Basis Of Decomposition Research Articles

Increase of Reliability of Anomalies Detection on Images at Formation of Their Feature Vectors in Wavelet Bases

The method of detection of life rafts and lifeboats in the water area of seas and oceans after shipwrecks based on the recognition of anomalies in the processed images, which increases the probability of recognition of monitoring objects, is proposed. The approach to solving such a problem is substantiated. The formulation of the problem of object recognition from the perspective of binary classification in the detection of anomalies is presented. The analytical expression for the decision-making algorithm is obtained. The possibility of formalization of image matrices in the form of histograms of color (brightness) intensity distributions is considered. The contrast of the feature space on their basis is estimated. It is suggested that the contrast of feature spaces be increased due to the secondary processing of histograms of distributions in the basis of multiple-scale wavelet decomposition. The possibility of realization of wavelet transformations on the basis of Haar functions and Gauss wavelets of the 1st and 2nd orders is considered. The mechanism of formation of secondary feature vectors from three-dimensional wavelet transforms by averaging their coefficients along the time shift axis is substantiated. It is shown that at the same dimensionality of histograms of brightness distribution with newly formed feature vectors, the latter provide higher contrast of feature spaces. It is recommended to use a Gaussian wavelet of the 2nd order for the formalization of images in jpeg format, which provides, other things being equal, a greater magnitude of differences for images containing anomalies. An approach to probabilistic evaluation of the algorithm for automatic image recognition is developed. The analytical expression is obtained and its constituent elements are justified. Graphical dependences of the probability of correct detection (recognition) of anomalies, depending on the size in relation to the total area of the frame and the dispersion of the underlying background are given. The results of the experiment on image recognition with a lifeboat in the ocean water area are presented. The directions of further research are defined.

A reduced-dimension method of Crank-Nicolson finite element solution coefficient vectors for the unsteady Burgers equation

This paper primarily focuses on the dimensionality reduction of finite element (FE) solution coefficient vectors for the unsteady Burgers equation, solved using the Crank-Nicolson FE (CNFE) method. The proper orthogonal decomposition (POD) basis is constructed from the snapshot matrix, which is formed using the first L solutions, where L is significantly smaller than the total number of time steps N of the CNFE method. By reconstructing the matrix form of the CNFE method, a reduced-dimension Crank-Nicolson finite element (RDCNFE) method is proposed and stability analysis and error estimates are discussed. Numerical tests are implemented to verify the theoretical results and demonstrate the high efficiency of the RDCNFE method.

Metal artifact reduction method based on single spectral CT (MARSS).

For patients with metal implants, computed tomography (CT) imaging results suffer from metal artifacts, which seriously affect image evaluation and even lead to misdiagnosis. Because spectral CT technology has the advantage of quantitative imaging, basis material decomposition, and so on, the current metal artifact reduction methods are utilizing spectral information to reduce metal artifacts with good results. However, they usually require projection data from multiple spectra or energy-windows, which is difficult to realize in conventional CT. To satisfy the status quo, the aim of this work is to propose a metal artifact reduction (MAR) method based on single spectral CT (MARSS). By incorporating prior information, the average density of some base materials, and a constrained image reconstruction model is established. It forces the solution spaces of the materials to be discrete and finite, making the model easier tosolve. The MARSS method uses the idea of discrete tomography to establish a constrained reconstruction model. By incorporating priori knowledge, the constraint forces the solution spaces of some materials to be discrete, which can effectively downsize the solution space and reduce the ill-posedness of this problem. Then, an iteration algorithm is developed to solve this model. This algorithm iterates alternately between reconstruction and discretization. It ensures that the solution spaces are discrete while the polychromatic projection of the reconstructed image converges to that of the scannedobject. The MRASS method significantly reduces artifacts and restores structures near metal to a large extent. Unlike the comparison MAR methods, it effectively prevents the introduction of new artifacts and distortion of thestructure. The MARSS method can achieve MAR based on single spectral CT. Subjective and quantitative evaluation of the results show that the method significantly improves image quality compared to competingmethods.

