ECE Ph.D. Qualifying Exam in "Communication, Networks, Signal, and Image Processing" (CS)

Question 1, August 2011, Part 1

$\color{blue}\text{Consider an image } f(x,y) \text{ with a forward projection}$

$\color{blue} p_{\theta}(r) = \mathcal{FP}\left \{ f(x,y) \right \}$

$\color{blue} = \int_{-\infty}^{\infty}{f \left ( r cos(\theta) - z sin(\theta),r sin(\theta) + z cos(\theta) \right )dz}.$

$\color{blue} \text{Let } F(\mu,\nu) \text{ be the continuous-time Fourier transform of } f(x,y) \text{ given by}$
$\color{blue} F(u,v) = \int_{-\infty}^{\infty}{\int_{-\infty}^{\infty}{f(x,y)e^{-j2\pi(ux,vy)}dx}dy}$

$\color{blue} \text{and let } P_{\theta}(\rho) \text{ be the continuous-time Fourier transform of } p_{\theta}(r) \text{ given by}$
$\color{blue} P_{\theta}(\rho) = \int_{-\infty}^{\infty}{p_{\theta}(r)e^{-j2\pi(\rho r)}dr}.$

$\color{blue}\text{a) Calculate the forward projection }p_{\theta}(r) \text{, for } f(x,y) = \delta(x,y).$

$\color{blue}\text{Solution 1:}$

$P_{\theta}(\rho) = \int_{-\infty}^{+\infty}{\delta(r cos\theta - z sin\theta, r sin\theta + z cos \theta) dz}$

$= z \text{ when } \left\{\begin{matrix} r cos\theta - z sin\theta = 0 \\ r sin \theta + z cos \theta = 0 \end{matrix}\right.$

$= \frac{r cos\theta}{sin \theta}, \theta > 0$