Lacunarity

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 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 Λ {displaystyle Lambda } or λ {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 E {displaystyle mathrm {E} } sizes, conventionally denoted ε {displaystyle varepsilon } s. For each ε {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 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. 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}} : 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: 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). Lacunarity analyses using the types of values discussed above have shown that data sets extracted from dense fractals, from patterns that change little when rotated, or from patterns that are homogeneous, have low lacunarity, but as these features increase, so generally does 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. 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. Other methods of assessing lacunarity from box counting data use the relationship between values of lacunarity (e.g., λ ε , g {displaystyle lambda _{varepsilon ,g}} ) and ε {displaystyle varepsilon } in different ways from the ones noted above. One such method looks at the ln {displaystyle ln } vs ln {displaystyle ln } plot of these values. According to this method, the curve itself can be analyzed visually, or the slope at g {displaystyle {mathit {g}}} can be calculated from the ln {displaystyle ln } vs ln {displaystyle ln } regression line. Because they tend to behave in certain ways for respectively mono-, multi-, and non-fractal patterns, ln {displaystyle ln } vs ln {displaystyle ln } lacunarity plots have been used to supplement methods of classifying such patterns.