*Forma,* Vol. 1 (No. 1), pp. 49-58, 1985

## On Mathematical Principles of Stereological Estimation

Ken-ichi Kanatani

Department of Computer Science, Gunma University

**Keywords: **
anisotropy, internal structure, stereology, sphere

**Abstract. **
Stereological procedures are classified according to the type of estimation, and their mutual relationships are discussed. Then, estimation of sphere size distribution from observation of material cross-sections is discussed from the viewpoint of numerical analysis, i.e., discrete approximation of the basic integral equation of Abel type. Finally, methods of estimating structural anisotropy due to internal distribution of line tissues and surfaces are considered. The anisotropy of structure is characterized by the distribution density, which is expressed in terms of what is called the "fabric tensors." The distribution density is related to observed data by what is called the "Buffon transform," and estimation is done by computing its inversion. Main emphasis is placed on the role played by mathematics.