Fractal Surface Dimension for Classification of Remotely Sensed Data
Fractal Surface Dimension
The familiar Euclidean dimension of the physical world is expressed as an integer such as in 2-dimensional space (flat surface0 and 3-dimensional space (solid object). The peculiar property of the fractal dimension is that it allows fractional dimension. This fractal dimension is defined as [1,5]
| D = |
log N
-------------
Log (1 / r) |
(1) |
Where D is the fractal dimension, N is the number of the scaled down copies (with linear dimension r) of the original objects which is needed to fill up the original objects. Taking the example of a flat square surface of unit area (1x1), if we scale down this unit area surface to a small square of size r x r, and then use this scaled down squared to fill up the original unit area surface, it is found that a total number o N (N = L2, where L = 1/r) scaled down squares are needed. For this flat surface case, Equation 1 gives
| D = |
log L2
-------------
Log L |
(1) |
Some fractal surfaces are generated using the midpoint displacement method [8], and shown in Figure 1. It is found that the fractal surface dimension corresponds closely to our general conception of the roughness of the surface.
Figure 1: Surfaces of different fractal dimensions
In order to apply the fractal analysis to the SAR images, it is necessary to construct a virtual 3-D surface from the image. This is realized by treating the intensity value of the image (gray level) for each pixel (with coordinated x and y) as the height h of the 3-D surface. The x and y coordinates of a particular point on the 3-D surface correspond respectively to the x and y coordinates of the pixels in the original SAR image. Covering method [7] is used to calculate the fractal surface dimension of each point on the constructed 3-D surface. A square window W of size L unit x L unit (centered at x0y0) is first chosen and the corresponding 3-D surface within the boundary of the window is constructed. This is followed by the normalization of the heights (h) of all the selected surface points within the window as shown below:
| hn = |
h*l
-------------
H |
(2) |
Where hn is the normalized height and H (=255) is the maximum gray level of 8-bit SAR image. The normalization is required to ensure that the calculated fractal surface dimension is within the limit of 3.0. After that, using the small square of size 1 unit x 1 unit as the basic measuring unit, the number of small squares needed to fill up the total 3-D surface within the window is calculated. The fractal surface dimension calculated using Equation 1 will then be assigned to the center pixel (x0y0), and by using the sliding window technique, the center of the window is moved to the adjacent pixel to calculate the fractal surface dimension of that pixel.
It is commonly found that small areas of different landuse regions are abundant in the SAR image. This is especially true for areas surrounding the urban district and areas with mixed plantation. In order to discriminate them from one another, it is reasonable to choose small window size (and thus small 3-D constructed surface) for the calculation of fractal surface dimension. However, due to the normalization process used in the conventional method elaborated above, the variation between the normalized heights (hn) of the pixels within the window becomes less distinct, and thus leads to the small difference in the calculated local fractal surface dimension of different classes. In addition, the fractal surface dimension calculated using the conventional method varies with the window size chosen. Therefore, a new method based on dimension normalization technique is developed. This modified method requires no height normalization of the constructed 3-D surface. The same windowing technique is applied and the original values of the intensity of the pixels are retained, and the 3-D virtual surface is constructed as stained above. The height normalization step is no longer performed. Using the same square of size 1 unit x 1 unit as the basic measuring unit, he total number of square (N') needed to cover the surface is calculated . This calculated N' is later normalized by the base surface area of size L unit x L unit. This flat base surface area has a physical dimension of 2 and is also equal to the window size. The normalized N' (N'n) can be related to the maximum gray level H by the power law [3] as shown below:
| N'n = |
N'
-------------
(L*L) |
= Hf (3) |
where f is the partial fractal surface dimension. Taking logarithm of both sides of Equation 3 and rearranging it, we have
| f = |
log(N'n)
-------------
log(H) |
(4) |
and the modified local fractal surface dimension Dd is obtained by adding the physical dimension of 2.0 to f, as follows:
Dd = 2.0 + f (5)
Results and Discussion
In this study, SAR images of Sungai Petani area with three polarizations (HH, VV, HV) taken during the GlobeSAR mission in November 1993 are used to calculate the fractal surface dimension (D) and the modified fractal surface dimension (Dd). The calculated D and Dd for window sizes from 3 x 3 to 25 x 25 for HV polarization are shown respectively in Figures 2a and 2b.
Figure 2a : Conventional fractal surface dimension versus window size for SAR HV image.
Figure 2b: Modified fractal surface dimension versus window size for SAR HV image.
