<title>Wavelet and fractal analysis of ground-vehicle images</title>

Abstract

A large number of terrain images were taken at Aberdeen Proving Grounds, some containing ground vehicles. Is it possible to screen the images for possible targets in a short amount of time using the fractal dimension to detect texture variations? The fractal dimension is determined using the wavelet transform for these visual images. The vehicles are positioned within the grass and in different locations. Since it has been established that natural terrain exhibits a statistical 1/f self-similarity property and the psychophysical perception of roughness can be quantified by the same self-similarity, fractal dimension estimates should vary only at texture boundaries and breaks in the tree and grass patterns. Breaks in the patterns are found using contour plots of the dimension estimates and are considered as perceptual texture variations. Variation in the dimension estimate is considered more important than the accuracy of the actual dimension number. Accurate variation estimates are found even with low resolution images. Mandelbrot defmed a fractal as a set whose Hausdorff-Besicovitch dimension strictly exceeds the topological dimension. This defmition of dimension has no application to fmite sets. There have been many other defmitions applied to the dimension to incorporate the idea of self-similar sets but their relationships are not always clear. For example, the Minkowski dimension is an upper bound for the Hausdorff dimension and the Box dimension is an upper bound for the Minkowski dimension. But when fmite sampling occurs, these inequalities may no longer hold.2 To avoid these difficulties, we will turn to a statistical defmition of a fractal defmed below.'We defme a statistically self-similar fractal function, y(t) as: y(t)=ay(at) witha>O. The equality is defined in terms of second order finite-dimensional statistics. The function y(t) is a zeromean Gaussian. It has been shown that these fractals are successful at modeling a number of natural phenomenon like texture variation in natural terrain.' These fractal textures and signals can be classified and segmented using both fractal dimension and lacunarity.

Suboptimal minimum cluster volume cover-based method for measuring fractal dimension

“…T HIS paper extends prior work [1], [2], [3] on developing more efficient and accurate methods for determining fractal dimension. Our interest in fractal dimension is primarily due to its ability to segment images into different textural regions.…”

“…Texture perception and, more specifically, texture roughness is a key cue feature used in recognition of objects [4], [5]. In [2], we began investigating the use of fractal dimension in segmenting images. Jardine [4] found that the HVS perceives changes in roughness of textures corresponding to the fractal dimension changes of random fractal textures.…”

Lacunarity definition for ramified data sets based on optimal cover

“…For examples, see Cantor set families D s = log(4)/log(3) where Table 1. [2,3], [3,2] and D s = log(9)/log(5) where N c,d = [3,3], [3,4], [4,3]. This is expected since only the gaps in the one dimension were changed, leaving the dimension with larger gaps still contained within the minimum spanning tree.…”

“…A number of mathematical approaches are currently used to quantify a data set's look, including fractal-based methods such as fractal dimension, lacunarity, and connectivity [1], as well as non-fractal-based methods such as variograms [2,3]. Many researchers have successfully used fractal dimension to measure a data set's look for target detection [4][5][6][7][8]. However, fractal dimension alone does not fully describe this visual look because it does not fully describe the space-filling characteristics of data.…”

“…Of all the methods for quantifying a data set's look [1][2][3][4][5][6][7][8], fractal dimension is perhaps the easiest to understand. However, as discussed above, fractal dimension does not fully quantify a texture's look-textures with identical fractal dimensions but small variations in lacunarity can vary substantially in their look.…”