Mathematical methods used in monofractal and multifractal analysis for the processing of biological and medical data and images

An object is self-similar if it can be decomposed into smaller copies of itself (the structure of the whole is contained in its parts) (Mandelbrot 1982; Falconer 2003). Not all irregular shapes found in nature are necessarily fractals (Losa 2009). Monofractal objects are mainly characterized by four properties: a) irregularity of the shape; b) self-similarity of the structure; c) non-integer or fractional dimension; d) complexity (Mandelbrot 1982; Reljin & Reljin 2002; Falconer 2003; Grizzi et al 2005). Because monofractals have the same scaling properties, characterized by a single singularity exponent, they are considered as homogeneous objects (Mandelbrot 1982). A monofractal is a set for which the Hausdorff-Besicovitch dimension (Dh) strictly exceeds its topological dimension (DT) (Mandelbrot 1982).