5. 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:

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

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.

• 5.1. Local Operators in Python
• 5.2. Linear Operators
  • 5.2.1. Definition
  • 5.2.2. Correlations and Convolutions
  • 5.2.3. Translation Invariance
  • 5.2.4. Linearity and Translation Invariance
  • 5.2.5. The Continuous Case
  • 5.2.6. Properties
  • 5.2.7. From Convolution Integrals to Convolution Sums
• 5.3. Percentile Filtering
  • 5.3.1. Definition
  • 5.3.2. Algorithms
    • 5.3.2.1. Theoretical Considerations
• 5.4. Bilateral Filtering
  • 5.4.1. Definition
  • 5.4.2. Algorithms
  • 5.4.3. Bilateral filtering of color images
  • 5.4.4. Theoretical Considerations