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Let <math>x</math> be an arbitrary vector
 
Let <math>x</math> be an arbitrary vector
 
  
 
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<math>
x^TYY^Tx=
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x^TYY^Tx=(Y^Tx)^T(Y^Tx)\geq 0
 
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</center>

Revision as of 01:43, 7 July 2019


ECE Ph.D. Qualifying Exam

Communication, Networking, Signal and Image Processing (CS)

Question 5: Image Processing

August 2016 (Published on Jul 2019)

Problem 1

  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 $

  1. 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 $

  1. 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} $

  1. 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

  1. 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 $.

  1. 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 $

  1. Derive expressions for $ V $ and $ \Sigma $ in terms of $ T $, and $ D $.
  1. Drive expressions for $ U $ in terms of $ Y $, $ T $, and $ D $.
  1. Derive expressions for $ E $ in terms of $ Y $, $ T $, and $ D $.
  1. If the columns of $ Y $ are images from a training database, then what name do we give to the columns of $ U $?

Alumni Liaison

Ph.D. on Applied Mathematics in Aug 2007. Involved on applications of image super-resolution to electron microscopy

Francisco Blanco-Silva