传统CV算法——仿射变换原理及应用

• 可以理解一下常规的翻转和平移。“线性变换”+“平移”
• 空间变换中的仿射变换对应着五种变换，平移，缩放，旋转，翻转，错切。而这五种变化由原图像转变到变换图像的过程，可以用仿射变换矩阵进行描述。而这个变换过程可以用一个3*3的矩阵与原图进行相乘得到。
• 仿射变换（Affine Transformation）其实是另外两种简单变换的叠加：一个是线性变换，一个是平移变换
• 共线性：若几个点变换前在一条线上，则仿射变换后仍然在一条线上
• 共线比例不变性：变换前一条线上两条线段的比例，在变换后比例不变
• 平行性：若两条线变换前平行，则变换后仍然平行。

矩阵显示

具体公式(可以看成线性变换R和平移变换T的叠加)：

[ cos ⁡ ( θ ) − sin ⁡ ( θ ) t x sin ⁡ ( θ ) cos ⁡ ( θ ) t y 0 0 1 ] [ x y 1 ] = [ x ′ y ′ 1 ] \left[ \begin{array}{ccc}{\cos (\theta)} & {-\sin (\theta)} & {t_{x}} \\ {\sin (\theta)} & {\cos (\theta)} & {t_{y}} \\ {0} & {0} & {1} \end{array}\right] \left[ \begin{array}{l}{x} \\ {y} \\ {1}\end{array}\right]=\left[ \begin{array}{c}{x^{\prime}} \\ {y^{\prime}} \\ {1}\end{array}\right] ​cos(θ)sin(θ)0​−sin(θ)cos(θ)0​tx​ty​1​ ​ ​xy1​ ​= ​x′y′1​ ​

其中 s , θ , t x , t y s, \theta, t_x, t_y s,θ,tx​,ty​变换量，角度经过变换后具体的公式为：
[ s cos ⁡ ( θ ) − s sin ⁡ ( θ ) t x s sin ⁡ ( θ ) s cos ⁡ ( θ ) t y 0 0 1 ] [ x y 1 ] = [ x ′ y ′ 1 ] \left[ \begin{array}{ccc}{s\cos (\theta)} & {-s\sin (\theta)} & {t_{x}} \\ {s\sin (\theta)} & {s\cos (\theta)} & {t_{y}} \\ {0} & {0} & {1} \end{array}\right] \left[ \begin{array}{l}{x} \\ {y} \\ {1}\end{array}\right]=\left[ \begin{array}{c}{x^{\prime}} \\ {y^{\prime}} \\ {1}\end{array}\right] ​scos(θ)ssin(θ)0​−ssin(θ)scos(θ)0​tx​ty​1​ ​ ​xy1​ ​= ​x′y′1​ ​

仿射变换基本公式：

f ( x ) = A x + b , x ∈ X f(x)=Ax+b,x∈X f(x)=Ax+b,x∈X

平移变换矩阵为:

[ 1 0 t x 0 1 t y 0 0 1 ] \left[ \begin{array}{ccc}{1} & {0} & {t_{x}} \\ {0} & {1} & {t_{y}} \\ {0} & {0} & {1} \end{array}\right] ​100​010​tx​ty​1​ ​
其中 t x , t y {t_{x}} ,{t_{y}} tx​,ty​控制平移的参数，向上向下平移参数

反射变换

[ − 1 0 0 0 1 0 0 0 1 ] \left[ \begin{array}{ccc}{-1} & {0} & {0} \\ {0} & {1} & {0} \\ {0} & {0} & {1} \end{array}\right] ​−100​010​001​ ​
控制矩阵的正负可以按照对应的坐标轴进行变换。具体如上图。

仿射变换，变化

import cv2
import numpy as np

src = cv2.imread("./images/1.jpg")

rows,cols,channel = src.shape


M = np.float32([[1,0,50],[0,1,50]])

# M = cv2.getRotationMatrix2D((cols/2,rows/2),angle=45,scale=0.7)
dst = cv2.warpAffine(src,M=M,dsize=(cols,rows))
cv2.imshow("src",src)

cv2.imshow("dst",dst)
cv2.waitKey(0)

• 示例2：

import cv2
import numpy as np

src = cv2.imread("./images/1.jpg")

rows,cols,channel = src.shape


# M = np.float32([[0.5,0,0],[0,0.5,0]])
# M = np.float32([[0.5,0,0],[0,0.5,0]])
M = np.float32([[-0.5,0,cols//2],[0,0.5,0]])

# M = cv2.getRotationMatrix2D((cols/2,rows/2),angle=45,scale=0.7)
dst = cv2.warpAffine(src,M=M,dsize=(cols,rows))
cv2.imshow("src",src)

cv2.imshow("dst",dst)
cv2.waitKey(0)

import cv2
import numpy as np
src = cv2.imread("./images/1.jpg")
rows,cols,channel = src.shape
# M = np.float32([[0.5,0,0],[0,0.5,0]])
# M = np.float32([[0.5,0,0],[0,0.5,0]])
# M = np.float32([[-0.5,0,cols//2],[0,0.5,0]])
M = np.float32([[1,0.5,0],[0,1,0]])

# M = cv2.getRotationMatrix2D((cols/2,rows/2),angle=45,scale=0.7)
dst = cv2.warpAffine(src,M=M,dsize=(cols,rows))
cv2.imshow("src",src)
cv2.imshow("dst",dst)
cv2.waitKey(0)

import cv2
import numpy as np

src = cv2.imread("./images/1.jpg")

rows,cols,channel = src.shape


# M = np.float32([[0.5,0,0],[0,0.5,0]])
# M = np.float32([[0.5,0,0],[0,0.5,0]])
# M = np.float32([[-0.5,0,cols//2],[0,0.5,0]])

# M = np.float32([[1,0.5,0],[0.5,1,0]])

M = cv2.getRotationMatrix2D((cols/2,rows/2),angle=45,scale=0.7)

dst = cv2.warpAffine(src,M=M,dsize=(cols,rows))
cv2.imshow("src",src)

cv2.imshow("dst",dst)
cv2.waitKey(0)