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. 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 Benoit 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).
Measuring lacunarity 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 Λ {\displaystyle \Lambda } or λ {\displaystyle \lambda } but there is no single standard and several different methods exist to assess and interpret lacunarity.
Box counting 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 E {\displaystyle \mathrm {E} } sizes, conventionally denoted ε {\displaystyle \varepsilon } s. For each ε {\displaystyle \varepsilon } , a box of size ε {\displaystyle \varepsilon } is placed successively on the image, in the end covering it completely, 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 ε {\displaystyle \varepsilon } in E {\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.
Calculations from box counting The data gathered for each ε {\displaystyle \varepsilon } are manipulated to calculate lacunarity. One measure, denoted here as λ ε {\displaystyle \lambda _{\varepsilon }} , is found from the coefficient of variation ( C V {\displaystyle {\mathit {CV}}} ), calculated as the standard deviation ( σ {\displaystyle \sigma } ) divided by the mean ( μ {\displaystyle \mu } ), for pixels per box. Because the way an image is sampled will depend on the arbitrary starting location, for any image sampled at any ε {\displaystyle \varepsilon } there will be some number ( G {\displaystyle {\mathit {G}}} ) of possible orientations, each denoted here by g {\displaystyle {\mathit {g}}} , that the data can be gathered over, which can have varying effects on the measured distribution of pixels. Equation 1 shows the basic method of calculating λ ε , g {\displaystyle \lambda _{\varepsilon ,g}} :
Probability distributions Alternatively, some methods sort the numbers of pixels counted into a probability distribution having B {\displaystyle B} bins, and use the bin sizes (masses, m {\displaystyle m} ) and their corresponding probabilities ( p {\displaystyle p} ) to calculate λ ε , g {\displaystyle \lambda _{\varepsilon ,g}} according to Equations 2 through 5:
Interpreting λ Lacunarity based on λ ε , g {\displaystyle \lambda _{\varepsilon ,g}} has been assessed in several ways including by using the variation in or the average value of λ ε , g {\displaystyle \lambda _{\varepsilon ,g}} for each ε {\displaystyle \varepsilon } (see Equation 6) and by using the variation in or average over all grids (see Equation 7).
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