I don't quite understand what is meant by the defintion of an ideally sampled image $b_s(r)$. The image is sampled based on a lattice $V$ ($k$ x $l$).
$b_s(r) = | det(V) | \sum_{\boldsymbol{k}=-\infty}^{\infty} b_\boldsymbol{k}\delta(r-V\boldsymbol{k})$ with $r = [ x ; y ] $, $\boldsymbol{k} = [k ; l]$
It is not clear to me why this is the defintion for a sampled image. Especially the reason of presence of the determinant is unclear.
An ideally sampled image is an image that is sampled by multiplication with a pulse train (a set of equally spaced Dirac delta functions). This means that the function is sampled at individual points. It's called "ideal" because it's impossible to do so in practice, typically when you sample you integrate over an area (think the pixel elements of a CCD camera, the same is true when sampling a signal). But using the ideal sampling scheme makes a whole lot of math easier, so we like to think of our data as point-sampled.
Your equation can be extended as
$$b_s(r) = |det(V)| \sum_{\boldsymbol{k}=-\infty}^{\infty} b_\boldsymbol{k}\delta(r-V\boldsymbol{k}) = |det(V)| \; b(r) \sum_{\boldsymbol{k}=-\infty}^{\infty} \delta(r-V\boldsymbol{k}) \; , $$
with $b(r)$ the continuous-domain image being sampled.
I presume that the $|det(V)|$ is there to preserve the integral. If $V$ is a diagonal matrix with the distances between the samples, such that $V\boldsymbol{k}$ gives the spatial location of the sample $\boldsymbol{k}$, then its determinant is the size of the rectangle in between four samples (in 2D). $V$ can be rotated to design rotated sampling grids, preserving its determinant.