(image-transformations)=
Image transformations
===

We can make use of image histograms in different ways. Using them we can
perform simple *image segmentation* or *image enhancement*.

To achieve a better understanding of how to change a histogram and the
affect it has on an image we must first take a look at so called *linear
transformation*.

````{prf:definition} Linear Transformation
A *linear transformation* in context of image histogram performs a
stretch/compression of the pixel intensity values from $[a,b]$ to
$[l,L]$:
```{math}
g(x) = \frac{f(x)-a}{b-a}\cdot(L-l)+l.
```

````
```{figure} ../atelier/img/01_linear_trans.png
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```

As one might see here we applied a linear transformation from the
interval $[a,b] = [75,225]$ to $[l,L] = [0,250]$. The applied
transformation would accordingly look like this:
```{math}
\hat{H}_{\theta} = \frac{H_{\theta}-75}{225-75}\cdot(250-0) + 0
```
Where there is a linear transformation one might expect a non linear
transformation. So we define the *non-linear gamma transformation*:

````{prf:definition} Non-linear Gamma Transformation
Contrary to the linear transformation which stretches or compresses more
so the values on the x-axis, the *gamma transformation*, which is
sometimes also called *gamma correction*, applies a stretch and
compression along the y-axis according to the choice of the parameter
$\gamma$. The gamma transformation is defined as follows:
```{math}
g(x)=255\cdot{\left(\frac{f(x)}{255}\right)}^\gamma \quad , \gamma>0.
```

````

In the following example we can see the impact of applying a gamma
transformation with the choice of the parameter $\gamma = 0.4$ has on
the original image. While the pixel intensity values on the left side of
the histogram get compressed into nothingness, the values stretch
upwards more as we move further to the right.

```{figure} ../atelier/img/01_gamma_beispiel.png
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```

The following graphic shows the different gamma functions.

```{figure} ../atelier/img/01_gamma_parameter.jpg
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```

As one might imagine by the form of the gamma function one can estimate
its impact it applies to a histogram. The higher our choice for $\gamma$
is, the stronger the pixel intensity values on the far right get
amplified while the values the further we move left get
compressed/lowered. Choosing smaller a $\gamma$ yields opposite results.
As already mentioned nearing the cumulative distribution to a line the
better the contrast of the image is. For this we have to equally
distribute the pixel intensity values across the histogram. For this we
require the following tool:

````{prf:definition} Equalization
For an image $f$ of size $M\times N$ pixels with gray-levels
$\{0,...,L-1\}$ the resulting function given by applying *equalization*
to a histogram is defined as:
```{math}
g_i = \lfloor (L-1) \sum_{k=0}^{f_i}h(k) \rfloor,
```
with $h(k)=\frac{|\{i\in\Omega|f_i=k\}|}{MN}$

````

The following graphic shows the effect of applying *equalization*.

```{figure} ../atelier/img/01_Equalization.png
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```

This example shows us exactly how the histogram changes and what effect
it has on the image. Equalization spreads the pixel intensity values
across the x-Axis without changing the individual values themselves.
Simply put, we just change their location. Until now we only took
monochrome images into account, though we can also apply our tools to
colored images. Take an RGB image for example, to properly analyze it
with histograms we would just need the histograms for red, green and
blue shades. Applying equalization yields the following results:

```{figure} ../atelier/img/01_RGB-EQ.png
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```

As we equalize the red, green and blue histogram we improve the contrast
from the original image on the left to the equalized image on the right.

```{figure} ../atelier/img/01_RGB-EX.png
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```