Documentation

1. Measure of fractal dimension

1.2 The counting methods

Now, there is seven methods to measure fractal dimension.

grid
radius mass
dilation
correlation
gaussian convolution
box-counting (testing)
network (testing)

1.2.1 Grid method

This is the most used method to estimate fractal dimension. The image is covered by a quadratical grid and the grid distance ε is then varied. Following the logic described earlier, for each value ε, the number of squares N(ε) containing any occupied point is counted. Usually, the value set of ε follows a power of 2.

Parameters : centre and size of the square zone to study

You can choose the centre by selecting a pixel of the image then click on the button "Cursor", or directly capturing the coordinates. You can also use the barycentre of the image. The size of the square define also the value set of ε, for example size 16 corresponds to the set (1, 2, 4, 8), size 72 corresponds to the set (1, 3, 9, 18, 36) and size 216 corresponds to the set (1, 3, 9, 27, 54, 108). Values sets are mixed with power of 2 and power of 3.

1.2.2 Radius mass method

This method refers to a specific point known as the counting centre and gives the law of distribution of the occupied sites around this point. A circle is drawn around this point, and the radius r is gradually increased. At each step, the total number of occupied points N(ε) inside the circle is counted. In this method, ε equals to 2.r+1.

Parameters : counting centre and form of the counting window (circle or square)
You can choose the counting centre by the same way as Grid method.

1.2.3 Correlation method

Each point of the image is surrounded with a small squared window. The number of occupied points inside each window is enumerated. This allows the mean number of points per window of that given size to be calculated. The same operation is applied for windows of increasing sizes. The X-axis of the graph represents the size of the side of the counting window ε = (2i+1). The Y-axis represents the mean number of counted points per window. (Because the theory underlying the correlation analysis considers the simultaneous presence of two points at a certain distance, i.e. the mean distance between a pair of built-up pixels, the correlation dimension is a second order fractal dimension. In a multi-fractal theoretical framework, this correlation dimension should be extended to a series of three, four or more points). In principle it is possible to choose any shape for the window, such as circle, hexagon, etc. However, since pixels are square-like, the choice of a square helps to avoid rounding errors.

Parameters : maximum window size (ε).

1.2.4 Dilation method

This method is based on the algorithm introduced by Minkowski and Bouligand to establish the dimension of an object using the measure theory approach. In this analysis each occupied point is surrounded by a square of size ε, the surface of which is considered to be completely occupied. The size of these squares is then gradually enlarged, and we measure the total surface A(ε) covered at each stage. As the squares are enlarged, any details smaller than ε are overlooked and we gradually obtain an approximation of the original form. Because more and more squares overlap, the total occupied surface for a particular value ε is less than what it would be if the same number of occupied points that make up the original form were surrounded individually. By dividing this total surface by the surface of a test square (ε²), we get an approximation of the number of elements N(ε) necessary to cover the whole.

Parameters : number of dilation

1.2.5 Gaussian convolution

When the image is reduced to a single curve, we can apply another counting method: the gaussian convolution. Contrary to the other methods, the gaussian convolution is applied on a curve and not on an image. At each iteration step, the curve is more and more smoothed. In this case, the structuring element (which increases at each iteration step) is the variance of the gaussian function used to smooth the curve. The X-axis represent the variance of the gaussian function and Y-axis the length of the curve (expressed in number of pixels) divided by the variance.

Parameters : number of step and maximal variance (in pixel).

1.2.6 Box method (testing)

This method consist in finding the least number of square of size ε needed to cover all black pixels. The algorithm converge to the minimum in infinite time, so the results are only approximation of the best coverage. It is a generalized version of grid method.

1.2.7 Network method (testing)

<- 1.1 Principles

1.3 The estimation module ->