几何变换类型

保距变换 isometry
相似变换 similarity
仿射变换 affine
射影变换 projective -> homography

What is Homography?

A Homography is a transformation ( a 3×3 matrix ) that maps the points in one image to the corresponding points in the other image.

单应性矩阵

homography只针对同一平面

How to calculate a Homography ?

摄影变换的自由度为8，一对点能产生两个方程，共需要4对对应点，即可求取H矩阵；如超过4对，通过最小二乘法或RANSAC求取最优参数

理论推导

单应性(homography)变换的推导 - flyinsky518 - 博客园

假设一对对应点 $a = (x,y,1)$ 和 $a^,=(x^,, y^,,1)$

有 $Ah=0$ ，其中

$A=\left[\begin{array}{ccccccccc}-x & -y & -1 & 0 & 0 & 0 & x x_{1} & y x_{1} & x_{1} \\ 0 & 0 & 0 & -x & -y & -1 & x y_{1} & y y_{1} & y_{1}\end{array}\right]$

$h=\left[h_{11}, h_{12}, h_{13}, h_{21}, h_{22}, h_{23}, h_{31}, h_{32}, h_{33}\right]^{T}$

当有n对点时， $A \in R^{2n\times9}$ ，对A进行SVD分解，即 $U * \sum * V^T$ ，取 $V$ 的最后一列求解h，再转换成 $3*3$ 矩阵即得到 $H$

[U,S,V]=svd(A);
h=V(:,9);
H= reshape(h,3,3);

工程实践

If you have more than 4 corresponding points, it is even better. OpenCV will robustly estimate a homography that best fits all corresponding points.
Usually, these point correspondences are found automatically by matching features like SIFT or SURF between the images.

'''
pts_src and pts_dst are numpy arrays of points
in source and destination images. We need at least
4 corresponding points.
'''
h, status = cv2.findHomography(pts_src, pts_dst)
'''
The calculated homography can be used to warp
the source image to destination. Size is the
size (width,height) of im_dst
'''
im_dst = cv2.warpPerspective(im_src, h, size)