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Radon Transform--Delta Function

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


$\displaystyle R(p, \tau)$ $\textstyle =$ $\displaystyle \int_{-\infty}^\infty \int_{-\infty}^\infty \delta(x-x_0)\delta(y-y_0)\delta[y-(\tau+px)]\,dy\,dx$  
  $\textstyle =$ $\displaystyle {1\over 2\pi} \int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty e^{-ik[y-(\tau+px)]} \delta(x-x_0)\delta(y-y_0)\,dk\,dy\,dx$  
  $\textstyle =$ $\displaystyle {1\over 2\pi} \int_{-\infty}^\infty e^{ik\tau}\left[{\int_{-\inft...
...ky}\delta(y-y_0)\,dy\int_{-\infty}^\infty e^{ikpx}\delta(x-x_0)\,dx}\right]\,dk$  
  $\textstyle =$ $\displaystyle {1\over 2\pi} \int_{-\infty}^\infty e^{ik\tau}e^{-iky_0}e^{ikpx_0}\,dk$  
  $\textstyle =$ $\displaystyle {1\over 2\pi} \int_{-\infty}^\infty e^{ik(\tau+px_0-y_0)}\,dk =\delta(\tau+px_0-y_0).$  




© 1996-9 Eric W. Weisstein
1999-05-25