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Revision as of 00:43, 7 July 2019
Communication, Networking, Signal and Image Processing (CS)
Question 5: Image Processing
August 2016 (Published on Jul 2019)
Problem 1
- Calculate 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=-\mu(x,y_0+n*\Delta d)\lambda_n $
- Calculate an expression for $ \hat{P}_n $, an estimate of the integral intensity in terms of $ \lambda_n $, $ \lambda_n^b $, and $ \lambda_b^d $.
$ \lambda_n=(\lambda_n^b-\lambda_n^d)e^{-\int_0^x \mu(t)dt} $
$ \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 $.
- Prove that both $ YY^T $ and $ Y^TY $ are both symmetric and positive semi-definite matrices.
To prove it is symmetric:
$ (YY^T)^T=YY^T $ To prove it is positive semi-definite:
Let $ x $ be an arbitrary vector
$ x^TYY^Tx=(Y^Tx)^T(Y^Tx)\geq 0 $
- Derive expressions for $ V $ and $ \Sigma $ in terms of $ T $, and $ D $.
- Drive expressions for $ U $ in terms of $ Y $, $ T $, and $ D $.
- Derive expressions for $ E $ in terms of $ Y $, $ T $, and $ D $.
- If the columns of $ Y $ are images from a training database, then what name do we give to the columns of $ U $?
