| Line 14: | Line 14: | ||
</math></span></font> | </math></span></font> | ||
| − | <math>\color{blue} | + | <math>\color{blue} |
y(m,n) = \sum_{k=-\infty}^{\infty}{\sum_{l=-\infty}^{\infty}{x(m-k,n-l)h(k,l)}}. | y(m,n) = \sum_{k=-\infty}^{\infty}{\sum_{l=-\infty}^{\infty}{x(m-k,n-l)h(k,l)}}. | ||
</math><br> | </math><br> | ||
| Line 21: | Line 21: | ||
\text{For parts a) and b) let} | \text{For parts a) and b) let} | ||
</math><br> | </math><br> | ||
| − | <math>\color{blue} | + | <math>\color{blue} |
h(m,n)=sinc(mT,nT), \text{where} T\leq1. | h(m,n)=sinc(mT,nT), \text{where} T\leq1. | ||
</math><br> | </math><br> | ||
| + | |||
| Line 30: | Line 31: | ||
<math>\color{blue}\text{b) Sketch the frequency response for } |\mu| < 2\pi \text{ and } |nu| < 2\pi \text{ when } T = \frac{1}{2} | <math>\color{blue}\text{b) Sketch the frequency response for } |\mu| < 2\pi \text{ and } |nu| < 2\pi \text{ when } T = \frac{1}{2} | ||
</math><br> | </math><br> | ||
| + | |||
<math>\color{blue} | <math>\color{blue} | ||
\text{For parts c), d), and e) let} | \text{For parts c), d), and e) let} | ||
</math><br> | </math><br> | ||
| − | <math>\color{blue} | + | <math>\color{blue} |
h(m,n)=sinc\left( \frac{(n+m)T}{\sqrt[]{2}},\frac{(n-m)T}{\sqrt[]{2}} \right) | h(m,n)=sinc\left( \frac{(n+m)T}{\sqrt[]{2}},\frac{(n-m)T}{\sqrt[]{2}} \right) | ||
</math><br> | </math><br> | ||
| Line 41: | Line 43: | ||
</math><br> | </math><br> | ||
| − | |||
| − | <math>\color{blue}\text{ | + | <math>\color{blue}\text{c) Calculate the frequency response, }H \left( e^{j\mu},e^{j\nu} \right).</math><br> |
| + | |||
| + | <math>\color{blue}\text{d) Sketch the frequency response for } |\mu| < 2\pi \text{ and } |nu| < 2\pi \text{ when } T = \frac{1}{2} | ||
</math><br> | </math><br> | ||
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</math></span></font> | </math></span></font> | ||
| − | <math>\color{blue} | + | <math>\color{blue} |
p_{\theta}(r) = \mathcal{FP}\left \{ f(x,y) \right \} | p_{\theta}(r) = \mathcal{FP}\left \{ f(x,y) \right \} | ||
</math><br> | </math><br> | ||
| − | + | <math>\color{blue} | |
= \int_{-\infty}^{\infty}{f \left ( r cos(\theta) - z sin(\theta),r sin(\theta) + z cos(\theta) \right )dz}. | = \int_{-\infty}^{\infty}{f \left ( r cos(\theta) - z sin(\theta),r sin(\theta) + z cos(\theta) \right )dz}. | ||
</math> | </math> | ||
| Line 68: | Line 71: | ||
\text{Let } F(\mu,\nu) \text{ be the continuous-time Fourier transform of } f(x,y) \text{ given by} | \text{Let } F(\mu,\nu) \text{ be the continuous-time Fourier transform of } f(x,y) \text{ given by} | ||
</math><br> | </math><br> | ||
| − | <math>\color{blue} | + | <math>\color{blue} |
F(u,v) = \int_{-\infty}^{\infty}{\int_{-\infty}^{\infty}{f(x,y)e^{-j2\pi(ux,vy)}dx}dy} | F(u,v) = \int_{-\infty}^{\infty}{\int_{-\infty}^{\infty}{f(x,y)e^{-j2\pi(ux,vy)}dx}dy} | ||
</math><br> | </math><br> | ||
| Line 75: | Line 78: | ||
\text{and let } P_{\theta}(\rho) \text{ be the continuous-time Fourier transform of } p_{\theta}(r) \text{ given by} | \text{and let } P_{\theta}(\rho) \text{ be the continuous-time Fourier transform of } p_{\theta}(r) \text{ given by} | ||
</math><br> | </math><br> | ||
| − | <math>\color{blue} | + | <math>\color{blue} |
P_{\theta}(\rho) = \int_{-\infty}^{\infty}{p_{\theta}(r)e^{-j2\pi(\rho r)}dr}. | P_{\theta}(\rho) = \int_{-\infty}^{\infty}{p_{\theta}(r)e^{-j2\pi(\rho r)}dr}. | ||
</math><br> | </math><br> | ||
| + | |||
| + | |||
<math>\color{blue}\text{a) Calculate the forward projection }p_{\theta}(r) \text{, for } f(x,y) = \delta(x,y). | <math>\color{blue}\text{a) Calculate the forward projection }p_{\theta}(r) \text{, for } f(x,y) = \delta(x,y). | ||
Revision as of 13:18, 27 July 2012
ECE Ph.D. Qualifying Exam in Communication Networks Signal and Image processing (CS), Question 5, August 2011
Question
Part 1. 50 pts
$ \color{blue}\text{Consider the following discrete space system with input } x(m,n) \text{ and output } y(m,n). $
$ \color{blue} y(m,n) = \sum_{k=-\infty}^{\infty}{\sum_{l=-\infty}^{\infty}{x(m-k,n-l)h(k,l)}}. $
$ \color{blue} \text{For parts a) and b) let} $
$ \color{blue} h(m,n)=sinc(mT,nT), \text{where} T\leq1. $
$ \color{blue}\text{a) Calculate the frequency response, }H \left( e^{j\mu},e^{j\nu} \right). $
$ \color{blue}\text{b) Sketch the frequency response for } |\mu| < 2\pi \text{ and } |nu| < 2\pi \text{ when } T = \frac{1}{2} $
$ \color{blue} \text{For parts c), d), and e) let} $
$ \color{blue} h(m,n)=sinc\left( \frac{(n+m)T}{\sqrt[]{2}},\frac{(n-m)T}{\sqrt[]{2}} \right) $
$ \color{blue} \text{where } T\leq1. $
$ \color{blue}\text{c) Calculate the frequency response, }H \left( e^{j\mu},e^{j\nu} \right). $
$ \color{blue}\text{d) Sketch the frequency response for } |\mu| < 2\pi \text{ and } |nu| < 2\pi \text{ when } T = \frac{1}{2} $
$ \color{blue}\text{e) Calculate } y(m,n) \text{ when } x(m,n)=1. $
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Part 2. 50 pts
$ \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{b) Calculate the forward projection }p_{\theta}(r) \text{, for } f(x,y) = \delta(x-1,y-1). $
$ \color{blue}\text{c) Calculate the forward projection }p_{\theta}(r) \text{, for } f(x,y) = rect \left(\sqrt[]{x^2+y^2} \right). $
$ \color{blue}\text{d) Calculate the forward projection }p_{\theta}(r) \text{, for } f(x,y) = rect \left(\sqrt[]{(x-1)^2+(y-1)^2} \right). $
$ \color{blue}\text{e) Describe in precise detail, the steps required to perform filtered back projection (FBP) reconstruction of } f(x,y). $
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