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Professor Hans Jørgen G. Gundersen MD, DMSc (1943–2021) was a pioneering stereologist whose work has inspired and influenced researchers across the world for almost half a century. He was a charismatic character and one of the founding fathers of modern stereology, whose achievements and contributions are fondly remembered below by colleagues and co-workers. It was an enormous pleasure to be in his company and although future generation will miss this opportunity, his work will live on, to inspire and influence future generations of researchers.

The isotropic Cavalieri design is based on a isotropically oriented set of parallel systematic sections a constant distance apart. Its advantage over the ordinary Cavalieri design is twofold - first, besides volume it allows the unbiased estimation of surface area, and second, the error variance predictor for the volume estimator is much simpler, involving only the surface area of the object, and the distance between sections. In an earlier paper, the two hemispheres of a rat brain were arranged perpendicular to each other before sectioning, aiming at reducing the error variance with respect to other arrangements (such as the aligned one) by exploiting an intuitively plausible antithetic effect. Because the total surface area of the objects is unchanged under any arrangements, however, the error variance predictor for the volume estimator does not depend on object shape, which looks intriguing. Using reconstructions of the mentioned hemispheres, we dilucidate the aparent paradox by means of automatic Monte Carlo replications of the relevant volume estimates under the antithetic and the aligned arrangements.

The nucleator is a design unbiased method of local stereology for estimating the volume of a bounded object. The only information required lies in the intersection of the object with an isotropic random ray emanating from a fixed point (called the pivotal point) associated with the object. For instance, the volume of a neuron can be estimated from a random ray emanating from its nucleolus. The nucleator is extensively used in biosciences because it is efficient and easy to apply. The estimator variance can be reduced by increasing the number of rays. In an earlier paper a systematic sampling design was proposed, and theoretical variance predictors were derived, for the corresponding volume estimator. Being the only variance predictors hitherto available for the nucleator, our basic goal was to check their statistical performance by means of Monte Carlo resampling on computer reconstructions of real objects. As a plus, the empirical distribution of the volume estimator revealed statistical properties of practical relevance.

The pivotal estimators of the surface area and the volume of a generic "particle" are based on a point sampled test line on an isotropic pivotal plane through a fixed point called the pivotal point. The purpose of this paper is to compare the precision of the pivotal estimator of surface area against the surfactor on the one hand, and of the pivotal estimator of volume against the nucleator on the other. For the sake of a tractable but informative model we compute exact variances for a spherical particle with an eccentric nucleolus.

Cavalieri sampling is often used to estimate the volume of an object with systematic sections a constant distance T apart. The variance of the corresponding estimator can be expressed as the sum of the extension term (which gives the overall trend of the variance and is used to estimate it), the 'Zitterbewegung' (which oscillates about zero) and higher order terms. The extension term is of order T2m+2 for small T, where m is the order of the first non-continuous derivative of the measurement function f, (namely of the area function if the target is the volume). A key condition is that the jumps of the mth derivative f (m) of f are finite. When this is not the case, then the variance exhibits a fractional trend, and the current theory fails. Indeed, in practice the mentioned trend is often of order T2q+2, typically with 0

The cochlear nuclei (CN) in the brainstem receive the input signals from the inner ear through the cochlear nerve, and transmit these signals to higher auditory centres. A variety of lesions of the cochlear nerve cause deafness. As reported in the literature, artificial removal of auditive input, or 'deafferentation', induces structural alterations in the CN. The purpose of this study was to estimate a number of relevant stereological parameters of the CN in control and deafferented Macaca fascicularis monkeys.

The tessellation studied here is motivated by some stereological applications of a new expression for the motion invariant density of straight lines in R3. The term 'pivotal' stems from the fact that the tessellation is constructed within a plane which is isotropic through a fixed, 'pivotal' origin. Consider either a stationary point process, or a stationary random lattice of points in that plane. Through each point event draw a straight line which is perpendicular to the axis determined by the origin and the point event. The union of all such lines (called p-lines) constitutes the mentioned tessellation. We concentrate on the pivotal tessellation based on a stationary and isotropic planar Poisson point process; we show that this tessellation is not stationary.

Population sizing from still aerial pictures is of wide applicability in ecological and social sciences. The problem is long standing because current automatic detection and counting algorithms are known to fail in most cases, and exhaustive manual counting is tedious, slow, difficult to verify and unfeasible for large populations. An alternative is to multiply population density with some reference area but, unfortunately, sampling details, handling of edge effects, etc., are seldom described. For the first time we address the problem using principles of geometric sampling. These principles are old and solid, but largely unknown outside the areas of three dimensional microscopy and stereology. Here we adapt them to estimate the size of any population of individuals lying on an essentially planar area, e.g. people, animals, trees on a savanna, etc. The proposed design is unbiased irrespective of population size, pattern, perspective artifacts, etc. The implementation is very simple-it is based on the random superimposition of coarse quadrat grids. Also, an objective error assessment is often lacking. For the latter purpose the quadrat counts are often assumed to be independent. We demonstrate that this approach can perform very poorly, and we propose (and check via Monte Carlo resampling) a new theoretical error prediction formula. As far as efficiency, counting about 50 (100) individuals in 20 quadrats, can yield relative standard errors of about 8% (5%) in typical cases. This fact effectively breaks the barrier hitherto imposed by the current lack of automatic face detection algorithms, because semiautomatic sampling and manual counting becomes an attractive option.