Fractal-Based Description

This paper addresses the problems of (1) representing natural shapes such as mountains, trees and clouds, and (2) computing such a description from image data. In order to solve these problems we must be able to relate natural surfaces to their images; this requires a good model of natural surface shapes. Fractal functions are good a choice for modeling natural surfaces because (1) many physical processes produce a fractal surface shape, (2) fractals are widely used as a graphics tool for generating naturallooking shapes, and (3) a survey of natural imagery has shown that the 3-D fractal surface model, transformed by the image formation process, furnishes an accurate description of both textured and shaded image regions. This characterization of image regions has been shown to be stable over transformations of scale and linear transforms of intensity. Much work has been accomplished that is relevant to computing 3-D information from the image data, and the computation of a 3-D fractal-based representation from actual image data has been demonstrated using an image of a mountain. This example shows the potential of a fractal-based representation for efficiently computing good 3-D representations of natural shapes, including such seemingly-difficult cases as mountains, clumps of leaves and clouds. I I N T R O D U C T I O N This paper addresses two related problems: (1) representing natural shapes such as mountains, trees and clouds, and (2) computing such a description from image data. The first step towards solving these problems, it appears, is to obtain a model of natural surface shapes. The task of finding such a model is extremely important to computer vision because we face problems that seem impossible to address with standard descriptive techniques. How, for instance, should we describe the shape of leaves on a tree? Or grass? Or clouds? When we attempt to describe such common, natural shapes using standard shape-primitive representations, the result is an unrealistically complicated model of something that, viewed introspectively, seems very simple. Furthermore, how can we extract 3-D information from the image of a textured surface when we have no models that describe natural surfaces and how they evidence themselves in the image? The lack of such a 3-D model has generally restricted image texture descriptions to being ad hoe statistical measures of the image intensity surface. A good model of natural surfaces together with the physics of image formation would provide the analytical tools necessary for relating natural surfaces to their images. The ability to relate image to surface can provide the necessary leverage for dealing appropriately with the problems of finding a good represent at ion for natural surfaces and computing such a description from the image data. Even shape-from-shading and surface-interpolation methods are limited by the lack of a 3-D model of natural surfaces. Currently all such methods employ the heuristic of "smoothness" to relate neighboring points on the surface. Such heuristics are applicable to many man-made surfaces, of course, but are demonstrably untrue of most natural surfaces. In order to apply such techniques to natural surfaces, therefore, we must find a heuristic that is true of natural surfaces. Finding such a heuristic requires recourse to a 3-D model of natural surfaces. Fractal functions seem to provide such a model of natural surface shapes. Fractals are a novel class of naturallyarising functions, discovered primarily by Benoit Mandelbrot. Mandelbrot and others [1,2,4] have shown that fractals are found widely in nature and that a number of basic physical processes, such as erosion and aggregation, produce fractal surfaces. Because fractals look natural to human beings, much recent computer graphics research has focused on using fractal processes to simulate natural shapes and textures (see Figure 1), including mountains, clouds, water, plants, trees, and primitive animals [3,4,5,6,7]. Additionally, we have recently conducted a survey of natural imagery and found that a fractal model of imaged 3-D surfaces furnishes an accurate description of both textured and shaded image regions, thus providing validation of this physics-derived model for both image texture and shading. The research reported herein was supported by the Defense Advanced Research Projects Agency under Contract No. MDA 903-83-C-0027; this contract is monitored by the U. S. Army Engineer Topographic Laboratory. Approved for public release, distribution unlimited.