多视几何_计算两幅图像之间的基础矩阵F和一副图像上的点在另一福图像上的极线L

红太狼 2022-12-29 02:26 88阅读 0赞

# 功能说明: #

*  输入：两帧图像的相对变换Tji =(R,t) (从图像i变换到图像j), 相机的内参K (假设两幅图像使用的相机是同一个), 输入图像i中的点m
 *  输出：计算图像i的**m**点, 在图像j上的极线**l'm** = (a,b,c), (则图像j上的极线方程为 ax + by +c = 0)

# 图示与原理： #

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70][]

基本矩阵 F 描述了图像点 m 与其对应极线 lm′ 间的对应关系：![20210317151639962.png][]。由于图像点 m 在第二幅图像上的对应点 m′ 在极线 lm′ 上，所以必有![20210317151741914.png][]。

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70 1][]

![20210317151925418.png][]

# 实现: #

def  calculate_epipolar_line (K, Tji, m):
        """
       calculate the epipolar line for point m
        Args:
            K (numpy, (3,3) ):  the intrinsic parameter of camera0 and camera1
            Tji (numpy, 4x4): the transformation from frame i(camera0) to  frame j(camera1)
            m (numpy, (2,) ): (x,y) point u in frame i(camera0)
    
        Returns:
            L(numpy, (3,1) ): the epipolar line of m in frame j
        """
        R =Tji[0:3,0:3]
        t = Tji[0:3, 3].reshape([3,1])
        MatK = mat(K)
        MatR = mat(R)
        
        # the epipolar point in frame i
        e = MatK * MatR.T * t   # shape: (3,1) #check ok
        # the antisymmetric matrix of e
        ex = mat(np.array([ [0, -e[2,0], e[1,0]],
                                                [e[2,0], 0, -e[0,0]],
                                                [-e[1,0], e[0,0], 0]]))
        # the fundamental matrix
        F = MatK.T.I * MatR * MatK.T * ex
        assert (matrix_rank(F) == 2) 
    
        m_h = np.array([[m[0]],[m[1]],[1]])
        L = F * m_h
    
        return L

[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70]: /images/20221120/8aa9f930cd1947269ccb6adbb2b55480.png
[20210317151639962.png]: https://img-blog.csdnimg.cn/20210317151639962.png
[20210317151741914.png]: https://img-blog.csdnimg.cn/20210317151741914.png
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ_size_16_color_FFFFFF_t_70 1]: https://img-blog.csdnimg.cn/20210317151846279.png?x-oss-process=image/watermark,type_ZmFuZ3poZW5naGVpdGk,shadow_10,text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3NoeWpoeXAxMQ==,size_16,color_FFFFFF,t_70
[20210317151925418.png]: https://img-blog.csdnimg.cn/20210317151925418.png