A projection-based time-segmented reduced order model for fluid-structure interactions

In this paper, a type of novel projection-based, time-segmented reduced order model (ROM) is proposed for dynamic fluid-structure interaction (FSI) problems based upon the arbitrary Lagrangian–Eulerian (ALE)-finite element method (FEM) in a monolithic frame, where spatially, each variable is separated from others in terms of their attribution (fluid/structure), category (velocity/pressure) and component (two/three dimension) while temporally, the proper orthogonal decomposition (POD) bases are constructed in some deliberately partitioned time segments tailored through extensive numerical trials. By the combination of spatial and temporal decompositions, the developed ROM approach enables prolonged simulations under prescribed accuracy thresholds. Numerical experiments are carried out to compare numerical performances of the proposed ROM with corresponding full-order model (FOM) by solving a two-dimensional FSI benchmark problem that involves a vibrating elastic beam in the fluid, where the performance of offline ROM on perturbed physical parameters in the online phase is investigated as well. Extensive numerical results demonstrate that the proposed ROM has a comparable accuracy to while much higher efficiency than the FOM. The developed ROM approach is dimension-independent and can be seamlessly extended to solve high dimensional FSI problems.

Zonal shape reconstruction for Shack-Hartmann sensors and deflectometry

Some metrological means, such as Shack-Hartmann, deflectometry sensors or fringe projection profilometry, measure the shape of an optical surface indirectly from slope measurements. Zonal shape reconstruction, a method to reconstruct shape with a high number of degrees of freedom, is used for all of these applications. It has risen in interest with the use of deflectometers for the acquisition of high resolution slope data for optical manufacturing, especially because shape reconstruction is limiting in terms of shape estimation error.Zonal reconstruction methods all rely on the choice of a data formation model, a basis on which the shape will be decomposed, and an estimator. In this paper, we first study the canonical Fried and Southwell models of the literature and analyze their limitations. We show that modeling the slope measurement by a point-wise derivative as they both do can induce a bias on the shape estimation, and that the bases on which the shape is decomposed are imposed because of this assumption.In the second part of this paper, we propose to build an unbiased model of the data formation, without constraints on the choice of the decomposition basis. We then compare these models to the canonical models of Fried and Southwell.Lastly, we perform a regularized MAP reconstruction, and compare the performance in terms of total shape error of this method to the state of the art for the Southwell and Fried models, first by simulation, then on experimental data. We demonstrate that the suggested method outperforms the canonical models in terms of total shape reconstruction error on a deflectometry measurement of the high-frequency content of a freeform mirror.

Proper orthogonal decomposition reduced-order model of the global oceans

AbstractA reduced-order model (ROM) of the global oceans is developed by projecting the hydrostatic Boussinesq equations of motion onto a proper orthogonal decomposition (POD) basis. Three-dimensional POD modes are calculated from the ocean fields of an ensemble climate reanalysis dataset. The coefficients in the POD ROM are calculated using a regression approach. The performance of various POD ROM configurations are assessed. Each configuration is derived from an alternate sea-water equation of state, linking the density and temperature fields. POD ROM variants incorporating an equation of state in which density is a quadratic function of temperature, are able to reproduce the statistics of the large-scale structures at a fraction of the computational cost required to numerically simulate this flow. Due to the speed and efficiency of calculation, such reduced-order models of the global geophysical system will enable researchers and policy makers to assess the physical risk for a broader range of potential future climate scenarios.

Markov generators as non-Hermitian supersymmetric quantum Hamiltonians: spectral properties via bi-orthogonal basis and singular value decompositions

