Multiple View Geometry in Computer Vision Chapter 10 Solutions

3D Reconstruction of Cameras and Structure

A 2 minute read, posted on 12 Jun 2020
Last modified on 12 Jun 2020

Tags computer vision, problem solution

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

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

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

$$\mathtt{l_1}^T\mathtt{v_1} = 0$$ $$\mathtt{l_2}^T\mathtt{v_2} = 0$$

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.

$$\mathtt{v_2 = H_{\infty}v_1}$$

Using this, we can rewrite the second constraint above as $$\mathtt{l_2}^T\mathtt{H_{\infty}v_1} = 0$$

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

$\mathtt{v_2}$ can then be determined by applying the infinite homography to $\mathtt{v_1}$.

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

$$\mathtt{x_1} \times \mathtt{v_1}$$ $$\mathtt{x_2} \times \mathtt{v_2}$$

comments powered by Disqus