On 2D generalization of Higuchi’s fractal dimension

• We propose a new numerical method for calculating 2D fractal dimension of a surface.
• This method is a generalization of Higuchi method for calculating fractal dimension.
• We tested exactness of our method by Weierstrass–Mandelbrot surfaces.
• The 2D fractal analysis method was applied to the collection of histological images.
• The proposed method efficiently differentiated phases of shoot organogenesis.

We propose a new numerical method for calculating 2D fractal dimension (DF) of a surface. This method represents a generalization of Higuchi’s method for calculating fractal dimension of a planar curve. Using a family of Weierstrass–Mandelbrot functions, we construct Weierstrass–Mandelbrot surfaces in order to test exactness of our new numerical method. The 2D fractal analysis method was applied to the set of histological images collected during direct shoot organogenesis from leaf explants. The efficiency of the proposed method in differentiating phases of organogenesis is proved.