The invariator Design: An update

The invariator is a method to generate a test line within an isotropically oriented plane through a fixed point, in such a way that the test line is effectively motion invariant in three dimensional space. Generalizations exist for non Euclidean spaces. The invariator design is convenient to estimate surface area and volume simultaneously. In recent years a number of new results have appeared which call for an updated survey. We include two new estimators, namely the a posteriori weighting estimator for surface area and volume, and the peak-and-valley formula for surface area

Stereology with cylinder probes

Intersection formulae of Croton type for general geometric probes are well known in integral geometry. For the special case of cylinders with non necessarily convex direktrix, however, no equivalent formula seems to exist in the literature. We derive such formula resorting to motion invariant probability elements associated with test systems, instead of using a traditional approach. Because cylinders are seldom used as probes in stereological practice, however, this note is mainly of a theoretical nature

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

A surface area estimator for three-dimensional convex sets, based on the invariator principle of local stereology, has recently motivated its generalization by means of new rotational Crofton-type formulae using Morse theory. We follow a different route to obtain related formulae which are more manageable and valid for submanifolds in constant curvature spaces. As an application, we obtain a simplified version of the mentioned surface area estimator for non-convex sets of smooth boundary.Work was supported by the UJI project P11B2012-24 and the PROMETEOII/2014/062 project

Efficient and Unbiased Estimation of Population Size

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

Occurrence and ecological risk assessment of organic UV filters in coastal waters of the Iberian Peninsula

This study aimed to assess the presence of 21 UVFs and metabolites in coastal regions of the Iberian Peninsula, to evaluate their environmental risk, and identify possible influential factors affecting their measured concentrations. Sampling was carried out in spring and summer to assess possible seasonal variations. UVFs were detected in 43 of the 46 sampling sites. Only 5 were found above LOD: BP4, OC, BP3 and metabolites BP1 and BP8. Samples collected in Mar Menor had the greatest variety of compounds per sample and the highest cumulative concentrations. The risk was characterized using Risk Quotients (RQ). BP1 showed a Low environmental Risk in 2 sites while for OC the RQ indicated a Moderate Risk in 22 points. The variables that contribute most to the variation are population density, sampling season, whether it was an open bay or not, and level of urbanization. The presence of WWTPs had a lower influence.We thank the Rey Juan Carlos University Internal Program for the Promotion and Development of Research and Innovation – Bridge Grants 2020; the Spanish Ministry of Education and Professional Development, and the “Salvador Madariaga” Program in Support of Research Stays for Professor and Senior Researchers in Foreign Research Centres. A. Rico acknowledges the funding provided by the Talented Researcher Support Programme – PlanGenT(CIDEGENT/2020/043) of the Generalitat Valenciana. G. Quintana acknowledges the Spanish Ministry of Science and Innovation (Project CEX2018-000794-S) for his fellowship.Peer reviewe

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

summary:A surface area estimator for three-dimensional convex sets, based on the invariator principle of local stereology, has recently motivated its generalization by means of new rotational Crofton-type formulae using Morse theory. We follow a different route to obtain related formulae which are more manageable and valid for submanifolds in constant curvature spaces. As an application, we obtain a simplified version of the mentioned surface area estimator for non-convex sets of smooth boundary

THE PIVOTAL TESSELLATION

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