The sifting property of the Dirac distribution

The ideal impulse in the image plane is defined using the Dirac distribution \(\delta(x,y)\)

\int_{-\infty}^{\infty}{\int_{-\infty}^{\infty}{\delta(x,y)\text{d}x\text{d}y=1} }

for \ all \ x,y\neq0, \delta(x,y)=0

It provides the value of the function \( f(x, y) \) at the point\( (\lambda, \mu) \).

\int_{-\infty}^{\infty}{\int_{-\infty}^{\infty}{f(x,y)\delta(x-\lambda,y-\mu)\text{d}x\text{d}y=f(\lambda,\mu)} }