Box-Counting Dimension with finite resolution

Question

1. Does the method of determining dimension of a shape via the Box-Counting dimension (Minkowski–Bouligand dimension) have to be performed on fractals (objects that look the same at all scales), or can it be performed on any shape? The coastline example seems to suggest that the method may be used to determine the dimension of any shape (since coastlines are not self-similar at all magnifications), but everything I've seen so far explicitly mentions the box-counting method as a tool for determining the dimensionality of fractals.

2. Must I necessarily send the limit of the size of the small boxes covering the shape to 0? I am interested in determining the (observed) dimension of an object given a finite resolution. (Ex: At low resolution, a ball of yarn will look 3D, but at high resolution, the ball of yarn is intrinsically 1D). My inclination is that if I am limited by a certain resolution, the (observed) dimensionality of the object may be a running quantity, but I can't find anything on the question one way or the other.

Answer 1

1. You can compute the box-counting dimension for any bounded subset of $\mathbb{R}^n$ (for unbounded sets, e.g., a line, it does not work as well). For non-fractal sets the result agrees with other concepts of dimension (dimension of a vector space, dimension of a manifold). E.g., a line segment has dimension $1$ because it's covered by $\lceil \epsilon^{-1}\rceil$ intervals of length $\epsilon$.

2. It is worthwhile to consider the behavior of sets on finite scale, although this gets further away from measure theory and closer to computational geometry. Specifically, if $N(\epsilon)$ is the size of a minimal $\epsilon$-net, one can think of $d(\epsilon) = -\log N(\epsilon)/\log \epsilon$ as "dimension at scale $\epsilon$". See $\epsilon$-net article on Wikipedia for a starting point; a search for this term brings up a few relevant articles. A natural question is: what should one do with this "scale-dependent dimension" other than pointing at it?

Answer 2

For completeness, box-counting dimension of a bounded set $E$ is defined by $$\dim(E) = \lim_{\varepsilon\rightarrow0} \frac{\log(N_{\varepsilon}(E))}{\log(1/\varepsilon)},$$ where $N_{\varepsilon}(E)$ represents the number of $\varepsilon$ mesh boxes (or squares or intervals, depending on the dimension of the ambient space) that intersect $E$.

1. There are a number of variations on the above definition, however. For example, $N_{\varepsilon}(E)$ might be defined as the minimum number of balls of radius $\varepsilon$ required to cover $E$ or the maximum number of disjoint balls of radius $\varepsilon$ with centers in $E$. There are other alternatives but they are all within a multiplicative factor of one another and, using the properties of the logarithm, it's not hard to show that they all yield the same value of the limit, if that limit exists.

   However, that limit might not exist. So the correct answer to your first question is that the basic idea of box-counting dimension may be applied to any bounded set but it might or might not yield a result. A generalization is to replace the limit with a limit superior or limit inferior. This yields two new definitions of dimension - the upper box-counting dimension and the lower box-counting dimension. These two are well defined for any bounded set and the (unqualified) box-counting dimension is well defined precisely when the upper and lower dimensions are equal.

2. As you point out, it can be difficult to estimate box-counting dimension in real world examples, as we typically only have the value of $N_{\varepsilon}(E)$ for finitely many values of $\varepsilon$ and we certainly can't compute it for $\varepsilon=0$. Furthermore, when it comes to "real world" examples, as far as we know, the box-counting idea breaks down at the sub-atomic level.

   The standard interpretation in the physics literature, as I understand it, is to presume that the relationship between $N_{\varepsilon}(E)$ and $\varepsilon$ should be maintained over a broad range of values. A standard way to compute the box-counting dimension is compute $N_{\varepsilon_k}(E)$ for some terms $\varepsilon_k$ chosen from a sequence that tends geometrically to zero. We then fit a line to the points in a log-log plot of $N_{\varepsilon}(E)$ versus $\varepsilon$. The box-counting dimension should be approximately the negative slope of that line.