Exploring the Influence of Initial Estimates on Iterative Maximum Likelihood Expectation-maximization in Tomographic Reconstruction.

Purpose: To explore the influence of initial guess or estimate (uniform as "ones" and "zeros" vs. filtered back projection [FBP] image) as an input image for maximum likelihood expectation-maximization (MLEM) tomographic reconstruction algorithm and provide the curves of error or convergence for each of these three initial estimates. Methods: Two phantoms, created as digital images, were utilized: one was a simple noiseless object and the other was a more complicated, noise-degraded object of the section of lower thorax in a matrix of 256 × 256 pixels. Both underwent radon transform or forward projection process and the corresponding sinograms were generated. For filtering during tomographic image reconstruction, ramp and Butterworth filters, as high-pass and low-pass ones, were applied to images. The second phantom (lower thorax) was radon-transformed and the resulting sinogram was degraded by noise. As initial guess or estimate images, in addition to FBP tomographic image, two uniform images, one with all pixels having a value of 1 ("ones") and the other with all having zero ("zeros"), were created. The three initial estimates (FBP, ones, and zeros) were reconstructed with iterative MLEM tomographic reconstruction (with 1, 2, 4, 8, 16, 32, and 64 iterations). The difference between the object and the updated slice was calculated at the end of each iteration (as error matrix), and the mean squared error (MSE) was computed and plotted separately or in conjunction with the MSE curves of other initial estimates. All computations were implemented in MATLAB software. Results: The results of ones and zeros seemed strikingly similar. The curves of uniform ones and uniform zeros were so close to each other that overlap near-perfectly. However, in the FBP slice as an initial estimate, the resulting tomographic slice was similar with a much higher extent to the object even after 1 or 2 iterations. The pattern of convergence for all three curves was roughly similar. The normalized MSE decreased sharply up to 5 iterations and then, after 10 iterations, the curves reached a plateau until 32 iterations. For the phantom of the lower thorax section with its noise-degraded sinogram, similar to the pattern observed for simple disk-shaped phantom, the curves (normalized MSE) fell sharply up to 10 iterations and then rapidly converged thereafter until 64 iterations. Conclusion: Similar results are observed when choosing different initial guesses or estimates (uniform vs. FBP) as the starting point, based on the error calculation using MSE. The algorithm converges almost similarly for all initial estimates. Therefore, selecting a uniform initial guess image can be an appropriate choice and may be preferred over an FBP image. Reducing the processing time can be a valid reason for this choice.

Numerical investigation on the convergence of self-consistent Schrödinger-Poisson equations in semiconductor device transport simulation.

Semiconductor devices at the nanoscale with low-dimensional materials as channels exhibit quantum transport characteristics, thereby their electrical simulation relies on the self-consistent solution of the Schrödinger-Poisson equations. While the non-equilibrium Green's function (NEGF) method is widely used for solving this quantum many-body problem, its high computational cost and convergence challenges with the Poisson equation significantly limit its applicability. In this study, we investigate the stability of the NEGF method coupled with various forms of the Poisson equation, encompassing linear, analytical nonlinear, and numerical nonlinear forms Our focus lies on simulating carbon nanotube field-effect transistors (CNTFETs) under two distinct doping scenarios: electrostatic doping and ion implantation doping. The numerical experiments reveal that nonlinear formulas outperform linear counterpart. The numerical one demonstrates superior stability, particularly evident under high bias and ion implantation doping conditions. Additionally, we investigate different approaches for presolving potential, leveraging solutions from the Laplace equation and a piecewise guessing method tailored to each doping mode. These methods effectively reduce the number of iterations required for convergence.

Motion Primitive Approach to Spacecraft Trajectory Design in a Multi-body System.