Continuity equations associated with continuous-time Markov processes can be considered as Euclidean Schrödinger equations, where the non-Hermitian quantum Hamiltonian H=divJ is naturally factorized into the product of the divergence operator div and the current operator J . For non-equilibrium Markov jump processes in a space of N configurations with M links and C=M−(N−1)⩾1 independent cycles, this factorization of the N × N Hamiltonian H=I†J involves the incidence matrix I and the current matrix J of size M × N, so that the supersymmetric partner H^=JI† governing the dynamics of the currents existing on the M links is of size M × M. To better understand the relations between the spectral decompositions of these two Hamiltonians H=I†J and H^=JI† with respect to their bi-orthogonal basis of right and left eigenvectors that characterize the relaxation dynamics towards the steady state and the steady currents, it is useful to analyze the properties of the singular value decomposition of the two rectangular matrices I and J of size M × N and the interpretations in terms of discrete Helmholtz decompositions. This general framework concerning Markov jump processes can be adapted to non-equilibrium diffusion processes governed by Fokker–Planck equations in dimension d, where the number N of configurations, the number M of links and the number C=M−(N−1) of independent cycles become infinite, while the two matrices I and J become first-order differential operators acting on scalar functions to produce vector fields.

Feasibility study of photon-counting CT for material identification based on YSO/SiPM detector: A proof of concept.

Current photon-counting computed tomography (CT) systems utilize semiconductor detectors, such as cadmium telluride (CdTe), cadmium zinc telluride (CZT), and silicon (Si), which convert x-ray photons directly into charge pulses. An alternative approach is indirect detection, which involves Yttrium Orthosilicate (YSO) scintillators coupled with silicon photomultipliers (SiPMs). This presents an attractive and cost-effective option due to its low cost, high detection efficiency, low dark count rate, and high sensor gain. This study aims to establish a comprehensive quantitative imaging framework for three-energy-bin proof-of-concept photon-counting CT based on YSO/SiPM detectors developed in our group using multi-voltage threshold (MVT) digitizers and assess the feasibility of this spectral CT for material identification. We developed a proof-of-concept YSO/SiPM-based benchtop spectral CT system and established a pipeline for three-energy-bin photon-counting CT projection-domain processing. The empirical A-table method was employed for basis material decomposition, and the quantitative imaging performance of the spectral CT system was assessed. This evaluation included the synthesis errors of virtual monoenergetic images, electron density images, effective atomic number images, and linear attenuation coefficient curves. The validity of employing A-table methods for material identification in three-energy-bin spectral CT was confirmed through both simulations and experimental studies. In both noise-free and noisy simulations, the thickness estimation experiments and quantitative imaging results demonstrated high accuracy. In the thickness estimation experiment using the practical spectral CT system, the mean absolute error for the estimated thickness of the decomposed Al basis material was 0.014±0.010mm, with a mean relative error of 0.66%±0.42%. Similarly, for the decomposed polymethyl methacrylate (PMMA) basis material, the mean absolute error in thickness estimation was 0.064±0.058mm, with a mean relative error of 0.70%±0.38%. Additionally, employing the equivalent thickness of the basis material allowed for accurate synthesis of 70keV virtual monoenergetic images (relative error 1.85%±1.26%), electron density (relative error 1.81%±0.97%), and effective atomic number (relative error 2.64%±1.26%) of the tested materials. In addition, the average synthesis error of the linear attenuation coefficient curves in the energy range from 40 to 150keV was 1.89%±1.07%. Both simulation and experimental results demonstrate the accurate generation of 70keV virtual monoenergetic images, electron density, and effective atomic number images using the A-table method. Quantitative imaging results indicate that the YSO/SiPM-based photon-counting detector is capable of accurately reconstructing virtual monoenergetic images, electron density images, effective atomic number images, and linear attenuation coefficient curves, thereby achieving precise material identification.

Human Serum Albumin Protein Corona in Prussian Blue Nanoparticles.

