Lacunarity

Lacunarity, from the Latin 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. Beyond being an intuitive measure of gappiness, lacunarity can quantify additional features of patterns such as "rotational invariance" and more generally, heterogeneity. This is illustrated in Figure 1 showing three fractal patterns. 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. The earliest reference to the term in geometry is usually attributed to Mandelbrot, who, in 1983 or perhaps as early as 1977, introduced it as, in essence, an adjunct to fractal analysis. Lacunarity analysis is now used to characterize patterns in a wide variety of fields and has application in multifractal analysis in particular (see Applications).

In many patterns or data sets, lacunarity is not readily perceivable or quantifiable, so computer-aided methods have been developed to calculate it. As a measurable quantity, lacunarity is often denoted in scientific literature by the Greek letters ${\displaystyle \Lambda }$ or ${\displaystyle \lambda }$ but it is important to note that there is no single standard and several different methods exist to assess and interpret lacunarity.

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. Similar to looking at a slide through a microscope with changing levels of magnification, box counting algorithms look at a digital image from many levels of resolution to examine how certain features change with the size of the element used to inspect the image. Basically, the arrangement of pixels is measured using traditionally square (i.e., box-shaped) elements from an arbitrary set of ${\displaystyle \mathrm {E} }$ sizes, conventionally denoted ${\displaystyle \varepsilon }$s. For each ${\displaystyle \varepsilon }$, the box is placed successively over the entire image, and each time it is laid down, the number of pixels that fall within the box is recorded. In standard box counting, the box for each ${\displaystyle \varepsilon }$ in ${\displaystyle \mathrm {E} }$ is placed as though it were part of a grid overlaid on the image so that the box does not overlap itself, but in sliding box algorithms the box is slid over the image so that it overlaps itself and the "Sliding Box Lacunarity" or SLac is calculated. Figure 2 illustrates both types of box counting.

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