Box counting is a method of gathering
data
In the pursuit of knowledge, data (; ) is a collection of discrete values that convey information, describing quantity, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols that may be further interpreted ...
for analyzing
complex patterns by breaking a
dataset A data set (or dataset) is a collection of data. In the case of tabular data, a data set corresponds to one or more database tables, where every column of a table represents a particular variable, and each row corresponds to a given record of the ...
, 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
Figure 1 is a Toronto, Ontario-based online social networking service for healthcare professionals to post and comment on medical images.
Figure 1 was founded in Toronto by Dr. Joshua Landy, Richard Penner and Gregory Levey. The platform launched ...
). 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
''Extracted'', also known as ''Extraction'' in the UK, is an independent 2012 American science fiction thriller directed and written by Nir Paniry. Sasha Roiz stars as a scientist whose consciousness becomes trapped in the mind of a convict (Dom ...
from
digital media
Digital media is any communication media that operate in conjunction with various encoded machine-readable data formats. Digital media can be created, viewed, distributed, modified, listened to, and preserved on a digital electronics device. ' ...
, although the
fundamental method can be used to investigate some patterns physically. The technique arose out of and is used in
fractal analysis
Fractal analysis is assessing fractal characteristics of data. It consists of several methods to assign a fractal dimension and other fractal characteristics to a dataset which may be a theoretical dataset, or a pattern or signal extracted from p ...
. It also has application in related fields such as
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. B ...
and
multifractal
A multifractal system is a generalization of a fractal system in which a single exponent (the fractal dimension) is not enough to describe its dynamics; instead, a continuous spectrum of exponents (the so-called singularity spectrum) is needed. ...
analysis.
The method
Theoretically, the intent of box counting is to quantify
fractal
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...
scaling, but from a practical perspective this would require that the scaling be known ahead of time. This can be seen in
Figure 1
Figure 1 is a Toronto, Ontario-based online social networking service for healthcare professionals to post and comment on medical images.
Figure 1 was founded in Toronto by Dr. Joshua Landy, Richard Penner and Gregory Levey. The platform launched ...
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
here for convenience, of sizes or calibres, which we will call the set of
s. Then these
-sized boxes are applied to the pattern and counted. To do this, for each
in
, a measuring element that is typically a 2-dimensional square or 3-dimensional box with side length corresponding to
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.
The data
The relevant features gathered during box counting depend on the subject being investigated and the type of analysis being done. Two well-studied subjects of box counting, for instance, are binary (meaning having only two colours, usually black and white)
and gray-scale
digital image
A digital image is an image composed of picture elements, also known as ''pixels'', each with ''finite'', '' discrete quantities'' of numeric representation for its intensity or gray level that is an output from its two-dimensional functions ...
s (i.e., jpegs, tiffs, etc.). Box counting is generally done on patterns
extracted
''Extracted'', also known as ''Extraction'' in the UK, is an independent 2012 American science fiction thriller directed and written by Nir Paniry. Sasha Roiz stars as a scientist whose consciousness becomes trapped in the mind of a convict (Dom ...
from such still images in which case the raw information recorded is typically based on features of pixels such as a predetermined colour value or range of colours or intensities. When box counting is done to determine a
fractal dimension
In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is meas ...
known as the
box counting dimension, the information recorded is usually either yes or no as to whether or not the box contained any pixels of the predetermined colour or range (i.e., the number of boxes containing relevant pixels at each
is counted). For other types of analysis, the data sought may be the number of pixels that fall within the measuring box,
the range or average values of colours or intensities, the spatial arrangement amongst pixels within each box, or properties such as average speed (e.g., from particle flow).
Scan types
Every box counting algorithm has a scanning plan that describes how the data will be gathered, in essence, how the box will be moved over the space containing the pattern. A variety of scanning strategies has been used in box counting algorithms, where a few basic approaches have been modified in order to address issues such as sampling, analysis methods, etc.
Fixed grid scans
The traditional approach is to scan in a non-overlapping regular grid or lattice pattern.