Prussian Blue nanoparticles (PBnps) are now popular in nanomedicine thanks to the FDA approval of PB. Despite the numerous papers suggesting or describing the in vivo use of PBnps, no studies have been carried out on the formation of a protein corona on the PBnp surface and its stabilizing role. In this paper, we studied qualitatively and quantitatively the corona formed by the most abundant protein of blood, human serum albumin (HSA). Cubic PBnps (41 nm side), prepared in citric acid solution at PB concentration 5 × 10-4 M, readily form a protein corona by redissolving ultracentrifuged PBnp pellets in HSA solutions, with CHSA ranging from 0.025 to 7.0 mg/mL. The basic decomposition of PBnp@HSA was studied in phosphate buffer at the physiological pH value of 7.4. Increased stability with respect to uncoated PBnps was observed at all concentrations, but a minimum CHSA value of 3.0 mg/mL was determined to obtain stability identical to that observed at serum-like HSA concentrations (35-50 mg/mL). Using a modified Lowry protocol, the quantity of firmly bound HSA in the protein corona (hard corona) was determined for all the CHSA used in the PBnp@HSA synthesis, finding increasing quantities with increasing CHSA. In particular, an HSA/PBnp number in the 1500-2300 range was found for CHSA 3.0-7.0 mg/mL, largely exceeding the 180 HSA/PBnp value calculated for an HSA monolayer on a PBnp. Finally, the stabilization brought by the HSA corona allowed us to carry out pH-spectrophotometric titrations on PBnp@HSA in the 3.5-9-0 pH range, revealing a pKa value of 6.68 for the water molecules bound to the Fe3+ centers on the PBnp surface, whose deprotonation is responsible for the blue-shift of the PBnp band from 706 nm (acidic solution) to 685 nm (basic solution).

Probabilistic maximization of time-dependent capacities in a gas network

AbstractThe determination of free technical capacities belongs to the core tasks of a gas network owner. Since gas loads are uncertain by nature, it makes sense to understand this as a probabilistic problem provided that stochastic modeling of available historical data is possible. Future clients, however, do not have a history or they do not behave in a random way, as is the case, for instance, in gas reservoir management. Therefore, capacity maximization becomes an optimization problem with uncertainty-related constraints which are partially of probabilistic and partially of robust (worst case) type. While previous attempts to solve this problem were devoted to models with static (time-independent) gas flow, we aim at considering here transient gas flow subordinate to the isothermal Euler equations. The basic challenge addressed in the manuscript is two-fold: first, a proper way of formulating probabilistic constraints in terms of the differential equations has to be provided. This will be realized on the basis of the so-called spherical-radial decomposition of Gaussian random vectors. Second, a suitable characterization of the worst-case load behaviour of future customers has to be found. It will be shown, that this is possible for quasi-static flow and can be transferred to the transient case. The complexity of the problem forces us to constrain ourselves in this first analysis to simple pipes or to a V-like structure of the network. Numerical solutions are presented and show that the differences between quasi-static and transient solutions are small, at least in these elementary examples.

Misinterpretations about CT numbers, material decomposition, and elemental quantification.

Quantitative CT imaging, particularly iodine and calcium quantification, is an important CT-based biomarker. This study quantifies sources of errors in quantitative CT imaging in both single-energy and spectral CT. This work examines the theoretical relationship between CT numbers, linear attenuation coefficient, and material quantification. We derive four understandings: (1) CT numbers are not proportional with element mass in vivo, (2) CT numbers are proportional with element mass only when contained in a voxel of pure water, (3) iodine-water material decomposition is never accurate in vivo, and (4) for error-free material decomposition a voxel must only consist of the basis decomposition vectors. Misinterpretation-based errors are calculated using the National Institute of Standards and Technology (NIST) XCOM database for: tissue chemical compositions, clinical concentrations of hydroxyapatite (HAP), and iodine. Quantification errors are also demonstrated experimentally using phantoms. In single-energy CT, misinterpretation-induced errors for HAP density in adipose, muscle, lung, soft tissue, and blood ranged from 0-132%, i.e., a mass error of 0-749 mg/cm3. In spectral CT, errors with iodine in the same tissues resulted in a range of < 0.1-33% error, resulting in a mass error of < 0.1-1.2 mg/mL. Our work demonstrates material quantification is fundamentally limited when measured in vivo due to measurement conditions differing from assumed and the errors are at or above detection limits for bone mineral density (BMD) and spectral iodine quantification. To define CT-derived biomarkers, the errors we demonstrate should either be avoided or built into uncertainty bounds. Improving error bounds in quantitative CT biomarkers, specifically in iodine and BMD quantification, could lead to improvements in clinical care aspects based on quantitative CT. CT numbers are only proportional with element mass only when contained in a voxel of pure water, therefore iodine-water material decomposition is never accurate in vivo. Misinterpretation-induced errors ranged from 0-132% for HAP density and < 0.1-33% in spectral CT with iodine. For error-free material decomposition, a voxel must only consist of the basis decomposition vectors.

