A Comparison of Property Estimators in Stereology and Digital Geometry

We consider selected geometric properties of 2D or 3D sets, given in form of binary digital pictures, and discuss their estimation. The properties examined are perimeter and area in 2D, and surface area and volume in 3D. We evaluate common estimators in stereology and digital geometry according to their multiprobe or multigrid convergence properties, and precision and efficiency of estimations. Keyword: property estimation, stereology, digital geometry. 1 The Estimation Problem A digital picture is a set of 2D or 3D digital data resultant from a process of digitization. Unlike sets in a continuous space, 2D or 3D data in digital pictures are presented in discrete form by a finite set of independent pixels or voxels. Perimeter and area in 2D, and surface area and volume in 3D are basic geometric properties which often need to be calculated. It is in general impossible to measure exact values (defined by sets in continuous space) of these properties if only a digital 2D or 3D picture is available. The precision of property estimators is important, and decisions need to be made about the type of estimator to be applied. [10] reviews the history and methods in the field of picture-based property estimators. However, it does not discuss in detail how stereology methods relate to methods popular in digital geometry. Gauss (1777-1855) studied the estimation of area by counting grid points, and this method is actually applied in stereology as well as in digital geometry. Thompson (1930) and Glagolev (1993) are cited in [14] for the origins of the point count method in quantitative microscopic analysis which is a predecessor of stereology. Both stereology as well as digital geometry are mainly oriented towards property estimations. Stereologists estimate geometric properties based on stochastic geometry and probability theory [13]. Key intentions are to ensure isotropic, uniform and random (IUR) object-probe interactions to ensure the unbias of estimations. The statistical behavior of property estimators is also a subject in digital geometry. But it seems that issues of algorithmic efficiency and multigrid convergence became more dominant in digital geometry. Both disciplines attempt to solve the same problem, and sometimes they follow the same principles, and in other cases they apply totally different methods. In this paper, a few property estimators of stereology and digital geometry are comparatively evaluated, especially according to their multiprobe or multigrid convergence behavior, precision and efficiency of estimations. 2 Stereology and Digital Geometry Digital geometry is the study of geometric properties of subsets of digital pictures. It includes ways of digitizing objects and also the estimation of their geometric properties based on the results of digitization (discrete data instead of continuous Euclidean data). Stereology is a way of estimating geometric properties of objects in a multidimensional space by observing its lower dimensional structures [15]. It is used broadly in some fields such as material science, biology and biomedicine for examining the microstructure of objects such as materials [14], biological tissues [13], and human organs. There are a number of commercial groups targeting computerized stereology to solve real-world problems, such as MicroBrightField Inc., Olympus Denmark (recently merged with Visiopharm), Kinetic Imaging Ltd., SPA Inc., R & M Biometrics Inc. These companies have made computer-based stereology systems (Stereo Investigator, CAST, Digital Stereology, Stereologer, Stereology Toolkit) either as a complete package, including advanced hardware (like a light microscope), accompanying software, or software toolkits for supporting stereological analysis. Since these systems are also computer-based, digital techniques must be used. The data which they process are digitized and discontinuous. The digitization model used in stereology may vary, but it is always within the scope of digital geometry. To the best of the author’s knowledge, no wide-scale comparison has ever been made so far in public for comparing accuracy or efficiency of stereology or digital geometry methods. From the theoretical point of view, one opportunity for comparison is to study how testing probe and resolution affect the accuracy of estimations. We define multiprobe convergence in stereology analogously to multigrid convergence in digital geometry. Both definitions can be generalized to cover not only the estimation of a single property such as length, but also of arbitrary geometric properties, including area in 2D, and surface area and volume in 3D.