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In this context, the present paper focus on the effects of spatial 
scales and radiometry of high spatial resolution satellites 
images in texture classification of urban areas with different 
inhabitability conditions. For this purpose, we calculated 
lacunarity measures of binary and grayscale images through 
different algorithms. Before introducing lacunarity, some 
characteristics of the fractal theory should be described. 
2. FRACTALITY 
As said above, any texture pattern is scale dependent. Actually, 
there is not a single or preferred scale to characterize a texture 
pattern. According to Mandelbrot (1982), the length of a 
coastline in a map depends on the ruler size used to measure it. 
In other words, if the coastline is measured with a large ruler, 
many details will not be calculated. Whereas if this same 
coastline is measured with a smaller ruler, a large number of 
measurement would be required, and bigger would be its length. 
The more accurately we try to measure the length of a fractal 
curve, the longer we find it. As the size of the ruler tends to 
zero, the length of the curve tends to infinite. 
Considering this relation, it is possible to generalize that any 
spatial pattern is governed by a scaling law: the bigger your size 
r is, the smaller will be its quantity N(r). Mathematically, this 
relationship can be represented by a logarithm graph which 
plots the pattern size (X axis) by its frequency (Y axis). The 
slope of the regression line in this graph corresponds to the 
fractal dimension D, and express the level of irregularity of the 
pattern or its space filling efficiency (Mandelbrot, 1982). 
Fractal dimension is based on the hypothesis that spatial 
patterns are self-similar, that is, they repeat themselves among 
many scales and exhibit a certain hierarchical dependency when 
they are simultaneously analyzed in different scales. This 
hierarchical dependency may provide valuable information to 
characterize such patterns. Many image texture analyses have 
been used to differentiate urban land-use classes, applying and 
comparing diverse fractal dimensions. 
There are many algorithms to estimate the fractal dimension of 
an image. One of the most applied is the Box-Counting 
algorithm. According to this algorithm, an image is 
systematically covered with grids. Each one is composed by 
adjacent boxes of size l/2" , where n takes increasing integer 
values from 0 to infinite. For each successive grid, the number 
of boxes N(n) of this size which are needed to cover the image 
are counted. The fractal dimension is defined as: 
D = fractal dimension 
n = any integer value 
N(r) = number of boxes of size r 
Fractal dimension (D) reveals the degree of gray tones 
concentration in an image (De Keersmaecer et al, 2003). The 
more uniform the spatial distribution of the mass (a set of 
selected pixels) in an image, the more close to 2 will be its 
fractal dimension. Otherwise, if the mass is concentrated in a 
single point of the image, D will be zero. Thus, in general, 
dispersed spatial patterns have values of D closer to 2 than 
aggregated ones. 
Despite its importance in understand spatial patterns, fractal 
dimension cannot provide a complete description of urban 
patterns because such patterns are not exactly self-similar. In 
other words: images are not really fractals. They do not exhibit 
the same structure at all scales. Images with similar fractal 
dimension can have different textures (Mandelbrot, 1982; Lin 
and Yang, 1986). Individual elements when aggregated gain 
new properties that cannot be explained by their original 
properties. This means that inference of results from different 
spatial aggregated levels may lead to ecological fallacy. 
3. LACUNARITY 
The concept of lacunarity was established and developed from 
the scientific need to analyze multi-scaling texture patterns in 
nature (mainly in medical and biological research), as a 
possibility to associate spatial patterns to several related 
diagnosis. Regarding texture analysis of urban spaces registered 
by satellite images, lacunarity is a powerful analytical tool as it 
is a multi-scalar measure, that is to say, it permits an analysis of 
density, packing or dispersion through scales. In the end, it is a 
measure of spatial heterogeneity, directly related to scale, 
density, emptiness and variance. It can also indicate the level of 
permeability in a geometrical structure. 
Lacunarity can be defined as a complementary measure of 
fractal dimension or the deviation of a geometric structure from 
its translational invariance (Gefen et al, 1984). It permits to 
distinguish spatial patterns through the analysis of their gap 
distribution in different scales (Plotnick et al, 1996). Gaps in an 
image can be understood as pixels with a specific value (e.g. 
foreground pixels in binary images) or a certain interval of 
values (in grayscale images). The higher the lacunarity of a 
spatial pattern, the higher will be the variability of its gaps in an 
image, and the more heterogeneous will be its texture. 
There are many algorithms to calculate lacunarity of an image. 
Among them, two algorithms have been commonly used: 
Gliding-Box and Differential Box-Counting. In the next section 
it will describe the basic characteristics of each one. 
3.1 Gliding-Box 
The Gliding-Box algorithm was proposed by Allain and Cloitre 
(1991). According to this algorithm a box of size r slides over 
an image. The number of gliding-box with radius r and mass M 
is defined as n(M,r). The probability distribution Q(Mj) is 
obtained by dividing n(M,r) by the total number of boxes.