Shape analysis using fractal dimension: a curvature based approach

The present work shows a novel fractal dimension method for shape analysis. The proposed technique extracts descriptors from the shape by applying a multiscale approach to the calculus of the fractal dimension of that shape. The fractal dimension is obtained by the application of the curvature scale-space technique to the original shape. Through the application of a multiscale transform to the dimension calculus, it is obtained a set of numbers (descriptors) capable of describing with a high precision the shape in analysis. The obtained descriptors are validated in a classification process. The results demonstrate that the novel technique provides descriptors highly reliable, confirming the precision of the proposed method.

💡 Research Summary

The paper introduces a novel shape‑analysis framework that combines curvature scale‑space (CSS) processing with fractal‑dimension estimation to produce a set of multiscale shape descriptors. Traditional fractal‑dimension methods—such as box‑counting or the Koch‑curve approach—operate on discrete representations and are highly sensitive to noise and the choice of scale. To overcome these drawbacks, the authors first extract the boundary of a binary shape and compute its curvature κ(s) at the original resolution. The boundary is then progressively smoothed by convolving it with Gaussian kernels of increasing standard deviation σ, generating a family of smoothed curves. For each σ, the absolute curvature |κσ(s)| is integrated (or averaged) along the contour, yielding a scale‑dependent function F(σ). By plotting log F against log σ and measuring the slope, a local fractal dimension D(σ) is obtained for every scale level.

Because a single scalar D(σ) does not capture the full multiscale behavior of a shape, the authors apply a multiscale transform (MST) to the D(σ) curve. Specifically, they differentiate D(σ) with respect to σ (or compute finite differences) to obtain a derivative signal D′(σ) that reflects how the fractal dimension changes as the shape is progressively smoothed. Sampling D′(σ) at uniformly spaced σ values produces a fixed‑length vector