lacunarity

The data gathered for each are manipulated to calculate lacunarity.

It also has application in related fields such as lacunarity and multifractal analysis.

One box counting technique using a "gliding" box calculates lacunarity according to:

Several types of fractal analysis are done, including box counting, lacunarity analysis, mass methods, and multifractal analysis.

In many patterns or data sets, lacunarity is not readily perceivable or quantifiable, so computer-aided methods have been developed to calculate it.

Beyond being an intuitive measure of gappiness, lacunarity can quantify additional features of patterns such as "rotational invariance" and more generally, heterogeneity.

Lacunarity analysis is now used to characterize patterns in a wide variety of fields and has application in multifractal analysis in particular (See Applications).

Below is a list of some fields where lacunarity plays an important role, along with links to relevant research illustrating practical uses of lacunarity.

When rotated 90 , the first two fairly homogeneous patterns do not appear to change, but the third more heterogeneous figure does change and has correspondingly higher lacunarity.

Another proposed way of assessing lacunarity using box counting, the Prefactor method, is based on the value obtained from box counting for the fractal dimension ().

One well-known method of determining lacunarity for patterns extracted from digital images uses box counting, the same essential algorithm typically used for some types of fractal analysis.

Other methods of assessing lacunarity from box counting data use the relationship between values of lacunarity (e.g., ) and in different ways from the ones noted above.

Because they tend to behave in certain ways for respectively mono-, multi-, and non-fractal patterns, vs lacunarity plots have been used to supplement methods of classifying such patterns.

From a practical perspective, multifractal analysis uses the mathematical basis of multifractal theory to investigate datasets, often in conjunction with other methods of fractal analysis and lacunarity analysis.

Indeed, as Mandelbrot originally proposed, lacunarity has been shown to be useful in discerning amongst patterns (e.g., fractals, textures, etc.) that share or have similar fractal dimensions in a variety of scientific fields including neuroscience.

Lacunarity, from the Latin lacuna meaning "gap" or "lake", is a specialized term in geometry referring to a measure of how patterns, especially fractals, fill space, where patterns having more or larger gaps generally have higher lacunarity.

As a measurable quantity, lacunarity is often denoted in scientific literature by the Greek letters or but it is important to note that there is no single standard and several different methods exist to assess and interpret lacunarity.

In some instances, it has been demonstrated that fractal dimensions and values of lacunarity were correlated, but more recent research has shown that this relationship does not hold for all types of patterns and measures of lacunarity.