From a book on imaging:
Using the sifting property of the $2D$ continuous impulse:
$$\int \int f(t,z)\delta(t-t_0, z-z_0) dt dz=f(t_0,z_0)$$
We can write a function $f(x,y)$ as: $$f(x,y) = \int \int f(a,b) \delta (x-a, y-b)\space da \space db$$
Can someone explain how the second formula can be derived from the first formula? I've been trying to figure a way to show this and nothing I've tried yielded any results.
The delta function is separable in each of its variable so $\delta(x,y) = \delta(x)\delta(y)$ (think about why this makes sense). Plugging this into the formula we have $$\begin{align*} f(t_0, z_0) &=\int \int f(t, z)\delta(t-t_0, z-z_0)dt dz\\ &=\int \delta(z-z_0)\left(\int f(t, z)\delta(t-t_0)dt\right) dz\\ &=\int f(t_0, z)\delta(z-z_0) dz\\ &=f(t_0, z_0) \end{align*} $$
Where the second line uses the separability of the Dirac delta, the third uses the sifting property of a 1D Dirac Delta, and the last line again uses the sifting property