How many people need to classify the same image? A method for optimizing volunteer contributions in binary geographical classifications

Involving members of the public in image classification tasks that can be tricky to automate is increasingly recognized as a way to complete large amounts of these tasks and promote citizen involvement in science. While this labor is usually provided for free, it is still limited, making it important for researchers to use volunteer contributions as efficiently as possible. Using volunteer labor efficiently becomes complicated when individual tasks are assigned to multiple volunteers to increase confidence that the correct classification has been reached. In this paper, we develop a system to decide when enough information has been accumulated to confidently declare an image to be classified and remove it from circulation. We use a Bayesian approach to estimate the posterior distribution of the mean rating in a binary image classification task. Tasks are removed from circulation when user-defined certainty thresholds are reached. We demonstrate this process using a set of over 4.5 million unique classifications by 2783 volunteers of over 190,000 images assessed for the presence/absence of cropland. If the system outlined here had been implemented in the original data collection campaign, it would have eliminated the need for 59.4% of volunteer ratings. Had this effort been applied to new tasks, it would have allowed an estimated 2.46 times as many images to have been classified with the same amount of labor, demonstrating the power of this method to make more efficient use of limited volunteer contributions. To simplify implementation of this method by other investigators, we provide cutoff value combinations for one set of confidence levels

Quasi-exactly solvable quartic potential

A new two-parameter family of quasi-exactly solvable quartic polynomial potentials $V(x)=-x^4+2iax^3+(a^2-2b)x^2+2i(ab-J)x$ is introduced. Until now, it was believed that the lowest-degree one-dimensional quasi-exactly solvable polynomial potential is sextic. This belief is based on the assumption that the Hamiltonian must be Hermitian. However, it has recently been discovered that there are huge classes of non-Hermitian, ${\cal PT}$-symmetric Hamiltonians whose spectra are real, discrete, and bounded below [physics/9712001]. Replacing Hermiticity by the weaker condition of ${\cal PT}$ symmetry allows for new kinds of quasi-exactly solvable theories. The spectra of this family of quartic potentials discussed here are also real, discrete, and bounded below, and the quasi-exact portion of the spectra consists of the lowest $J$ eigenvalues. These eigenvalues are the roots of a $J$th-degree polynomial.Comment: 3 Pages, RevTex, 1 Figure, encapsulated postscrip

New Quasi-Exactly Solvable Sextic Polynomial Potentials

A Hamiltonian is said to be quasi-exactly solvable (QES) if some of the energy levels and the corresponding eigenfunctions can be calculated exactly and in closed form. An entirely new class of QES Hamiltonians having sextic polynomial potentials is constructed. These new Hamiltonians are different from the sextic QES Hamiltonians in the literature because their eigenfunctions obey PT-symmetric rather than Hermitian boundary conditions. These new Hamiltonians present a novel problem that is not encountered when the Hamiltonian is Hermitian: It is necessary to distinguish between the parametric region of unbroken PT symmetry, in which all of the eigenvalues are real, and the region of broken PT symmetry, in which some of the eigenvalues are complex. The precise location of the boundary between these two regions is determined numerically using extrapolation techniques and analytically using WKB analysis