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#Specify the size of <math>YY^t</math> and <math>Y^tY</math>. Which matrix is smaller | #Specify the size of <math>YY^t</math> and <math>Y^tY</math>. Which matrix is smaller | ||
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| + | <center> | ||
| + | Y is of size <math>p \times N</math>, so the size of <math>YY^t</math> is <math>p \times p</math> | ||
| + | |||
| + | Y is of size <math>p \times N</math>, so the size of <math>Y^tY</math> is <math>N \times N</math> | ||
| + | |||
| + | Obviously, the size of <math>Y^tY</math> is much smaller, since N << p. | ||
| + | </center> | ||
Revision as of 19:24, 9 July 2019
Communication, Networking, Signal and Image Processing (CS)
Question 5: Image Processing
August 2016 (Published in Jul 2019)
Problem 1
- Calcualte an expression for $ \lambda_n^c $, the X-ray energy corrected for the dark current
$ \lambda_n^c=\lambda_n^b-\lambda_n^d $
- Calculate an expression for $ G_n $, the X-ray attenuation due to the object's presence
$ G_n = \frac{d\lambda_n^c}{dx}=-\mu (x,y_0+n * \Delta d)\lambda_n^c $
- Calculate an expression for $ \hat{P}_n $, an estimate of the integral intensity in terms of $ \lambda_n $, $ \lambda_n^b $, and $ \lambda_n^d $
$ \lambda_n = (\lambda_n^b-\lambda_n^d) e^{-\int_{0}^{x}\mu(t)dt}d)\lambda_n^c $
$ \hat{P}_n = \int_{0}^{x}\mu(t)dt= -log(\frac{\lambda_n}{\lambda_n^b-\lambda_n^d}) $
- For this part, assume that the object is of constant density with $ \mu(x,y) = \mu_0 $. Then sketch a plot of $ \hat{P}_n $ versus the object thickness, $ T_n $, in mm, for the $ n^{th} $ detector. Label key features of the curve such as its slope and intersection.
Problem 2
- Specify the size of $ YY^t $ and $ Y^tY $. Which matrix is smaller
Y is of size $ p \times N $, so the size of $ YY^t $ is $ p \times p $
Y is of size $ p \times N $, so the size of $ Y^tY $ is $ N \times N $
Obviously, the size of $ Y^tY $ is much smaller, since N << p.
