There are many definitions of the fractal dimension of an object, including box dimension, Bouligand-Minkowski dimension, and intersection dimension. Although they are all equivalent in the continuous domain, they differ substantially when discretized and applied to digitized data. We show that the standard implementations of these definitions on self-aSne curves with known fractal dimension (Weierstrass-Mandelbrot, Kiesswetter, fractional Brownian motion) yield results with significant errors. An analysis of the source of these errors leads to a new algorithm in one dimension, called the variation method, which yields accurate results. The variation method uses the notion of e oscillation to measure the amplitude of the one-dimensiona1 function in an e neighborhood. The order of growth of the integral of the e oscillation (called the e variation), as e tends toward zero, is directly related to the fractal dimension. In this paper, we present the variation method for one-dimensional {1D) profiles and show that, in the limit, it is equivalent to the classical boxcounting method. The result is an algorithm for reliably estimating the fractal dimension of 1D profiles; i.e. , graphs of functions of a single variable. The algorithm is tested on profiles with known fractal dimension.

Fractal objects derive from many interface phenomena, as they arise in, for example, materials science, chemistry and geology. Hence the problem of estimating fractal dimension becomes of both theoretical and practical importance. Existing algorithms implement the standard definitions of fractal dimension directly, but, as we show, often give unreliable results when applied to digitized and quantized data. We present a new algorithm for estimating the fractal dimension of surfaces - the variation method - that is more reliable and robust than the standard ones. It is based on a new definition of fractal dimension particularly suited for graphs of functions. The variation method is validated with both fractional brownian surfaces and Takagi surfaces, two classes of mathematical objects with known fractal dimension, and is shown to give more accurate results than the classical algorithms. Finally, our new algorithm is applied to data from sand-blasted metal surfaces.

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