Radon Transform--Delta Function

For a Delta Function at $(x_0,y_0)$,

$$R(p, \tau) = \int_{-\infty}^\infty \int_{-\infty}^\infty \delta(x-x_0)\delta(y-y_0)\delta[y-(\tau+px)]\,dy\,dx$$

$$= {1\over 2\pi} \int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty e^{-ik[y-(\tau+px)]} \delta(x-x_0)\delta(y-y_0)\,dk\,dy\,dx$$

$$= {1\over 2\pi} \int_{-\infty}^\infty e^{ik\tau}\left[{\int_{-\inft... ...ky}\delta(y-y_0)\,dy\int_{-\infty}^\infty e^{ikpx}\delta(x-x_0)\,dx}\right]\,dk$$

$$= {1\over 2\pi} \int_{-\infty}^\infty e^{ik\tau}e^{-iky_0}e^{ikpx_0}\,dk$$

$$= {1\over 2\pi} \int_{-\infty}^\infty e^{ik(\tau+px_0-y_0)}\,dk =\delta(\tau+px_0-y_0).$$