Multi-Scale Fractal Analysis of Image Texture and Pattern Charles W. Emerson, Nina Slu-Ngan Lam, and Dale A. Quattrochi Abstract Generalization Analyses of the fractal dimension of Normalized Difference Maps and other models of physical phenomena are simpli- Vegetation Index (NDVI] images of homogeneous land covers fied abstractions of reality and necessarily involve some de- near Huntsville, Alabama revealed that the fractal dimension gree of generalization. When performed correctly, generaliza- of an image of an agricultural land cover indicates greater tion both reduces the volume of data that must be stored and complexity as pixel size increases, a forested land cover analyzed and clarifies the analysis itself by separating signal gradually grows smoother, and an urban image remains from noise. For geographical analyses, a key concept in the roughly self-similar over the range of pixel sizes analyzed (10 generalization process is scale (Quatttochi, 1993). Cao and to 80 meters). A similar analysis of Landsat Thematic Map- Lam (1997) outline various measures of scale: per images of the East Humboldt Range in Nevada taken Cartographic scale - proportion of distance on a map to the four months apart show a more complex relation between corresponding distance on the ground pixel size and fractal dimension. The major visible difference Geographic (observational) scale - size or spatial extent of between the spring and late summer NDvI images is the ab- the study sence of high elevation snow cover in the summer image. Operational scale - the spatial domain over which certain This change significantly alters the relation between fractal processes operate in the environment dimension and pixel size. The slope of the fractal dimension- Measurement (resolution) scale - the smallest distinguish- resolution relation provides indications of how image classi- able object or parts of an object fication or feature identification will be affected by changes Landscape processes are generally hierarchical in pattern in sensor spatial resolution. and structure, and the study of the relation between the pat- terns at different levels in this hierarchy may provide a bet- Introduction ter understanding of the scale and resolution problem (Batty For as long as computers have been used to analyze geo- and Xie, 1996; Cao and Lam, 1997). Thus, an analyst must graphical data sets, spatial data have placed heavy demands first understand the research question and the spatial domain on the data processing and storage capabilities of hardware of the process being measured (including the spatial organi- and software. As the geographical and temporal coverage, the zation of the features of interest) in order to determine the spectral and spatial resolution, and the number of individual extent of the required input data and the cartographic scale sensors increase, the sheer volume and complexity of avail- of the output maps. The research question and the support- able data sets will continue to tax hardware and software. ing inputs and outputs necessary to address the question, to- The increasing importance of networking with the require- gether with the availability of data and the capabilities of the ment to move data sets between different servers and clients analyst (knowledge, hardware, and software), then determine makes the data volume problem particularly acute. Analyti- what resolution is needed in the input data. cal techniques such as stochastic simulation, wavelet decom- position of images into different space and scale compo- Fractals nents, and geostatistics, which all require large amounts of Quantifying the complex interrelation between these notions storage as well as fast processors and networks, also tax of size, generalization, and precision has proven to be a diffi- available hardware, software, and network services. Mitigat- cult task, although several measures such as univariate and ing this problem requires using data efficiently, that is, using multivariate statistics, spatial autocorrelation indices such as data at the appropriate scale and resolution to adequately Moran's I or Geary's C, and local variance within a moving characterize phenomena, thus providing accurate answers to window (Woodcock and Strahler, 1987) provide some under- the questions being asked. standing 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 resolu- tion (Goodchild and Mark, 1987; Goodchild and Klinkenberg, C.W. Emerson is with the Department of Geography, Geol- 1993; Lam and Quattrochi, 1992; Mark and Aronson, 1984). ogy, and Planning, Southwest Missouri State University, Fractals embody the concept of self-similarity, in which the Springfield, MO 65804. ([email protected]). spatial behavior or appearance of a system is largely inde- pendent of scale (Burrough, 1993). Self-similarity is defined N.S.N. Lam is with the Department of Geography and An- thropology, Louisiana State University, Baton Rouge, LA Photogrammetric Engineering & Remote Sensing, 70803. Vol. 65, No. 1, January 1999, pp. 