The Hurst exponent H describing scaling behavior in random processes was introduced in Section 10.4. H can also be determined as a local property when a single window of variable length t remains centered on the observational point in time to. In this case, H is referred to as the local Holder exponent as opposed to the Hurst exponent, which applies to a global property. The rescaled range analysis yields one value of the Holder exponent for each observational point in time and therefore provides a time series H(t).
In images, time corresponds to the Euclidean distance between pixels. It is therefore possible to determine local scaling properties in binary and gray-scale images by analyzing the image values in expanding circles centered on the point for which the scaling property is to be determined. For this purpose, the gray-scale mass dimension is calculated for each pixel of the original image for a small region around the pixel. The scaling range (the range of radii for which the enclosed sum of image values is computed) varies with the total image size and the size of the features of interest, but typical radii range from 2 to 15 pixels. The algorithm to determine the local mass dimension is very similar to Algorithm 10.3, with the difference that the main loop of Algorithm 10.3 is embedded in another double loop over all of the image’s pixels, and instead of using the central pixel (xc, yc), each individual pixel, addressed by the outer double loop, is used as the center for the mass dimension computation.
The effect of the local Holder operator is demonstrated in Figure 10.27. The test image (Figure 10.27A) consists of four square patches of fractional Brownian noise (from left to right and top to bottom with H = 0.2, H = 0.4, H = 0.6, H = 0.8) over a background of white noise, with all areas having the same mean value and standard deviation. The local Holder operator (Figure 10.27B) reveals the local scaling behavior. The areas with a high value of H can be identified clearly. An image (generally, any object) with regions that exhibit different scaling behavior is called multifractal and can be examined further by analyzing the histogram (the probability distribution) of the Holder exponents.
Use of the Holder operator with binary images provides information on the scaling behavior of the shape outlines. When applied to gray-scale images, the Holder operator provides information on the scaling behavior of the texture. Whereas the analysis of time series is established, application of the Holder operator requires special considerations that result from the nature of two-dimensional images. Strictly, the Hurst and Holder exponents are defined only for random processes with long-term persistence. Therefore, the Holder operator is best suited for analysis of the noise content of images. Furthermore, the numerical range of the exponents obtained from the Holder operator can exceed the theoretical range of 0 < H < 1. It is easy to construct cases where the central pixel lies in a dark image region and is surrounded by successively brighter areas as the Euclidean distance from the central point increases. In Figure 10.27B, values of up to 2.6 exist. This would translate into negative and thus invalid fractal dimensions. Finally, the two-dimensional nature of the expanding circles influences the measured mass as it averages more and more pixels with expanding radius. The larger circles contain enough pixels to eliminate many details by averaging. In the extreme case, the averaging behavior causes the scaling properties to degenerate to those of a flat surface. An unsuitable situation that fits this description can be recognized because the log-transformed data points of the mass m(r) and its corresponding radius r no longer lie on a straight line. A linear fit would yield a poor correlation coefficient. It is therefore advisable to compute the correlation coefficient for each fit and reject any pixel with a poor correlation coefficient. Rejection criteria should be made strict, such as acceptance only of pixels with r2 > 0.8. Furthermore, when analyzing the noise content of biomedical images, only few steps with small radii should be made to reduce the influence of large-area averaging.
FIGURE 10.27 Demonstration of the effect of the local Holder operator. Image A shows four patches of fractional Brownian noise over a background of Gaussian (white) noise. Image B shows the result after the application of the Holder operator. Brighter areas indicate a higher exponent H.
To provide one example of texture analysis with the Holder operator, Figure 10.28 shows two examples from the UIUC texture database1 with their associated histograms of the Holder exponent distribution. Preprocessing of the images is important. The influence of noise needs to be minimized by using noise reduction and filtering steps, and contrast should be maximized and standardized: for example, by local histogram equalization.
FIGURE 10.28 Comparison of the distribution of local Holder exponents in images of two different textures: carpet (A) and corduroy (B) (see topic 8). The corresponding histograms of the local Holder exponent are shown in parts C and D, respectively. The local Holder operator may serve as an image-enhancement operator, and histogram analysis (described in Section 8.1) can be applied to obtain quantitative descriptors of the texture.
Different notions for the estimation of local Holder exponents are possible. The rescaled range analysis can be applied directly to images. In this case, the minimum and maximum values (and therefore the range) are determined as a function of the circle radius. Equation (10.14) applies directly, whereby the radius r is used instead of the time window t . A particularly efficient algorithm was proposed by Russ61 in which an octagonal neighborhood with a total of 37 pixels is defined. For each of these pixels, the Euclidean distance to the central pixel is known (Figure 10.29). Based on this neighborhood definition, each pixel that surrounds the central pixel in a 7 x 7 box is labeled with a Euclidean distance to the central pixel, and a table is built that contains the distance and their associated ranges. The table contains seven rows
and two columns. The first column holds the distance.
The second column is filled with the neighborhood range by considering larger and larger distances from row to row. Consequently, the first row contains the range of the four pixels labeled "1" in Figure 10.29. The second row contains the range of the previous four pixels and the four pixels labeled
and so on. Once the table is filled, nonlinear regression yields the exponent H in Equation (10.14). The neighborhood range can be extended arbitrarily by extending the size of the octagon. However, local rescaled range analysis, similar to the local mass dimension, does not necessarily yield valid values for the Holder exponent. Consider the neighborhood given in Table 10.2 as an example. If the bold pixel is chosen as the central pixel, the range increases almost linearly with the Euclidean distance. However, if the central pixel to the right is chosen, the last three steps do not show an increase of the range. Although the regression would still output a value for H and a reasonable correlation coefficient, the scaling behavior has changed for the last three entries of the table (Figure 10.30), and it is debatable whether the resulting Holder exponent should be accepted as valid.
FIGURE 10.29 Octagonal neighborhood and the associated Euclidean distances to the central pixel (shaded).
