# 5. Local OperatorsΒΆ

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:

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

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 $$\beta$$:

$\op T( \alpha f + \beta g) = \alpha \op T f + \beta \op T g$

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 $$\wedge$$):

$\op E ( (\alpha + f) \wedge (\beta + g) ) = (\alpha + \op E f) \wedge (\beta + \op E g)$

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

$\op D ( (\alpha + f) \vee (\beta + g) ) = (\alpha + \op D f) \vee (\beta + \op D g)$

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.