Evaluation of a Method Combining Physical Experiment and Data Insertion For the Assessment of PET-CT Systems

Background: To evaluate and compare Positron Emission Tomography (PET) devices among them, tests are performed on phantoms that generally consist in simple geometrical objects, fillable with radiotracers. On one hand, those tests bring a control over the experiment through the operator preparation but on the other hand, they are limited in terms of reproducibility, repeatability and are time-consuming, in particular, if several replications are required. To overcome these restrictions, we designed a method combining physical experiment and data insertion that aims to avoid experimental repetitions while testing multiple configurations for the performance evaluation of PET scanners.Methods: Based on the National Electrical Manufacturers Association Image Quality standard, four experiments, with different spheres-to-background ratios: 2:1, 4:1, 6:1 and 8:1, were performed. An additional acquisition was done with a radioactive background and no activity within the spheres. It was created as a baseline to artificially simulate the radioactive spheres and reproduce initial experiments. Standard sphere set was replaced by smaller target sizes (4, 5, 6, 8, 10 and 13 mm) to match current detectability performance of PET scanners. Images were reconstructed following standard guidelines, i.e. using OSEM algorithm, and an additional BPL reconstruction was performed. We visually compared experimental and simulated images. We measured the activity concentration values into the spheres to calculate the mean and maximum recovery coefficient (RCmean and RCmax ) which we used in a quantitative analysis.Results: No significant visual discrepancies were identified between experimental and simulated series. Mann-Whitney U tests comparing simulated and experimental distributions showed no statistical differences for both RCmean (P value = 0.611) and RCmax (P value = 0.720). Spearman tests revealed high correlation for RCmean (ρ = 0.974, P value < 0.001) and RCmax (ρ = 0.974, P value < 0.001) between both datasets. According to Bland-Altman plots, we highlighted slight shifts in RCmean and RCmax of respectively 2.1 ± 16.9 % and 3.3 ± 22.3 %.Conclusions: The method produced realistic results compared to experimental data. Known synthesized information fused with original data allows full exploration of the system's capabilities while avoiding the limitations associated with repeated experiments.

Characterization of Lagrangian Submanifolds by Geometric Inequalities in Complex Space Forms

In this paper, we give an estimate of the first eigenvalue of the Laplace operator on a Lagrangian submanifold M n minimally immersed in a complex space form. We provide sufficient conditions for a Lagrangian minimal submanifold in a complex space form with Ricci curvature bound to be isometric to a standard sphere S n . We also obtain Simons-type inequality for same ambient space form.

The spinor and tensor fields with higher spin on spaces of constant curvature

In this article, we give all the Weitzenböck-type formulas among the geometric first-order differential operators on the spinor fields with spin $$j+1/2$$ j + 1 / 2 over Riemannian spin manifolds of constant curvature. Then, we find an explicit factorization formula of the Laplace operator raised to the power $$j+1$$ j + 1 and understand how the spinor fields with spin $$j+1/2$$ j + 1 / 2 are related to the spinors with lower spin. As an application, we calculate the spectra of the operators on the standard sphere and clarify the relation among the spinors from the viewpoint of representation theory. Next we study the case of trace-free symmetric tensor fields with an application to Killing tensor fields. Lastly we discuss the spinor fields coupled with differential forms and give a kind of Hodge–de Rham decomposition on spaces of constant curvature.

Underwater High-Precision 3D Reconstruction System Based on Rotating Scanning

This paper presents an underwater high-precision line laser three-dimensional (3D) scanning (LLS) system with rotary scanning mode, which is composed of a low illumination underwater camera and a green line laser projector. The underwater 3D data acquisition can be realized in the range of field of view of 50° (vertical) × 360° (horizontal). We compensate the refraction of the 3D reconstruction system to reduce the angle error caused by the refraction of light on different media surfaces and reduce the impact of refraction on the image quality. In order to verify the reconstruction effect of the 3D reconstruction system and the effectiveness of the refraction compensation algorithm, we conducted error experiments on a standard sphere. The results show that the system’s underwater reconstruction error is less than 0.6 mm within the working distance of 140 mm~2500 mm, which meets the design requirements. It can provide reference for the development of low-cost underwater 3D laser scanning system.

Homology of warped product submanifolds in the unit sphere and its applications

In this work, several pinched conditions on the Laplacian and gradient of the warping function are found in consideration of warped product submanifolds structure that force to homology groups vanish with no stable currents. Also, it is proved that a warped product pointwise semi-slant submanifold [Formula: see text] that is compact and oriented in an odd-dimensional spheres [Formula: see text] and [Formula: see text], has no stable integral [Formula: see text]-currents and [Formula: see text]-currents, respectively, and their homology groups are null, provided squared norm of the gradient for warping function satisfies some extrinsic restrictions including the Laplacian of the warping function, pointwise slant functions in addition to dimension of fiber of warped product immersions. Moreover, under assumption of extrinsic condition on the warping function, it is show [Formula: see text] being homeomorphic to a standard sphere [Formula: see text] with [Formula: see text] and homotopic to a standard sphere [Formula: see text] with [Formula: see text]. Further, the same results are generalized for contact CR-warped product submanifolds of same ambient spaces.

The normalized Ricci flow and homology in Lagrangian submanifolds of generalized complex space forms

In this paper, we prove that a simply connected Lagrangian submanifold in the generalized complex space form is diffeomorphic to standard sphere [Formula: see text] and the normalized Ricci flow converges to a constant curvature metric, provided its squared norm of the second fundamental form satisfies some upper bound depending only on the squared norm of the mean curvature vector field, the constant sectional curvature, and the dimension of the Lagrangian immersion of the ambient space. Next, we conclude that stable currents do not exist and homology groups vanish in a compact real submanifold of the general complex space form, provided that the second fundamental form satisfies some extrinsic conditions. We show that our results improve some previous results.

On Differential Equations Characterizing Legendrian Submanifolds of Sasakian Space Forms

In this paper, we give an estimate of the first eigenvalue of the Laplace operator on minimally immersed Legendrian submanifold N n in Sasakian space forms N ˜ 2 n + 1 ( ϵ ) . We prove that a minimal Legendrian submanifolds in a Sasakian space form is isometric to a standard sphere S n if the Ricci curvature satisfies an extrinsic condition which includes a gradient of a function, the constant holomorphic sectional curvature of the ambient space and a dimension of N n . We also obtain a Simons-type inequality for the same ambient space forms N ˜ 2 n + 1 ( ϵ ) .

Pointwise Convergence of Solutions to the Schrödinger Equation on Manifolds

Let $(M^{n},g)$ be a Riemannian manifold without boundary. We study the amount of initial regularity required so that the solution to a free Schrödinger equation converges pointwise to its initial data. Assume the initial data is in $H^{\unicode[STIX]{x1D6FC}}(M)$. For hyperbolic space, the standard sphere, and the two-dimensional torus, we prove that $\unicode[STIX]{x1D6FC}>\frac{1}{2}$ is enough. For general compact manifolds, due to the lack of a local smoothing effect, it is hard to improve on the bound $\unicode[STIX]{x1D6FC}>1$ from interpolation. We managed to go below 1 for dimension ${\leqslant}$ 3. The more interesting thing is that, for a one-dimensional compact manifold, $\unicode[STIX]{x1D6FC}>\frac{1}{3}$ is sufficient.