Abstract
In this chapter, we explore a new model to calculate the fractal dimension of a subset with respect to a fractal structure. The new definition we provide presents better analytical properties than box dimension and can be calculated with easiness. It is worth mentioning that such a fractal dimension will be formulated as a discretization of Hausdorff dimension. Interestingly, we shall prove that it equals box dimension for Euclidean subsets endowed with their natural fractal structures. Therefore, it becomes a middle definition of fractal dimension which inherits some of the advantages of classical Hausdorff dimension and can be also calculated in empirical applications.