\documentclass{uebungsblatt}
\author{Marek Kubica, kubica@in.tum.de}
\fach{Biological Imaging}
\blatt{Midterm}
\gruppe{Bachelor}
\usepackage{graphicx}

\begin{document}

\aufgabe

In X-Ray CT Filtered Backprojection is used to reconstruct the image. To do
that, the images taken from each angle get projected into a virtual copy of
the space in the angle it was taken. As no depth information is available, the
projections go through the complete virtual copy of the space. The points in
which most of the projections overlap are the places where the object actually
is, the remaining noise gets filtered out.

With MRI, the resonance of atoms is exploited by sending RF energy through
the tissue. The rays encode different information which can be measured and
then reconstructed using Fourier transformations.

The MRI is supposedly faster, because it has to run only once for the data,
unlike the backprojection, which has to run for every angle again.

\aufgabe

We know the thickness of the tissue (2cm). Assuming that the tissue is muscle
we can determine from a table that the attenuation coefficient $\mu$ is $0.25$
when put through a 50 keV CT.

\begin{equation}
\frac{I}{I_0} = e^{-\mu x} = e^{-0.25 \cdot 2} \approx 0.6
\end{equation}

\aufgabe

We assume that the tissue was a mouses upper torso, so we can take the values
for $\mu_a$ and $\mu_s$ from the In-Vivo book, Chapter 5.1, Table 5.1.3 as
$\mu_a = 0.25~cm^{-1}$ and $\mu_s = 20~cm^{-1}$.

So using the formula from Beer-Lamberts law, we can calculate the scattering:

\begin{equation}
\frac{I_s}{I_0} = e^{-\mu_s x} = e^{-20 \cdot 2} \approx 4.25 \cdot 10^{-18}
\end{equation}

\begin{equation}
\frac{I_a}{I_0} = e^{-\mu_a x} = e^{-0.25 \cdot 2} \approx 0.37
\end{equation}

Now we would have to subtract $I_s$ from $I_a$ but as $I_a$ is very small, it
would not make a difference.

With a laser (we assume a HeNe laser with a light beam of $632 nm$) the $\mu_a$
would sink, according to Figure 5.1.14, as well as the $\mu_s$, because lasers
tend to have a lower spread than normal light.

\aufgabe

The result of X-Ray is higher because it gets absorbed less than light at 750 nm
by the tissue.

\aufgabe

Conventional microscopy relies on the reflection of light from the surface of
tissues and a system of lenses to magnify.

Confocal microscopy uses an additional pinhole so that light that is not
reflected from the focal point gets blocked and does not light up the image
gathering process, thus giving better quality pictures.

Two-photon microscopy works by sending two photons separately that together
have enough energy for the molecule to be hit to send out light. Other
molecules on the way only get hit my one photon at a time, so they stay dark.

Two-photon microscopy can image deeper, because it uses wavelengths between 700
and 1200 nm, which allows up to fivefold deeper tissue penetration than
confocal microscopy.

\aufgabe

Graphing the attenuation intensity for each angle:

\includegraphics{mri}

(0, 90 and 45 degrees respectively)

To construct the image, one would create images with darker and lighter lines,
depending on the intensity at that point and then overlay the graphics to form
an image. Note that 3 measurements does not lead to an accurate image,
realistically there would be a lot more measurements to form an overlay that
resembles reality more.

\end{document}
