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