Box counting

Box counting is a method of gathering data for analyzing complex patterns by breaking a dataset, object, image, etc. into smaller and smaller pieces, typically "box"-shaped, and analyzing the pieces at each smaller scale. The essence of the process has been compared to zooming in or out using optical or computer based methods to examine how observations of detail change with scale. In box counting, however, rather than changing the magnification or resolution of a lens, the investigator changes the size of the element used to inspect the object or pattern (see Figure 1). Computer based box counting algorithms have been applied to patterns in 1-, 2-, and 3-dimensional spaces. The technique is usually implemented in software for use on patterns extracted from digital media, although the fundamental method can be used to investigate some patterns physically. The technique arose out of and is used in fractal analysis. It also has application in related fields such as lacunarity and multifractal analysis.

Theoretically, the intent of box counting is to quantify fractal scaling, but from a practical perspective this would require that the scaling be known ahead of time. This can be seen in Figure 1 where choosing boxes of the right relative sizes readily shows how the pattern repeats itself at smaller scales. In fractal analysis, however, the scaling factor is not always known ahead of time, so box counting algorithms attempt to find an optimized way of cutting a pattern up that will reveal the scaling factor. The fundamental method for doing this starts with a set of measuring elements—boxes—consisting of an arbitrary number, called ${\displaystyle \mathrm {E} }$ here for convenience, of sizes or calibres, which we will call the set of ${\displaystyle \epsilon }$s. Then these ${\displaystyle \epsilon }$-sized boxes are applied to the pattern and counted. To do this, for each ${\displaystyle \epsilon }$ in ${\displaystyle \mathrm {E} }$, a measuring element that is typically a 2-dimensional square or 3-dimensional box with side length corresponding to ${\displaystyle \epsilon }$ is used to scan a pattern or data set (e.g., an image or object) according to a predetermined scanning plan to cover the relevant part of the data set, recording, i.e.,counting, for each step in the scan relevant features captured within the measuring element.

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