​]⎣ ⎡​10Δx​01Δy​001​⎦ ⎤​

$x^{\prime }=x+\Delta x$
$y^{\prime }=y+\Delta y$

镜像变换

图像宽$w$，高$h$
左上角为原点，向下为$x$轴，向右为$y$轴

水平镜像变换

[ x ′ y ′ 1 ] = [ − 1 0 w 0 1 0 0 0 1 ] [ x y 1 ]

= ⎣ ⎡​x′y′1​⎦ ⎤​=⎣ ⎡​−100​010​w01​⎦ ⎤​⎣ ⎡​xy1​⎦ ⎤​

[ x ′ y ′ 1 ] = [ x y 1 ] [ − 1 0 0 0 1 0 w 0 1 ]

= [x′​y′​1​]=[x​y​1​]⎣ ⎡​−10w​010​001​⎦ ⎤​

水平镜像变换：先绕$y$轴翻转，再沿$x$轴移动
[ − 1 0 w 0 1 0 0 0 1 ] = [ 1 0 w 0 1 0 0 0 1 ] [ − 1 0 0 0 1 0 0 0 1 ]

= ⎣ ⎡​−100​010​w01​⎦ ⎤​=⎣ ⎡​100​010​w01​⎦ ⎤​⎣ ⎡​−100​010​001​⎦ ⎤​

[ − 1 0 0 0 1 0 w 0 1 ] = [ − 1 0 0 0 1 0 0 0 1 ] [ 1 0 0 0 1 0 w 0 1 ]

= ⎣ ⎡​−10w​010​001​⎦ ⎤​=⎣ ⎡​−100​010​001​⎦ ⎤​⎣ ⎡​10w​010​001​⎦ ⎤​

$x^{\prime }=w-x$
$y^{\prime }=y$

垂直镜像变换

[ x ′ y ′ 1 ] = [ 1 0 0 0 − 1 h 0 0 1 ] [ x y 1 ]

= ⎣ ⎡​x′y′1​⎦ ⎤​=⎣ ⎡​100​0−10​0h1​⎦ ⎤​⎣ ⎡​xy1​⎦ ⎤​

[ x ′ y ′ 1 ] = [ x y 1 ] [ 1 0 0 0 − 1 0 0 h 1 ]

= [x′​y′​1​]=[x​y​1​]⎣ ⎡​100​0−1h​001​⎦ ⎤​

垂直镜像变换：先绕$x$轴翻转，再沿$y$轴移动
[ 1 0 0 0 − 1 h 0 0 1 ] = [ 1 0 0 0 1 h 0 0 1 ] [ 1 0 0 0 − 1 0 0 0 1 ]

= ⎣ ⎡​100​0−10​0h1​⎦ ⎤​=⎣ ⎡​100​010​0h1​⎦ ⎤​⎣ ⎡​100​0−10​001​⎦ ⎤​

[ 1 0 0 0 − 1 0 0 h 1 ] = [ 1 0 0 0 − 1 0 0 0 1 ] [ 1 0 0 0 1 0 0 h 1 ]

= ⎣ ⎡​100​0−1h​001​⎦ ⎤​=⎣ ⎡​100​0−10​001​⎦ ⎤​⎣ ⎡​100​01h​001​⎦ ⎤​

$x^{\prime }=x$
$y^{\prime }=h-y$

注：线性代数有$xAB=x(AB)$

恒等变换

[ x ′ y ′ 1 ] = [ 1 0 0 0 1 0 0 0 1 ] [ x y 1 ]

= ⎣ ⎡​x′y′1​⎦ ⎤​=⎣ ⎡​100​010​001​⎦ ⎤​⎣ ⎡​xy1​⎦ ⎤​

[ x ′ y ′ 1 ] = [ x y 1 ] [ 1 0 0 0 1 0 0 0 1 ]

= [x′​y′​1​]=[x​y​1​]⎣ ⎡​100​010​001​⎦ ⎤​

$x^{\prime }=x$
$y^{\prime }=y$

旋转变换

绕原点逆时针旋转

[ x ′ y ′ 1 ] = [ x y 1 ] [ c o s θ s i n θ 0 − s i n θ c o s θ 0 0 0 1 ]

= [x′​y′​1​]=[x​y​1​]⎣ ⎡​cosθ−sinθ0​sinθcosθ0​001​⎦ ⎤​

[ x ′ y ′ 1 ] = [ c o s θ − s i n θ 0 s i n θ c o s θ 0 0 0 1 ] [ x y 1 ]

= ⎣ ⎡​x′y′1​⎦ ⎤​=⎣ ⎡​cosθsinθ0​−sinθcosθ0​001​⎦ ⎤​⎣ ⎡​xy1​⎦ ⎤​

$x^{\prime }=cos\theta x-sin\theta y$
$y^{\prime }=sin\theta x+cos\theta y$

绕原点顺时针旋转

[ x ′ y ′ 1 ] = [ x y 1 ] [ c o s θ − s i n θ 0 s i n θ c o s θ 0 0 0 1 ]

= [x′​y′​1​]=[x​y​1​]⎣ ⎡​cosθsinθ0​−sinθcosθ0​001​⎦ ⎤​

[ x ′ y ′ 1 ] = [ c o s θ s i n θ 0 − s i n θ c o s θ 0 0 0 1 ] [ x y 1 ]

= ⎣ ⎡​x′y′1​⎦ ⎤​=⎣ ⎡​cosθ−sinθ0​sinθcosθ0​001​⎦ ⎤​⎣ ⎡​xy1​⎦ ⎤​

$x^{\prime }=cos\theta x+sin\theta y$
$y^{\prime }=-sin\theta x+cos\theta y$

绕任意点旋转

先将该点平移到原点，再绕原点进行旋转，最后将该点平移回去。

错切变换，偏移变换

水平错切变换

[ x ′ y ′ 1 ] = [ 1 0 0 s h 1 0 0 0 1 ] [ x y 1 ]

= ⎣ ⎡​x′y′1​⎦ ⎤​=⎣ ⎡​1sh​0​010​001​⎦ ⎤​⎣ ⎡​xy1​⎦ ⎤​

[ x ′ y ′ 1 ] = [ x y 1 ] [ 1 s h 0 0 1 0 0 0 1 ]

= [x′​y′​1​]=[x​y​1​]⎣ ⎡​100​sh​10​001​⎦ ⎤​

$x^{\prime }=x$
$y^{\prime }=s_{h}x+y$

垂直错切变换

[ x ′ y ′ 1 ] = [ 1 s v 0 0 1 0 0 0 1 ] [ x y 1 ]

= ⎣ ⎡​x′y′1​⎦ ⎤​=⎣ ⎡​100​sv​10​001​⎦ ⎤​⎣ ⎡​xy1​⎦ ⎤​

[ x ′ y ′ 1 ] = [ x y 1 ] [ 1 0 0 s v 1 0 0 0 1 ]

= [x′​y′​1​]=[x​y​1​]⎣ ⎡​1sv​0​010​001​⎦ ⎤​

$x^{\prime }=x+s_{v}y$
$y^{\prime }=y$