To illustrate,
Figure 2
Figure may refer to:
General
*A shape, drawing, depiction, or geometric configuration
*Figure (wood), wood appearance
*Figure (music), distinguished from musical motif
*Noise figure, in telecommunication
*Dance figure, an elementary dance patter ...
a shows the typical pattern used in software that calculates box counting dimensions from patterns extracted into binary digital images of contours such as the fractal contour illustrated in
Figure 1
Figure 1 is a Toronto, Ontario-based online social networking service for healthcare professionals to post and comment on medical images.
Figure 1 was founded in Toronto by Dr. Joshua Landy, Richard Penner and Gregory Levey. The platform launched ...
or the classic example of the coastline of Britain often used to explain the method of finding a
box counting dimension. The strategy simulates repeatedly laying a square box as though it were part of a grid overlaid on the image, such that the box for each
never overlaps where it has previously been (see
Figure 4). This is done until the entire area of interest has been scanned using each
and the
relevant information has been recorded.
When used to find a
box counting dimension, the method is modified to find an
optimal covering.
Sliding box scans
Another approach that has been used is a sliding box algorithm, in which each box is slid over the image overlapping the previous placement.
Figure 2
Figure may refer to:
General
*A shape, drawing, depiction, or geometric configuration
*Figure (wood), wood appearance
*Figure (music), distinguished from musical motif
*Noise figure, in telecommunication
*Dance figure, an elementary dance patter ...
b illustrates the basic pattern of scanning using a sliding box. The fixed grid approach can be seen as a sliding box algorithm with the increments horizontally and vertically equal to
. Sliding box algorithms are often used for analyzing textures in
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. B ...
analysis and have also been applied to
multifractal analysis
A multifractal system is a generalization of a fractal system in which a single exponent (the fractal dimension) is not enough to describe its dynamics; instead, a continuous spectrum of exponents (the so-called singularity spectrum) is needed ...
.
Subsampling and local dimensions
Box counting may also be used to determine local variation as opposed to global measures describing an entire pattern. Local variation can be assessed after the data have been gathered and analyzed (e.g., some software colour codes areas according to the fractal dimension for each subsample), but a third approach to box counting is to move the box according to some feature related to the pixels of interest. In
local connected dimension box counting algorithms, for instance, the box for each
is centred on each pixel of interest, as illustrated in
Figure 2
Figure may refer to:
General
*A shape, drawing, depiction, or geometric configuration
*Figure (wood), wood appearance
*Figure (music), distinguished from musical motif
*Noise figure, in telecommunication
*Dance figure, an elementary dance patter ...
c.
Methodological considerations
The implementation of any box counting algorithm has to specify certain details such as how to determine the actual values in
, including the minimum and maximum sizes to use and the method of incrementing between sizes. Many such details reflect practical matters such as the size of a digital image but also technical issues related to the specific analysis that will be performed on the data. Another issue that has received considerable attention is how to approximate the so-called "optimal covering" for determining
box counting dimensions and assessing
multifractal scaling.
Edge effects
One known issue in this respect is deciding what constitutes the edge of the useful information in a digital image, as the limits employed in the box counting strategy can affect the data gathered.
Scaling box size
The algorithm has to specify the type of increment to use between box sizes (e.g., linear vs exponential), which can have a profound effect on the results of a scan.
Grid orientation
As
Figure 4 illustrates, the overall positioning of the boxes also influences the results of a box count. One approach in this respect is to scan from multiple orientations and use averaged or optimized data.
To address various methodological considerations, some software is written so users can specify many such details, and some includes methods such as smoothing the data after the fact to be more amenable to the type of analysis being done.
See also
*
Fractal analysis
Fractal analysis is assessing fractal characteristics of data. It consists of several methods to assign a fractal dimension and other fractal characteristics to a dataset which may be a theoretical dataset, or a pattern or signal extracted from p ...
*
Fractal dimension
In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is meas ...
*
Minkowski–Bouligand dimension
450px, Estimating the box-counting dimension of the coast of Great Britain
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 se ...
*
Multifractal
A multifractal system is a generalization of a fractal system in which a single exponent (the fractal dimension) is not enough to describe its dynamics; instead, a continuous spectrum of exponents (the so-called singularity spectrum) is needed. ...
analysis
*
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. B ...
References
{{Fractals, state=collapsed
Fractals
Dimension theory