Multi-Scale Fractal Analysis of Image Texture and Pattern

Analyses of the fractal dimension of Normalized Difference Vegetation Index (NDVI] images of homogeneous land covers near Huntsville, Alabama revealed that the fractal dimension of an image of an agricultural land cover indicates greater complexity as pixel size increases, a forested land cover gradually grows smoother, and an urban image remains roughly self-similar over the range of pixel sizes analyzed (10 to 80 meters). A similar analysis of Landsat Thematic Mapper images of the East Humboldt Range in Nevada taken four months apart show a more complex relation between pixel size and fractal dimension. The major visible difference between the spring and late summer NDvI images is the absence of high elevation snow cover in the summer image. This change significantly alters the relation between fractal dimension and pixel size. The slope of the fractal dimensionresolution relation provides indications of how image classification or feature identification will be affected by changes in sensor spatial resolution. Introduction For as long as computers have been used to analyze geographical data sets, spatial data have placed heavy demands on the data processing and storage capabilities of hardware and software. As the geographical and temporal coverage, the spectral and spatial resolution, and the number of individual sensors increase, the sheer volume and complexity of available data sets will continue to tax hardware and software. The increasing importance of networking with the requirement to move data sets between different servers and clients makes the data volume problem particularly acute. Analytical techniques such as stochastic simulation, wavelet decomposition of images into different space and scale components, and geostatistics, which all require large amounts of storage as well as fast processors and networks, also tax available hardware, software, and network services. Mitigating this problem requires using data efficiently, that is, using data at the appropriate scale and resolution to adequately characterize phenomena, thus providing accurate answers to the questions being asked. C.W. Emerson is with the Department of Geography, Geology, and Planning, Southwest Missouri State University, Springfield, MO 65804. ([email protected]). N.S.N. Lam is with the Department of Geography and Anthropology, Louisiana State University, Baton Rouge, LA 70803. D.A. Quattrochi is with the National Aeronautics and Space Administration, Global Hydrology and Climate Center, HR20, George C. Marshall Space Flight Center, Marshall Space Flight Center, AL 35812. PHOTOGRAMMETRIC ENGINEERING 81 REMOTE SENSING Generalization Maps and other models of physical phenomena are simplified abstractions of reality and necessarily involve some degree of generalization. When performed correctly, generalization both reduces the volume of data that must be stored and analyzed and clarifies the analysis itself by separating signal from noise. For geographical analyses, a key concept in the generalization process is scale (Quatttochi, 1993). Cao and Lam (1997) outline various measures of scale: Cartographic scale proportion of distance on a map to the corresponding distance on the ground Geographic (observational) scale size or spatial extent of the study Operational scale the spatial domain over which certain processes operate in the environment Measurement (resolution) scale the smallest distinguishable object or parts of an object Landscape processes are generally hierarchical in pattern and structure, and the study of the relation between the patterns at different levels in this hierarchy may provide a better understanding of the scale and resolution problem (Batty and Xie, 1996; Cao and Lam, 1997). Thus, an analyst must first understand the research question and the spatial domain of the process being measured (including the spatial organization of the features of interest) in order to determine the extent of the required input data and the cartographic scale of the output maps. The research question and the supporting inputs and outputs necessary to address the question, together with the availability of data and the capabilities of the analyst (knowledge, hardware, and software), then determine what resolution is needed in the input data. Fractals Quantifying the complex interrelation between these notions of size, generalization, and precision has proven to be a difficult task, although several measures such as univariate and multivariate statistics, spatial autocorrelation indices such as Moran's I or Geary's C, and local variance within a moving window (Woodcock and Strahler, 1987) provide some understanding of these interactions under given assumptions and within limits of certainty. An important area of research on this topic employs the concept of fractals (Mandelbrot, 1983) to determine the response of measures to scale and resolution (Goodchild and Mark, 1987; Goodchild and Klinkenberg, 1993; Lam and Quattrochi, 1992; Mark and Aronson, 1984). Fractals embody the concept of self-similarity, in which the spatial behavior or appearance of a system is largely independent of scale (Burrough, 1993). Self-similarity is defined Photogrammetric Engineering & Remote Sensing, Vol. 65, No. 1, January 1999, pp. 51-61. 0099-1112/99/6501-0051$3.00/0 O 1999 American Society for Photogrammetry and Remote Sensing l a n u a r y 1999 51 as a property of curves or surfaces where each part is indistinguishable from the whole, or where the form of the curve or surface is invariant with respect to scale. It is impossible to determine the size of a self-similar feature from its form; thus, photographs of geological strata usually include some object of known size for reference. An ideal fractal (or monofractal) curve or surface has a constant dimension over all scales (Goodchild, 1980), although it may not be an integer value. This is in contrast to Euclidean or topological dimensions, where discrete one, two, and three dimensions describe curves, planes, and volumes, respectively. Theoretically, if the digital numbers of a remotely sensed image resemble an ideal fractal surface, then, due to the selfsimilarity property, the fractal dimension of the image will not vary with scale and resolution. However, most geographical phenomena are not strictly self-similar at all scales (Goodchild and Mark, 1987), but they can often be modeled by a stochastic fractal in which the scaling and self-similarity properties of the fractal have inexact patterns that can be described by statistics