Research on the identification method based on energy distribution of adaptive WPD for railway tunnel lining void

ABSTRACT As the main disease of railway tunnels, void disease affects the safe operation of railways. The traditional detection technology recognition void relies on the intuitive characteristics of the signal, there are still problems such as poor void edge recognition capability and low intelligence level. To solve this problem, this study used Comsol software to establish a sound-solid coupling finite element model of the tunnel, extracted the acoustic signals of voids under various conditions, analysed the acoustic signal characteristics under different conditions and proposed the acoustic sensitive frequency band of tunnel lining void. The test model of Partial tunnel lining is established by the method of layering pouring. The wavelet packet algorithm is improved by the maximum cross correlation coefficient, the optimal wavelet basis decomposition algorithm is proposed. The layer number of the wavelet packet decomposition (WPD) algorithm was calculated based on the sensitive frequency band and the energy distribution of the sub-signal is obtained. Then, the energy distribution of the sensitive frequency band is used to estimate the void boundary. The experimental results show that the acoustic frequency band of the tunnel void mainly distributes in the range of 0–10,000 Hz and the sensitive frequency band of the void concentrated around 0–3000 Hz. According to the sampling frequency and the sensitive frequency band of the void, the sub-signal energy distribution of each layer is obtained. The energy distribution of sub-signal 1 has strong void boundary recognition ability. The research results will be beneficial to the damage detection and state assessment of tunnel lining.

Shape prediction in continuous roll forming of sheet metal based on rigid rolls

The continuous roll forming based on rigid rolls (CRFRR) is an innovative process for rapid fabrication of doubly curved surfaces of sheet metal. The curved parts with different shapes and sizes can be obtained by adjusting the generatrix radius of the two rigid rolls or controlling the rolling reduction. In this paper, the principle of CRFRR is introduced and the process that a 3D surface of sheet metal with positive or negative Gaussian curvature is formed using two rigid rolls is analyzed based on the distribution of exit velocity. The exit velocity of material after the sheet metal passes through the roll gap is decomposed into two portions: a linearly distributed velocity and a very small additional velocity. On the basis of exit velocity decomposition and shallow shell theory, an analytical method for predicting the shape of formed surface in CRFRR process is proposed for the first time. The primary part of curvature of the formed 3D surface in longitudinal direction is determined by the linear portion of exit velocity and that in transverse direction is determined by the generatrix radius of the two rigid rolls. The curvature change in the final shape of 3D surface is induced by the additional portion of exit velocity and it is solved from equilibrium equation of shallow shell. The final curvature of the formed surface is acquired through superimposing the primary curvature and the curvature change. Numerical simulations and forming experiments have been carried out on convex and saddle-shaped parts, and the results confirm the validity and accuracy of presented method.

Uncertainty qualification in seismic analysis of concrete dams based on model order reduction accelerated stochastic SBFEM

A pioneering framework for uncertainty quantification in seismic analysis of concrete dams rooted in the stochastic scaled boundary finite element method (SSBFEM) is proposed. For seismic analysis of dams, octree decomposition is used for mesh generation. SSBFEM is developed to predict the structural responses with randomly distributed material properties. The script in this paper can integrate octree mesh and SSBFEM into ABAQUS user element (UEL). Implementing octree SSBFEM on commercial software provides accurate and convenient full order models for uncertainty quantification. Monte Carlo simulation (MCs) is deployed to calculate the statistical characteristics of structural responses under random variables. Additionally, singular value decomposition (SVD) and radial basis function (RBF) are leveraged to refine traditional MCs. The solution space of MCs is decomposed into lower-order subspaces. Any structural responses can be rapidly evaluated from linear combinations of subspaces. Particularly, the method has been successfully applied to seismic analysis of concrete dams with different random material properties. Finally, several illustrative examples are provided to validate the accuracy and efficacy of the algorithm.

