3.2 2D Variation Procedure

The 2D variation procedure is an analysis of a pixel's environment at different distances [3]. These distances are defined by a square of size . Fig. 6 shows a distance metric for three square sizes , and .

Figure 6: 2D variation procedure

The squares with different sizes are running pixel by pixel across the image matrix from left to right and from top to bottom (fig. 6).

For binary images the numbers of squares containing a part of the structure are counted at the defined sizes . Since the squares are counted pixel by pixel, the number is of approximately the same size as the area of squares for the box counting procedure in sect. 3.1. Thus the calculation of the binary 2D dimension is given in a similar way as the box dimension in equ. (3):

where is again the slope of the regression line in the Richardson-Mandelbrot plot. Fig. 7 shows this plot for the Sierpinsky Gasket (comp. fig. 4).

Figure 7: Richardson-Mandelbrot plot for the number of occupied boxes counted with the 2D variation procedure at different box sizes (in pixels)

It is obvious in fig. 7 that the number of occupied boxes reaches a saturation value for bigger square sizes. This is due to an edge problem since the procedure cannot start at the pixels at the edges of the image. The bigger the square size the smaller the possible number of boxes to be counted anyway. Thus this procedure is not as effective as the box counting procedure (see sect. 3.1). Nevertheless the 2D variation procedure makes sense for discrimination problems. For small box sizes the 2D dimension can be calculated by the slope of the regression line in fig. 7 according to equ. (4):

Only squares with size were taken for the calculation, bigger square sizes have been excluded. The Hausdorff-Besicovitch dimension is overestimated by the 2D dimension . The slope in fig. 7 becomes smaller and smaller for bigger square sizes under consideration and vice versa. Thus it is necessary to be aware of the range of the distance metric if comparisons of different structures are made (see sect. 4).

The 2D variation procedure yields better results for greyscale images. The algorithm determines the minimum and maximum grey values within the square of size and assignes them to the central pixel respectively. Thus one gets both a twodimensional maximum and minimum function for each square size and the difference in volume between the maximum and minimum function is determined for the entire image.

In the Richardson-Mandelbrot plot the dependence of this volume should be linear with the square size , resulting in a power law:

The 2D dimension is again calculated using the slope of the Richardson-Mandelbrot plot. Half of the slope is subtracted from :

This fitting is performed for a better comparism of with the triangular prism surface area dimension in sect. 3.3.

Fig. 8 shows the Richardson-Mandelbrot plot for a 2D analysis of fig. 2.

Figure 8: Richardson-Mandelbrot plot for the 2D variation analysis of a greyscale image

The linear dependence for greyscale images is much better than for binary images (comp. fig. 7). This is due to the algorithms purpose to analyze elevation profiles rather than structures in space. The dimension is according to equ. (7):