Back-projected ray in homogeneous coordinates

[tdomhan]

suppose I got a projective camera model. for this model I would like to back-project a ray through a point in the image plane. I know that the equation for this is the following:
$$
y(\lambda) = P^+_0 x_0 + \lambda c_0
$$
where $P^+_0$denotes the pseudoinverse of the camera matrix.
$x_0$ the point on the image plane and $c_0$ the center of the camera. (This is taking from the book "Multiple View Geometry in Computer Vision" page 162)

Now I don't fully get this equation. I get that $P^+_0 x_0$ results in a point on the line we are looking for. Hence we have two points that we can use for constructing a line. However I don't get the parametrization using $\lambda$. Why is the equation not in the form like:
$$y(\lambda) = (1-\lambda) a + \lambda b$$

Any help in understanding the original equation of the resulting ray would be appreciated! :D

 

[pat_connell]

The equation you provided is a parametrization of the line through two points, like the equation you provided. In this case, $P^+_0 x_0$ is the point $a$ in your equation and $c_0$ is the point $b$. The parameter $\lambda$ is used to calculate a point on the line for any value of $\lambda$. For example, if $\lambda = 0$, then $y(0) = P^+_0 x_0$ and if $\lambda = 1$, then $y(1) = c_0$. Thus, the equation is a way of expressing the line as a function of $\lambda$.