51-61. D.A. Quattrochi is with the National Aeronautics and Space Administration, Global Hydrology and Climate Center, HR20, 0099-1112/99/6501-0051$3.00/0 George C. Marshall Space Flight Center, Marshall Space O 1999 American Society for Photogrammetry Flight Center, AL 35812. and Remote Sensing PHOTOGRAMMETRIC ENGINEERING 81 REMOTE SENSING lanuary 1999 51 as a property of curves or surfaces where each part is indis- the same area but with different cartographic scales and reso- tinguishable from the whole, or where the form of the curve lutions can indicate the heterogeneity of the scene under ob- or surface is invariant with respect to scale. It is impossible servation. The higher the resulting textural parameter, the to determine the size of a self-similar feature from its form; greater the degree of contrast or heterogeneity in the image. thus, photographs of geological strata usually include some When texture analysis is applied to images of the same scene object of known size for reference. An ideal fractal (or mono- with different resolutions, the computed indices can be com- fractal) curve or surface has a constant dimension over all pared with regard to the changes that result from scaling. It scales (Goodchild, 1980), although it may not be an integer is argued that the highest texture index indicates the highest value. This is in contrast to Euclidean or topological dimen- variation and, thus, the resolution level at which most pro- sions, where discrete one, two, and three dimensions de- cesses operate (Cao and Lam, 1997). The texture analysis scribe curves, planes, and volumes, respectively. method also operates on the assumption that the variability Theoretically, if the digital numbers of a remotely sensed of the geographic data changes with scale and resolution, image resemble an ideal fractal surface, then, due to the self- and the scale at which the maximum variability occurs is re- similarity property, the fractal dimension of the image will lated to the operational domain of the processes depicted in not vary with scale and resolution. However, most geograph- the image. By f,inding the maximum variability of the data ical phenomena are not strictly self-similar at all scales set, one could find the operational domain of the geographic (Goodchild and Mark, 1987), but they can often be modeled phenomenon, thus aiding the selection of the appropriate by a stochastic fractal in which the scaling and self-similarity resolution and spatial extent needed in the input imagery. properties of the fractal have inexact patterns that can be de- scribed by statistics such as trail lengths, area-perimeter ra- Ovetview of the Methodology tios. s~atialautocovariances. and rank-order or freauencv A software package known as the Image Characterization and dis&idutions (Burrough, 199.3). Stochastic fractal sets refa Modeling System (ICAMS)(Quattrochi et al., 1997) was used the monofractal self-similiarity assumption and measure to explore: many scales and resolutions ih order io represent the varying How changes in sensor spatial resolution affect the computed form of a phenomenon as a function of local variables across fractal dimension of ideal fractal sets, space (De Cola, 1993). How fractal dimension is related to surface texture, and Multifractal fields are those in which the scaling proper- How changes in the relation between fractal dimension and ties of the field are characterized by a scaling exponent func- resolution are related to differences in images collected at tion. Rather than being described by a single fractal dimen- different dates. sion, a multifractal field can be thought of as a hierarchy of ICAMS provides the ability to calculate the fractal dimen- sets corresponding to the regions exceeding fixed thresholds sion of remotely sensed images using the isarithm method (De Cola, 1993). If E,(x) is a value in a multifractal field, then (Lam and De Cola, 1993) (described below) as well as the the probability of finding the a value of E, greater than a variogram (Mark and Aronson, 1984) and triangular prism given scale-dependent threshold A is expressed as methods [Clarke, 19861. ICAMS also allows calculation of ba- sic descriptive statistiis, spatial statistics, and textural meas- ures such as the local variance (Woodcock and Strahler, where y is the order of singularity (Pecknold et al., 1997). A 1987), and contains utilities for aggregating images and gen- is a resolution (as expressed as the square root of the ratio of erating specially characterized images, such as the Normal- the two-dimensional area to the areas of the smallest object ized Difference Vegetation Index (NDW). represented in the image). -c(y) is the codimension function, The ICAMS software was verified using simulated images which describes the sparseness of the field intensities. This of ideal fractal surfaces with specified dimensions. ICAMS equation describes how histograms of the density of interest was also used to analyze real imagery obtained by an air- vary with map resolution.