A reduced-order model for nonlinear radiative transfer problems based on moment equations and POD-Petrov-Galerkin projection of the normalized Boltzmann transport equation

A data-driven projection-based reduced-order model (ROM) for nonlinear thermal radiative transfer (TRT) problems is presented. The TRT ROM is formulated by (i) a hierarchy of low-order quasidiffusion (aka variable Eddington factor) equations for moments of the radiation intensity and (ii) the normalized Boltzmann transport equation (BTE). The multilevel system of moment equations is derived by projection of the BTE onto a sequence of subspaces which represent elements of the phase space of the problem. Exact closure for the moment equations is provided by the Eddington tensor. A Petrov-Galerkin (PG) projection of the normalized BTE is formulated using a proper orthogonal decomposition (POD) basis representing the normalized radiation intensity over the whole phase space and time. The Eddington tensor linearly depends on the solution of the normalized BTE. By linear superposition of the POD basis functions, a low-rank expansion of the Eddington tensor is constructed with coefficients defined by the PG projected normalized BTE. The material energy balance (MEB) equation is coupled with the effective gray low-order equations which exist on the same dimensional scale as the MEB equation. The resulting TRT ROM is structure and asymptotic preserving. A detailed analysis of the ROM is performed on the classical Fleck-Cummings (F-C) TRT multigroup test problem in 2D geometry. Numerical results are presented to demonstrate the ROM's effectiveness in the simulation of radiation wave phenomena. The ROM is shown to produce solutions with sufficiently high accuracy while using low-rank approximation of the normalized BTE solution. Essential physical characteristics of supersonic radiation wave are preserved in the ROM solutions.

Turbulence induced by a swarm of rising bubbles from coarse-grained simulations

We performed numerical simulations of a homogeneous swarm of bubbles rising at large Reynolds number, $Re=760$ , with volume fractions ranging from 1 % to 10 %. We consider a simplified model in which the interfaces are not resolved, but which allows us to simulate flows with a large number of bubbles and to emphasize the interactions between bubble wakes. The liquid phase is described by solving, on an Eulerian grid, the Navier–Stokes equations, including sources of momentum which model the effect of the bubbles. The dynamics of each bubble is determined within the Lagrangian framework by solving an equation of motion involving the hydrodynamic forces exerted by the fluid accounting for the correction of the fictitious self-interaction of a bubble with its own wake. The comparison with experiments shows that this coarse-grained simulations approach can reliably describe the dynamics of the resolved flow scales. We use conditional averaging to characterize the mean bubble wakes and obtain in particular the typical shear imposed by the rising bubbles. On the basis of the spectral decomposition of the energy budget, we observe that the flow is dominated by production at large scales and by dissipation at small scales and we rule out the presence of an intermediate range in which the production and dissipation are locally in balance. We propose that the $k^{-3}$ subrange of the energy spectra results from the mean shear rate imposed by the bubbles, which controls the rate of return to isotropy.

Investigations on Superbase Mediated Reactivity of N‐Tosylhydrazones with Aza‐ortho‐Quinone Methide Precursors

We have encountered a superbase‐mediated chemoselective reaction of N‐tosylhydrazones with aza‐ortho‐quinone methide precursors. When tosylhydrazone was treated with ortho‐aminobenzyl alcohol in super basic conditions (KOH+DMSO), we observed the formation of 2‐substituted quinoline. The reaction was found to be general, and by this method, mono‐, di‐ and tri‐substituted quinolines could be made. We could prove experimentally and theoretically that the reaction proceeded via the formation of an azine from the basic decomposition of N‐tosylhydrazones. Finally, the reaction of tosylhydrazone with N‐(2‐(chloromethyl) phenyl)‐4‐methylbenzenesulfonamide (aza‐ortho‐quinone methide precursor) under super basic conditions afforded hydrazine substituted sulfonamides.

