Multiple View Geometry in Computer Vision Chapter 10 Solutions

I. Using only (implicit) image relations (i.e. without an explicit 3D reconstruction) and given the images of a line $L$ and point $X$ (not on $L$) in two views, together with $H_{\infty }$ between the two views, compute the image of the line in 3-space parallel to $L$ and through $X$. Other examples of this implicit approach to computation are given in [Zeller-96].

The image of a line parallel to $L$ will intersect the image of $L$ in the vanishing point of both lines. So, if we find the vanishing point $v$, we can draw the required line connecting $v$ and the image of $X$.

Let the vanishing points of $L$ in the two images be $v_1$ and $v_2$. Let the images of $L$ be $l_1$ and $l_2$. Then the two vanishing points must satisfy the following conditions.

$${l_1}^T{v}_1=0$$ $${l_2}^T{v}_2=0$$

As the infinite homography relates the images of a point on the plane at infinity, the two vanishing points being the images of a point on ${\pi }_{\infty }$ must satisfy.

$$v_2=H_{\infty }{v}_1$$

Using this, we can rewrite the second constraint above as $${l_2}^T{H}_{\infty }{v}_1=0$$

As $l_1$, $l_2$ and $H_{\infty }$ are known quantities, we can solve for $v_1$, a homogeneous point having two degrees of freedom, using the following two linear constraints $${l_1}^T{v}_1=0$$ $${l_2}^T{H}_{\infty }{v}_1=0$$

$v_2$ can then be determined by applying the infinite homography to $v_1$.

Finally, if $x_1$ and $x_2$ are the images of $X$, we can determine the required lines using

$$x_1\times v_1$$ $$x_2\times v_2$$