Image Representation
====================


Coordinates and Indices
-----------------------

Accustomed to the choice of coordinate axes in mathematics you might
think that the axes for an image $f(x,y)$ are the same: the x-axis
running from left to right, the y-axis from bottom to top with the
origin in the lower left corner. This is **not** true for digital
images in general (the one exception i know of is Microsofts Device
Independent Bitmap).
   
In order to get an understanding of the many different formats for
digital images we have to distinguish:
   
- the axes with the coordinates along those axes, and
- the indices into the array representing all samples.
   
We will use numpy and matplotlib throughout to set our definition for
these lecture notes.
   
We will use this image as leading example:
   
.. math::
   \begin{array}{cccc}
   0& 1& 2& 3\\
   4& 5& 6& 7\\
   8& 9&10&11
   \end{array}
   
In numpy this is easy to generate and plot with matplotlib:

.. ipython:: python

   f = np.arange(12).reshape(3,4)
   plt.imshow(f, cmap='gray', interpolation='nearest');

   @savefig imageaxes.png width=8cm align=center
   plt.show()

The sample points are in the middle of the squares (NOT pixels) and we
see that the origin with coordinates is chosen to be at the top
left. The x-coordinate axis runs from left to right and the
y-coordinate axis runs from top to bottom.

In this coordinate axes system we can draw a line from $(0,2)$ to
$(3,0)$

.. ipython:: python

   f = np.arange(12).reshape(3,4)
   plt.imshow(f, cmap='gray', interpolation='nearest');

   plt.plot([0,3], [2,0], lw=3);

   @savefig imageaxesplot.png width=8cm align=center
   plt.show()

Carefully note that the coordinate system is a **left-handed**
one. You have to turn the x-axis *clockwise* to the y-axis. The
classical choice with x axis from left to right and y-axis from bottom
to top results in a right-handed coordinate system. Be aware that in a
left handed system things are a bit different then you might expect. A
rotation over a positive angle $\phi$ characterized with rotation
matrix:

.. math::
   R = \matvec{cc}{\cos\phi & -\sin\phi\\
                   \sin\phi & \cos\phi}

is turning a vector *counter clockwise* in a right handed coordinate
system, but *clockwise* in a left handed system.


Being familiar with numpy arrays you have probably understood that in
case we want to index the array with the coordinates $(x,y)$ (with $x$
and $y$ integer valued of course) that we need to write :code:`f[y,x]`.

**So, the first index in the image array is the y-coordinate, and the
second index is the x-coordinate.**


Color Images
------------

Let's read a color image from disk and display it:

.. ipython:: python

   f = plt.imread('python/data/images/peppers.png')
   plt.imshow(f);

   @savefig peppersrgb.png width=10cm align=center
   plt.show();

   print(f.shape)

The shape of a color image array thus is $M\times N \times 3$. It is
also possible that a 4-th *channel* is used that encodes the
transparency mask when displaying an image. Be sure to check and not
assume things that are not nescessarily true.

Each of the planes :code:`f[:,:,c]` with :code:`c` equal to 0, 1 or 2
gives one of the color planes. Each color plane is a scalar image and
can be displayed as such:

.. ipython:: python

   plt.imshow(f[:,:,1], cmap='gray'); # the green channel is a good approximation of the luminance
  
   @savefig peppersgreenchannel.png width=10cm align=center
   plt.show()



Domain Iterators
----------------