Estimating the surface area of nonconvex particles from central planar sections

Abstract

SummaryIn this paper, we present a new surface area estimator in local stereology. This new estimator is called the 'Morse-type surface area estimator' and is obtained using a two-stage sampling procedure. First a plane section through a fixed reference point of a three-dimensional structure is taken. In this section plane, a modification of the area tangent count method is used. The Morse-type estimator generalizes Cruz-Orive's pivotal estimator for convex objects to nonconvex objects. The advantages of the Morse-type estimator over existing local surface area estimators are illustrated in a simulation study. The Morse-type estimator is well suited for computer-assisted confocal microscopy and we demonstrate its practicability in a biological application: the surface area estimation of the nuclei of giant-cell glioblastoma from microscopy images. We also present an interactive software that allows the user to efficiently obtain the estimator.

On semiautomatic estimation of surface area

“…Alternatively, the method of surface area estimation based on the invariator principle and using the point‐sampled test lines, given in Cruz‐Orive (), could be used. The upcoming paper Thórisdóttir & Kiderlen () will deal with this topic.…”

New rotational integrals in space forms, with an application to surface area estimation

“…From equation (4.18) we obtain a simplified version of the Morse-type surface area estimator presented in equation (7) from [16]. We simply associate signed indexes to the critical points of the height function, without resorting to the concept of Euler characteristic.…”

“…which constitutes the aforementioned modification of equation (7) from [16]. The surface area estimator given by equation (3.2) from [5] corresponds to the case in which Y is a convex set, whereby m = 2.…”

“…Further, in [16], the mentioned new generalization has been adapted to estimate the surface area in three-dimensional microscopy. In the present paper, we follow a different route to obtain related generalizations which are more manageable and valid for submanifolds in space forms of constant curvature.…”

“…In the present paper, we follow a different route to obtain related generalizations which are more manageable and valid for submanifolds in space forms of constant curvature. In particular, we arrive at a surface area representation which constitutes a simplification of the analogous formula given in [15], [16]. The subsequent estimation procedure is thereby simplified-the basic task consists in identifying the local maxima and minima of the height function of the pivotal section.…”