such as trail lengths, area-perimeter ratios. s~a t ia l autocovariances. and rank-order or freauencv dis&idutions (Burrough, 199.3). Stochastic fractal sets r e f a the monofractal self-similiarity assumption and measure many scales and resolutions ih order io represent the varying form of a phenomenon as a function of local variables across space (De Cola, 1993). Multifractal fields are those in which the scaling properties of the field are characterized by a scaling exponent function. Rather than being described by a single fractal dimension, a multifractal field can be thought of as a hierarchy of sets corresponding to the regions exceeding fixed thresholds (De Cola, 1993). If E,(x) is a value in a multifractal field, then the probability of finding the a value of E, greater than a given scale-dependent threshold A is expressed as where y is the order of singularity (Pecknold et al., 1997). A is a resolution (as expressed as the square root of the ratio of the two-dimensional area to the areas of the smallest object represented in the image). -c(y) is the codimension function, which describes the sparseness of the field intensities. This equation describes how histograms of the density of interest vary with map resolution. Pattern and Texture In image interpretation, pattern is defined as the overall spatial form of related features, and the repetition of certain forms is a characteristic pattern found in many cultural objects and some natural features. Texture is the visual impression of coarseness or smoothness caused by the variability or uniformity of image tone or color (Avery and Berlin, 1992). A potential use of fractals concerns the analysis of image texture (De Cola, 1989; de Jong and Burrough, 1995). In these situations, it is commonly observed that the degree of roughness, or large brightness differences in short spatial intervals, in an image or surface is a function of scale and not of experimental technique. Very often, attempts to increase the precision of measurements by working at greater detail only result in the discovery of additional features that complicate analysis at the new scale (Burrough, 1993). The fractal dimension of remote sensing data could yield quantitative insight on the spatial complexity and information content contained within these data (Lam, 1990). Thus, remote sensing data acquired from different sensors and at differing spatial and spectral resolutions could be compared and evaluated based on fractal measurements (Jaggi et al., 1993). Texture can be measured using a variety of other indices such as variance, range, or standard deviation within a moving window. The differences in texture for images covering the same area but with different cartographic scales and resolutions can indicate the heterogeneity of the scene under observation. The higher the resulting textural parameter, the greater the degree of contrast or heterogeneity in the image. When texture analysis is applied to images of the same scene with different resolutions, the computed indices can be compared with regard to the changes that result from scaling. It is argued that the highest texture index indicates the highest variation and, thus, the resolution level at which most processes operate (Cao and Lam, 1997). The texture analysis method also operates on the assumption that the variability of the geographic data changes with scale and resolution, and the scale at which the maximum variability occurs is related to the operational domain of the processes depicted in the image. By f,inding the maximum variability of the data set, one could find the operational domain of the geographic phenomenon, thus aiding the selection of the appropriate resolution and spatial extent needed in the input imagery. Ovetview of the Methodology A software package known as the Image Characterization and Modeling System (ICAMS) (Quattrochi et al., 1997) was used to explore: How changes in sensor spatial resolution affect the computed fractal dimension of ideal fractal sets, How fractal dimension is related to surface texture, and How changes in the relation between fractal dimension and resolution are related to differences in images collected at different dates. ICAMS provides the ability to calculate the fractal dimension of remotely sensed images using the isarithm method (Lam and De Cola, 1993) (described below) as well as the variogram (Mark and Aronson, 1984) and triangular prism methods [Clarke, 19861. ICAMS also allows calculation of basic descriptive statistiis, spatial statistics, and textural measures such as the local variance (Woodcock and Strahler, 1987), and contains utilities for aggregating images and generating specially characterized images, such as the Normalized Difference Vegetation Index (NDW). The ICAMS software was verified using simulated images of ideal fractal surfaces with specified dimensions. ICAMS was also used to analyze real imagery obtained by an aircraft-mounted high-resolution scanner and multi-date satellite imagery. The fractal dimension for areas of homogeneous land cover in the vicinity of Huntsville, Alabama was computed to investigate the relation between texture and resolution, and a multi-tem~oral analvsis was carried out on ands sat Thematic ~ a ~ i e r image; of the East Humboldt Range area in Nevada. Normalized Difference Vegetation Indexlmm) images were used in each case, because ICAMS requires an eight-bit, single function data set, and because NDW can readily be interpreted in terms of the urbanlrurallforest contrasts in the Huntsville area and the in terms of the seasonal changes in vegetation pattern in the Nevada images. lsarithm Method The isarithm or line-divider method for calculating fractal dimension was used in this analysis due to its robustness, its accuracy, and its relative lack of sensitivity to input parameters. In this method, the fractal dimension of a curve (in a two-dimensional case) is measured using different step sizes that represent the segments necessary to traverse a curve (Lam and De Cola, 1993). For an irregular curve, as the step sizes become smaller, the complexity and length of the stepped representation of the curve increases. The fractal dimension D is derived from the equation D = log Nllog (1lG) (2) where G is the step size and N is the number of steps re52 l a n u a r y 1999 PHOTOCRAMMETRIC ENGINEERING & REMOTE SENSING quired to traverse the curve. If we plot the logarithm of the number of segments needed to traverse a curve for a range of step sizes versus the length of the curve and perform a linear regression, we get