The increasing number and variety of spacecraft that are expected to operate within cislunar space and other multi-body gravitational environments throughout the solar system necessitates the continued development of strategies for rapid trajectory design and design space exploration. In the field of robotics, similar needs have been addressed using motion primitives that capture the fundamental building blocks of motion and are used to rapidly construct complex paths. Inspired by this concept, this paper leverages motion primitives to construct a framework for rapid and informed spacecraft trajectory design in a multi-body gravitational system. First, motion primitives of fundamental solutions, e.g., selected periodic orbits and their stable and unstable manifolds, are generated via clustering to form a discrete summary of segments of the phase space. Graphs of motion primitives are then constructed and searched to produce primitive sequences that form candidate initial guesses for transfers of distinct geometries. Continuous transfers are computed from each initial guess using multi-objective constrained optimization and collocation. This approach is demonstrated by constructing an array of geometrically distinct transfers between libration point orbits in the Earth-Moon circular restricted three-body problem with impulsive maneuvers.

Improved Self-Calibration of a Multilateration System Based on Absolute Distance Measurement.

Multilateration tracking systems (MLTSs) are used in industrial three-dimensional (3D) coordinate measuring applications. For high-precision measurement, system parameters must be calibrated properly in advance. For an MLTS using absolute distance measurement (ADM), the conventional self-calibration method significantly reduces estimation efficiency because all system parameters are estimated simultaneously using a complicated residual function. This paper presents a novel self-calibration method that optimizes ADM to reduce the number of system parameters via highly precise and separate estimations of dead paths. Therefore, the residual function to estimate the tracking station locations can be simplified. By applying a suitable mathematical procedure and solving the initial guess problem without the aid of an external device, estimation accuracy of the system parameters is significantly improved. In three self-calibration experiments, with ADM repeatability of approximately 3.4 µm, the maximum deviation of the system parameters estimated by the proposed self-calibration method was 68.6 µm, while the maximum deviation estimated by the conventional self-calibration method was 711.9 µm. Validation of 3D coordinate measurements in a 1000 mm × 1000 mm × 1000 mm volume showed good agreement between the proposed ADM-based MLTS and a commercial laser tracker, where the maximum difference based on the standard deviation was 17.7 µm. Conversely, the maximum difference was 98.8 µm using the conventional self-calibration method. These results confirmed the efficiency and feasibility of the proposed self-calibration method.

In Vivo Estimation of Head Tissue Conductivities Using Bound Constrained Optimization.

The conductivity of head tissues was noninvasively estimated using electrical impedance tomography technique. Instead of using conventional unconstrained optimization method to estimate the conductivities, a constrained method with the scaled-logistic function was employed to improve the very high sensitivity of the skull region resulting in accuracy and robustness improvement. Estimation of five conductivities i.e. scalp, skull, cerebrospinal fluid (CSF), grey matter (GM), and white matter (WM) conductivity was investigated by simulation on random and low-value initial guesses. Simulation results showed that the performance of the unconstrained method depended directly to the difference between the exact skull conductivity value and the initial guess value of the skull conductivity. However, the approached constrained method was independent of the guess selection. It can reduce the sensitivity of the skull region by 126 times and reduce the condition number of the sensitivity matrix by 13-17 times. The estimation resulted in only positive and in-range of reported conductivity values. The estimation error of the skull conductivity decreased by 15% and the robustness increased by 2 times. However, the estimation of the CSF, the WM, and the GM may be not reliable due to the very low sensitivity of these regions in both methods.

The ultrasound elastography inverse problem and the effective criteria.

The elastography (elasticity imaging) is one of the recent state-of-the-art methods for diagnosis of abnormalities in soft tissue. The idea is based on the computation of the tissue elasticity distribution. This leads to the inverse elasticity problem; in that, displacement field and boundary conditions are known, and elasticity distribution of the tissue is aimed for computation. We treat this problem by the Gauss-Newton method. This iterative method results in an ill-posed problem, and therefore, regularization schemes are required to deal with this issue. The impacts of the initial guess for tissue elasticity distribution, contrast ratio between elastic modulus of tumor and normal tissue, and noise level of the input data on the estimated solutions are investigated via two different regularization methods. The numerical results show that the accuracy and speed of convergence vary when different regularization methods are applied. Also, the semi-convergence behavior has been observed and discussed. At the end, we signify the necessity of a clever initial guess and intelligent stopping criteria for the iterations. The main purpose here is to highlight some technical factors that have an influence on elasticity image quality and diagnostic accuracy, and we have tried our best to make this article accessible for a broad audience.