Local Operators
===============

Local operators are the most important operators in image
processing. To calculate the value in one point in the output image a
local operator looks at all values of that point in the input
image. Let $\set N(\v x)$ be a neighborhood of point $\v x$. A local
operator $f\rightarrow \op T f$ considers all tuples $(\v y,
f(\v y))$ for $y\in\set N(\v x)$ and calculates a new value based on
all these tuples. In general we have:

.. math::
   g(\v x) = (\op T f)(\v x) = t( \{(\v y, f(\v y)) \bigm| \v y \in
   \set N(\v x) \} )

A simple example is local average filter. We take $\set N(\v x) = \{\v
y \bigm| \|\v y-\v x\| \leq R\}$ that is all points with distance less
then or equal $R$ to $\v x$. The function $t$ in this case takes
the average value of all $f(\v y)$ within the neighborhood of $\v x$
and assigns the average to $g(\v x)$. For an image defined on a
continuous domain the average is an integration, for a discrete
(sampled) image the average is a summation.

The definition given above for a local operator encompasses almost
everything that you might think of. But as often in science the power
of a theory is based on restricting the possibilities. In this chapter
we will consider:

* **Translation Invariant Operators**. For these operators the exact
  position of the neighborhood in the image is irrelevant. The
  neighborhood of a point $\v x_1$ is a translated version of the
  neighborhood of a point $\v x_2$. Let $\set N$ denote some subset of
  the domain of the image (most often centred at and containing the
  origin) then for a translation invariant operator $\set N(\v x) =
  \set N+\v x$ where $\set N +\v x$ is the translation of $\set N$
  over vector $\v x$.

Furthermore we will consider the classes of linear and morphological
operators. 

* **Linear Operators**. A linear operator is an operator $\op T$ such
  that for all images $f$ and $g$ and constants $\alpha$ and $\beta$:

  .. math::
     \op T( \alpha f + \beta g) = \alpha \op T f + \beta \op T g

  We say that a linear operator *distributes* over a weighted sum of
  images.

* **Morphological Operators**. There are two basic types of
  morphological operators: erosions and dilations. An erosion is an
  operator that distributes over a pointwise minimum of images
  (denoted with $\wedge$):

  .. math::
     \op E ( (\alpha + f) \wedge (\beta + g) ) = 
     (\alpha + \op E f) \wedge (\beta + \op E g)

  A dilation $\op D$ is defined as an operator that distributes over
  a pointwise maximum of images:

  .. math::
     \op D ( (\alpha + f) \vee (\beta + g) ) = 
     (\alpha + \op D f) \vee (\beta + \op D g)
     
  Observe that weighting for morphological images is additive
  whereas for linear operators weighting is multiplicative.

For both these classes a lot of theory is available. In these lecture
notes we only give a very brief introduction to both types of
operators. 

And of course a lot of local operators that are neither pure linear or
pure morphological are being used in image processing. In this chapter
we only give a few examples: the percentile filter, the bilateral filter
and the kuwahara filter.

.. toctree::

   pythonlocalops.rst
   linearoperators.rst
   morphologicaloperators.rst
   percentilefilter.rst
   bilateralfilter.rst
   kuwaharafilter.rst