box-counting dimension

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box-counting dimension
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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.

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In fractal geometry, the Minkowski-Bouligand dimension,
also known as Minkowski dimension or box-counting dimension,
is a way of determining the fractal dimension of a set S in a Euclidean space Rn,
or more generally in a metric space (X, d).
It is named after the German mathematician Hermann Minkowski and
the French mathematician Georges Bouligand.


To calculate this dimension for a fractal S,
imagine this fractal lying on an evenly spaced grid,
and count how many boxes are required to cover the set.

The box-counting dimension is calculated by seeing how this number changes
as we make the grid finer by applying a box-counting algorithm.

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