Fractal dimension is an important parameter to characterize size distribution and shape features of particles. However, it still remains unclear whether the fractal dimensions measured with different methods for the same particles will have the identical results, or whether the similar quantitative relationship exists between D-3 and D-2 or D-3 and D-1 as stated in the product-sum theorem. Silicon dioxide and potassium feldspar were selected for PDS measurements and 2DSPI extractions in this paper. By introducing the concept of fractal size, D-3, D-2 and D-1 are defined in a unified way, and their relationships are obtained under the defined assumptions including the product-sum theorem. The two fractional mathematical models and the three new algorithms including the power spectrum method presented in this paper provide the solutions to the problems encountered in the measurement of fractal dimensions. The Gauss-Newton iterative method of R-R distribution is superior to the existing fitted method. The four relationships of the fractal dimensions have been verified both experimentally and by the computational models. The results show that the relationships among D-3, D-2 and D-1 are rational and valid. (C) 2015 Elsevier B.V. All rights reserved.