Difference between revisions of "ECE637 tomographic reconstruction convolution back projection S13 mhossain" - Rhea

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-[[Category:ECE637]]
-[[Category:sLecture]]
-[[Category:image processing]]
-[[Category:lecture notes]]
-[[ECE637_Bouman_lectures_Image_Processing_sLecture_mhossain|sLecture]]
-:↳ [[ECE637_tomographic_reconstruction_S13_mhossain|Topic 2: Tomographic Reconstruction]]
-::↳ [[ECE637_tomographic_reconstruction_intro_S13_mhossain|Intro]]
-::↳ [[ECE637_tomographic_reconstruction_CT_S13_mhossain|CT]]
-::↳ [[ECE637_tomographic_reconstruction_PET_S13_mhossain|PET]]
-::↳ [[ECE637_tomographic_reconstruction_coordinate_rotation_S13_mhossain|Co-ordinate Rotation]]
-::↳ [[ECE637_tomographic_reconstruction_radon_transform_S13_mhossain|Radon Transform]]
-::↳ [[ECE637_tomographic_reconstruction_fourier_slice_theorem_S13_mhossain|Fourier Slice Theorem]]
-::↳ [[ECE637_tomographic_reconstruction_convolution_back_projection_S13_mhossain|Convolution Back Projection]]
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-<center><font size= 4>
-'''[[ECE637_Bouman_lectures_Image_Processing_sLecture_mhossain|The Bouman Lectures on Image Processing]]'''
-</font size>
-A sLecture by [[user:Mhossain | Maliha Hossain]]
-<font size= 3> Subtopic 3: Convolution Back Projection </font size>
-© 2013
-</center>
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-= Excerpt from Prof. Bouman's Lecture =
-----
-=Accompanying Lecture Notes=
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-introduction
-Convolution back projection (CBP) or forward back projection.
-do not need any calculation in frequency domain so it is faster
-definition?
-Summary of CBP algorithm
-# Measure projections <math>p_{\theta}(r)</math>.
-# Filter the projections to obtain <math>g_{\theta}(r) = h(r)*p_{\theta}(r)</math>.
-# Back project filtered projections <br/>
-<math>f(x,y) = \int_0^{\pi}g_{\theta}(x\cos(\theta)+x\sin(\theta))d\theta</math>
-find images like the copyrighted one using impulse?
-In order to compute the inverse CSFT of <math>F(u,v)</math> in polar coordinates, we must use the Jacobian of the polar coordinate transformation.<br/>
-<math>du dv = |\rho|d\theta d\rho</math>
-where <math>u = \rho\cos(\theta)</math> and <math>v = \rho\sin(\theta)</math>
-This results in the expression <br/>
-<math>f(x,y) = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}F(u,v)e^{2\pi j(xu+yv)}dudv</math>
-Using the Fourier slice theorem, we can replace F(x,y) with its representation in polar coordinates, <math>P_{\theta}(\rho)</math><br/>
-<math>\begin{align}
-\Rightarrow f(x,y) &= \int_{-\infty}^{\infty}\int_0^{\pi}P_{\theta}(r)e^{2\pi j(x\rho\cos(\theta) +y\rho\sin(\theta))}dudv \\
-&= \int_0^{\pi}\underbrace{[\int_{-\infty}^{\infty}|\rho|P_{\theta}e^{2\pi j\rho(x\cos(\theta) +y\sin(\theta))}d\rho]}_{g_{\theta}(x\cos(\theta) + y\sin(\theta))}d\theta
-\end{align}</math>
-Then <math>g_{\theta}(t)</math> is given by<br/>
-<math>\begin{align}
-g_{\theta}(t) &= \int_{\infty}^{\infty}|\rho|P_{\theta}(\rho)e^{2\pi j\rho t}d\rho \\
-&= CTFT^{-1}\{|\rho|P_{\theta}(\rho)\} \\
-&= h(t) * p_{\theta}(r)
-\end{align}</math>
-where <math>h(t) = CTFT^{-1}\{|\rho|\}</math>, and <br/>
-<math>f(x,y) = \int_0^{\pi}g_{\theta}(x\cos(\theta) + y\sin(\theta))d\theta</math>
-==A closer look at Projection Filter==
-. At each angle, projections are filtered.<br/>
-<math>g_{\theta}(r) = h(r)*p_{\theta}(r) \ </math>.
-. the frequency response of the filter is given by <br/>
-<math>H(\rho) = |\rho|</math></br>
-it's like a high pass filter.
-. But the real filters must be bandlimited to <math>|\rho|</math> ≤ <math>f_c</math> for some cutoff frequency <math>f_c</math>.
-Fig 1: Frequency response of <math>H(\rho) = |\rho|</math>
-So<br/>
-<math>\begin{align}
-H(\rho) &= f_c[rect(f/(2f_c))-\wedge(f/f_c)] \\
-h(r) &= f_c^2[2sinc(t2f_c)-sinc^2(tf_c)]
-\end{align}</math>
-Back projection function is <br/>
-<math>f(x,y) = \int_0^{\pi}b_{\theta}(x,y)d\theta</math>
-where <br/>
-<math> b_{\theta}(x,y) = g_{\theta}(x\cos(\theta)+y\sin(\theta))</math>
-Consider the set of points <math>(x,y)</math> such that <br/>
-<math>r = x\cos(\theta)+y\sin(\theta)</math>
-This set looks like
-Fig 2: Graphical representation of the set <math>\{(x,y):r=x\cos(\theta)+y\sin(\theta)\}</math>
-Along this line, <math>b_{\theta}(x,y) = g_{\theta}(r)</math>.
-For each angle <math>\theta</math> back projection is constant along the lines of angle <math>\theta</math> and take on value <math>g_{\theta}(r)</math>.
-Fig 3: Geometric interpretation
-Complete back projection is formed by integrating (summing) back projections for angles ranging from <math>0</math> to <math>\pi</math>.<br/>
-<math>\begin{align}
-f(x,y) &= \int_0^{\pi}b_{\theta}(x,y)d\theta \\
-&\approx \frac{\pi}{M} \sum_{m=0}^{M-1} b_{\frac{m\pi}{M}}(x,y)
-\end{align}</math>
-Back projection "smears" values of <math>g(r)</math> back over the image, and then adds smeared images for each angle.
-[[Image:FST_fig1_mh.jpeg|400px|thumb|left|Fig 1: <math>P_{\theta}(\rho)</math> is <math>F(u,v)</math> in polar coordinates]]
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-== References ==
-* C. A. Bouman. ECE 637. Class Lecture. Digital Image Processing I. Faculty of Electrical Engineering, Purdue University. Spring 2013.
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-==[[Talk:ECE637_tomographic_reconstruction_convolution_back_projection_S13_mhossain|Questions and comments]]==
-If you have any questions, comments, etc. please post them on [[Talk:ECE637_tomographic_reconstruction_convolution_back_projection_S13_mhossain|this page]]
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-[[ECE637_Bouman_lectures_Image_Processing_sLecture_mhossain|Back to the "Bouman Lectures on Image Processing" by Maliha Hossain]]

Difference between revisions of "ECE637 tomographic reconstruction convolution back projection S13 mhossain" - Rhea

Revision as of 16:25, 1 June 2013

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