Box counting

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)[2] and gray-scale[5] digital images (i.e., jpegs, tiffs, etc.). Box counting is generally done on patterns extracted 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 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 ${\displaystyle \epsilon }$ is counted). For other types of analysis, the data sought may be the number of pixels that fall within the measuring box,[4] 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).[5][6][7][8]

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.

Figure 2a. Boxes laid over an image as a fixed grid.

Figure 3. Retinal vasculature revealed through box counting analysis; colour-coded local connected fractal dimension analysis done with FracLac freeware for biological image analysis.

Figure 4. It takes 12 green but 14 yellow boxes to completely cover the black pixels in these identical images. The difference is attributable to the position of the grid, illustrating the importance of grid placement in box counting.

The traditional approach is to scan in a non-overlapping regular grid or lattice pattern.[3][4] To illustrate, Figure 2a 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 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 ${\displaystyle \epsilon }$ never overlaps where it has previously been (see Figure 4). This is done until the entire area of interest has been scanned using each ${\displaystyle \epsilon }$ and the relevant information has been recorded.[9] [10] When used to find a box counting dimension, the method is modified to find an optimal covering.

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 2b 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 ${\displaystyle \epsilon }$. Sliding box algorithms are often used for analyzing textures in lacunarity analysis and have also been applied to multifractal analysis.[2][8][11][12][13]

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 ${\displaystyle \epsilon }$ is centred on each pixel of interest, as illustrated in Figure 2c.[7]

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.

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.

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.[17][18]

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.[19]