m |
m |
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Line 58: | Line 58: | ||

<math>Y</math> is of size <math>p\times N</math>, so the size of <math>Y^TY</math> is <math>N\times N</math>. | <math>Y</math> 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. | + | Obviously, the size of <math>Y^TY</math> is much smaller, since <math>N<<p</math>. |

</center> | </center> | ||

# Prove that both <math>YY^T</math> and <math>Y^TY</math> are both symmetric and positive semi-definite matrices. | # Prove that both <math>YY^T</math> and <math>Y^TY</math> are both symmetric and positive semi-definite matrices. | ||

+ | |||

+ | <center> | ||

+ | To prove it is symmetric: | ||

+ | |||

+ | <math> | ||

+ | (YY^T)^T=YY^T | ||

+ | </math> | ||

+ | To prove it is positive semi-definite: | ||

+ | |||

+ | Let <math>x</math> be an arbitrary vector | ||

+ | |||

+ | |||

+ | <math> | ||

+ | x^TYY^Tx= | ||

+ | </math> | ||

+ | </center> | ||

# Derive expressions for <math>V</math> and <math>\Sigma</math> in terms of <math>T</math>, and <math>D</math>. | # Derive expressions for <math>V</math> and <math>\Sigma</math> in terms of <math>T</math>, and <math>D</math>. |

## Revision as of 01:42, 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= $

- 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 $?