As shown in Figure $3,$ a moving camera takes two images of the same object at two poses. Two coordinate frames represented by $F_a$ and $F_b$ are attached to the projection centre of the camera at the two poses, respectively. Seven feature points $O_1,O_2,O_3,O_4,O_5,O_6$ and $O_7$ are on three different lines on the same plane. Let $H$ denote the Euclidean homography matrix from $F_a$ to $F_b.$

$($a$)$ Six feature points $O_1,O_2,O_3,O_4,O_5$ and $O_6$ can be detected in the image taken at the pose attached to $F_a.$ Their corresponding normalized coordinates in $F_a$ are given by:

$m_{1a}=[a_{1x},a_{1y},1{]}^T,m_{2a}=[a_{2x},a_{2y},1{]}^T,m_{3a}=[a_{3x},a_{3y},1{]}^T,m_{4a}=[a_{4x},a_{4y},1{]}^T,m_{5a}=[a_{5x},a_{5y},1{]}^T,m_{6a}=[a_{6x},a_{6y},1{]}^T.$

Six feature points $O_1,O_3,O_4,O_5,O_6$ and $O_7$ can be detected in the image taken at the pose attached to $F_b.$ Their corresponding normalized coordinates in $F_b$ are given by:

$m_{1b}=[b_{1x},b_{1y},1{]}^T,m_{3b}=[b_{3x},b_{3y},1{]}^T,m_{4b}=[b_{4x},b_{4y},1{]}^T,m_{5b}=[b_{5x},b_{5y},1{]}^T,$

$m_{6b}=[b_{6x},b_{6y},1{]}^T,m_{7b}=[b_{7x},b_{7y},1{]}^T.$

Let $h_{33}$ denote the third row third column element of the matrix $H.$ Find the scaled homography matrix $\frac{H}{h_{33}}.$

$(15$ Marks$)$

$($b$)$ Scaled homography matrix can be used to extract rotation matrix information. When computing a rotation matrix from a scaled homography matrix, the solution may not be unique during the initial computation. How can you determine the correction solution?

$(5$ Marks$)$

$4$