The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B3b. Beijing 2008 
Lacunarity at scale r is defined as the mean-square deviation of 
the variation of mass distribution probability Q(M,r) divided by 
its square mean. 
I(r) 
2' u M 2 Q(M.r) 
m 
_ M 
(2) 
where L(r) = lacunarity at box size r 
M = mass or pixels of interest 
Q(M,r)~ probability of M in box size r 
The Gliding-Box algorithm when applied to binary images 
(images with only 1 bit) counts only the foreground pixels. This 
is because each pixel in a binary image can only have one of 
two possible values (either background or foreground). Whereas 
in greyscale images, one pixel can have many values. In an 8 
bits image, for instance, each pixel can have 2 X values. In this 
case it measures the average intensity of pixels per box which is 
the difference between the maximum and minimum intensity 
value at each box of size r (Karperien, 2007). 
The vertical bars within the column represent the pixel gray 
level values. The red square represent the central pixel of the 
window which is assigned the lacunarity value. The minimum 
pixel value is inside box 1 (which correspond to box u of 
equation 3). The maximum pixel value is inside box 3 (box v in 
equation 3). So, the relative height of the column at this 
particular location will be: 3 - 1 + 1 = 3. 
Figure 1. Example of the DBC algorithm 
3.2 Differential Box Counting 
The Differential Box-Counting (DBC) algorithm was proposed 
by Dong (2000) based on the Gliding-Box algorithm described 
before, and the Differential Box-Counting algorithm proposed 
by Sarkar and Chaudhuri (1992) to fractal dimension estimation. 
According to this algorithm, a gliding-box of size r is placed at 
the upper comer of an image window of size W x W. The 
window size W should be an odd number to allow the computed 
value to be assigned to a central pixel, and r < W. Depending 
on the pixel values within the r x r gliding-box, a column with 
more than one cube may be necessary to cover the maximum 
pixel value by stacking cube boxes on the top of each other. If 
the minimum and maximum pixel values within a given column 
fall in cubic box u and v, respectively. Then, the relative height 
of the column will be (Myint et al, 2006): 
n X i J) = v-K + 1 (3) 
where n r (i, j) = relative height of column at i and j 
V = cubic box with maximum pixel value 
U - cubic box with minimum pixel value 
When the gliding-box slides over the W x W image window, the 
mass will be: 
M r = Y J n r(i,j) (4) 
i.j 
where M r = mass of the grayscale image 
n (f j) = relative height of column at i and j 
Then, the mass M in equation 2 is replaced by M r in equation 3 
to obtain the lacunarity in the W x W window. The lacunarity 
value is assigned to the central pixel of the window and the W x 
W window slides throughout the whole image. 
Figure 1 illustrates an example of a column with three cubic 
boxes r x r x r (boxes 1,2, and 3) on a image window W x W. 
4. EXPERIMENTS 
The image samples selection firstly requires the construction of 
a methodology to identify urban areas with high and low 
inhabitability conditions. A socioeconomic index was created 
from Census 2000 data of the city of Recife (Brazil). The index 
values range between 0 and 1. The closer to 1 an urban area is, 
the higher its inhabitability conditions. Then, these values were 
geo-referenced to each census sector’s centroid of the city. 
Finally, these centroid’s points were interpolated by Ordinary 
Kriging, generating a raster surface which was used as 
reference to the image samples selection (Barros Filho, 2006). 
Then a RGB composition of a © Quickbird image, with spatial 
resolution of 0.70 meters, was laid over the raster surface 
described above, and 30 image samples with 250 x 250 meters 
were selected from this image. Fifteen of them are from urban 
areas with high inhabitability condition, while the other fifteen 
are from areas with low inhabitability conditions. These image 
samples were then enhanced through a histogram equalization 
and converted to grayscale and binary images. 
Appendixes A and B show, respectively, all the grayscale and 
binary image samples generated from the 30 RGB images of 
urban areas with high (images with prefix A) and low (images 
with prefix B) inhabitability condition. As we can see in those 
images, the grayscale image offer more information than binary 
ones. For instance, we can clearly distinguish the shadows of 
the buildings from the trees and the roads in the grayscale 
images, while this is not possible to do in the binary images of 
the same areas. Moreover, buildings are more clearly identified 
in grayscale images than in the binary ones. Some black areas 
of the binary images which seem to be non-built areas are 
actually rooftops with different materials. 
Finally, Gliding-Box and Differential Box-Counting lacunarity 
were estimated for 11 box sizes (in pixels): 2 x 2, 4 x 4, 5 x 5, 8 
x 8, 10 x 10, 16 x 16, 20 x 20, 32 x 32, 40 x 40, 80 x 80, and 
160 x 160. The maximum box size corresponds to 45% of the 
image size. The boxes were set up to slides 2 by 2 pixels (on the