Abstract. Examples of Kähler metrics of constant scalar curvature are relatively scarce. Over the past two decades, several workers in geometry and physics have used symmetry reduction to construct complete Kähler metrics of constant scalar curvature by ODE methods. One fruitful idea-the "Calabi ansatz"-is to begin with an Hermitian line bundle p : (L, h) → (M, g M ) over a Kähler manifold, and to search for Kähler forms ω = p * ω M + dd c f (t) in some disk subbundle, where t is the logarithm of the norm function and f is a function of one variable.Our main technical result (Theorem A) is the calculation of the scalar curvature for an arbitrary Kähler metric g arising from the Calabi ansatz. This suggests geometric hypotheses (which we call "σ-constancy") to impose upon the base metric g M and Hermitian structure h in order that the scalar curvature of g be specified by solving an ODE. We show that σ-constancy is "necessary and sufficient for the Calabi ansatz to work" in the following sense. Under the assumption of σ-constancy, the disk bundle admits a one-parameter family of complete Kähler metrics of constant scalar curvature that restrict to g M on the zero section (Theorems B and D); an analogous result holds for the punctured disk bundle (Theorem C). A simple criterion determines when such a metric is Einstein. Conversely, in the absence of σ-constancy the Calabi ansatz yields at most one metric of constant scalar curvature, in either the disk bundle or the punctured disk bundle (Theorem E).Many of the metrics constructed here seem to be new, including a complete, negative Einstein-Kähler metric on the disk subbundle of a stable vector bundle over a Riemann surface of genus at least two, and a complete, scalar-flat Kähler metric on C 2 .
Small animal SPECT imaging systems have multiple potential applications in biomedical research. Whereas SPECT data are commonly interpreted qualitatively in a clinical setting, the ability to accurately quantify measurements will increase the utility of the SPECT data for laboratory measurements involving small animals. In this work, we assess the effect of photon attenuation, scatter and partial volume errors on the quantitative accuracy of small animal SPECT measurements, first with Monte Carlo simulation and then confirmed with experimental measurements. The simulations modeled the imaging geometry of a commercially available small animal SPECT system. We simulated the imaging of a radioactive source within a cylinder of water, and reconstructed the projection data using iterative reconstruction algorithms. The size of the source and the size of the surrounding cylinder were varied to evaluate the effects of photon attenuation and scatter on quantitative accuracy. We found that photon attenuation can reduce the measured concentration of radioactivity in a volume of interest in the center of a rat-sized cylinder of water by up to 50% when imaging with iodine-125, and up to 25% when imaging with technetium-99m. When imaging with iodine-125, the scatter-to-primary ratio can reach up to approximately 30%, and can cause overestimation of the radioactivity concentration when reconstructing data with attenuation correction. We varied the size of the source to evaluate partial volume errors, which we found to be a strong function of the size of the volume of interest and the spatial resolution. These errors can result in large (>50%) changes in the measured amount of radioactivity. The simulation results were compared with and found to agree with experimental measurements. The inclusion of attenuation correction in the reconstruction algorithm improved quantitative accuracy. We also found that an improvement of the spatial resolution through the use of resolution recovery techniques (i.e. modeling the finite collimator spatial resolution in iterative reconstruction algorithms) can significantly reduce the partial volume errors.
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