Fractal Dimension Computation for 1D Curves

The box-counting dimension algorithm is a widely used method for computing fractal dimensions of 1D curves. Its core principle involves covering the curve with "boxes" of different scales and observing how the number of required boxes changes with scale variation. The main algorithm steps are: First, cover the curve using a series of squares (boxes) with different side lengths, counting the minimum number of boxes needed to cover the curve at each scale. Then analyze the relationship between box count and scale using logarithmic coordinates, where the slope obtained through linear regression provides the estimated fractal dimension value for the curve. This method is particularly suitable for 1D curves exhibiting self-similar characteristics, such as coastlines, time-series signals, and biological waveforms. As a quantifier of curve complexity, fractal dimension finds significant applications in signal processing, geographic information systems, and biomedical engineering. In practical implementations, code should include scale range validation to ensure reasonable selection and avoid accuracy issues from excessively large or small scales. The algorithm demonstrates sensitivity to noise, potentially requiring smoothing preprocessing routines. Key programming considerations include efficient box-counting functions, logarithmic transformation modules, and robust linear regression components for slope calculation.