Nice Error Basis and Quantum Channel

The Weyl operators give a convenient basis of [Formula: see text] which is also orthonormal with respect to the Hilbert-Schmidt inner product. The properties of such a basis can be generalised to the notion of a nice error basis (NEB), as introduced by E. Knill [3]. We can use an NEB of [Formula: see text] to construct an NEB for [Formula: see text], the space of linear maps on [Formula: see text]. Any linear map will then correspond to a [Formula: see text] coefficient matrix in the basis decomposition with respect to such an NEB of [Formula: see text]. Positivity, complete (co)positivity or other properties of a linear map can be characterised in terms of such a coefficient matrix.

Spectral information content of Compton scattering events in silicon photon counting detectors.

Silicon (Si) is a possible sensor material for photon counting detectors (PCDs). A major drawback of Si is that roughly two-thirds of x-ray interactions in the diagnostic energy range are Compton scattering. Because Compton scattering is an energy-insensitive process, it is commonly assumed that Compton events retain little spectral information. To quantify how much information can be recovered from Compton scattering events in models of Si PCDs. We built a simplified model of Si interactions including two interaction mechanisms: photoelectric effect and Compton scattering. We considered three different binning options that represent strategies for handling Compton events: in Compton censoring, all events under 38keV (the maximum energy possible from Compton scattering for a 120keV incident photon) were discarded; in Compton counting, all events between 1 and 38keV were placed into a single bin; in Compton binning, all events were placed into energy bins of uniform width. These were compared to the ideal detector, which always recorded the correct energy (i.e., 100% photoelectric effect). Every photon was assumed to interact once and only once with Si, and the energy bin width was 5keV. In the primary analysis, the Si detector was irradiated with a 120kV spectrum filtered by 30cm of water, with 99.5% of the arriving spectrum above 38keV so that there was good separation between photoelectric effect and Compton scattering, and the figures of merit were the Cramér-Rao lower bound (CRLB) of the variance of iodine and water basis material decomposition images, as well as the CRLB of virtual monoenergetic images (i.e., linear combinations of material images) that maximize iodine CNR or water CNR. We also constructed a local linear estimator that attains the CRLB. In secondary analyses, we applied other sources of spectral distortion: (1) a nonzero minimum energy threshold; (2) coarser, 10keV energy bins; and (3) a model of charge sharing. With our chosen spectrum, 67% of the interactions were Compton scattering. Consistent with this, the material decomposition variance for the Compton censoring model, averaged over both basis materials, was 258% greater than the ideal detector. If Compton events carried no spectral information, the Compton counting model would show similar variance. Instead, its basis material variance was 103% greater than the ideal detector, implying that Compton counts indeed carry significant spectral information. The Compton binning model had a basis material variance 60% greater than the ideal detector. The Compton binning model was not affected by a 5keV minimum energy threshold, but the variance increased from 60% to 107% when charge sharing was included and to 78% with coarser energy bins. For optimized CNR images, the average variance was 149%, 12%, and 10% higher than the ideal detector for the Compton censoring, counting, and binning models, reinforcing the hypothesis that Compton counts are useful for detection tasks and that precise energy assignments are not necessary. Substantial spectral information remains after Compton scattering events in silicon PCDs.

Tracking Performance of Feedback Systems Over a Fading Channel With Limited Bandwidth.

In this study, we investigated the optimal tracking performance (OTP) of feedback control systems with limited bandwidth and colored noise in a fading channel. For the steady state of the feedback control systems, an equivalent average channel (EAC) model was developed by retaining the effects of the first and second moments of the multiplicative channel output, and on the basis of the coprime decomposition, all-pass factorization, and Youla parameterization of controllers, exact expressions for the OTP were derived by designing two compensators. The expressions quantitatively show the relationship between the OTP and inherent features of the plant. Specifically, the directions and locations of unstable poles (UPs) and nonminimum phase (NMP) zeros adversely affect the tracking performance. Furthermore, the bandwidth limitation and the presence of colored noise also degrade the tracking performance. Finally, our conclusions were verified by considering a numerical arithmetic example.