Iterative Least-squares Algorithm Research Articles

A stopping criterion for iterative proton CT image reconstruction based on correlated noise properties.

Whereas filtered back projection algorithms for voxel-based CT image reconstruction have noise properties defined by the filter, iterative algorithms must stop at some point in their convergence and do not necessarily produce consistent noise properties for images with different degrees of heterogeneity. A least-squares iterative algorithm for proton CT (pCT) image reconstruction converges toward a unique solution for relative stopping power (RSP) that optimally fits the protons. We present a stopping criterion that delivers solutions with the property that correlations of RSP noise between voxels are relatively low. This provides a method to produce pCT images with consistent noise properties useful for proton therapy treatment planning, which relies on summing RSP along lines of voxels. Consistent noise properties will also be useful for future studies of image quality using metrics such as contrast to noise ratio, and to compare RSP noise and dose of pCT with other modalities such as dual-energyCT. With simulated and real images with varying heterogeneity from a prototype clinical proton imaging system, we calculate average RSP correlations between voxel pairs in uniform regions-of-interest versus distance between voxels. We define a parameter r, the remaining distance to the unique solution relative to estimated RSP noise, and our stopping criterion is based on r falling below a chosenvalue. We find large correlations between voxels for larger values of r, and anticorrelations for smaller values. For r in the range of 0.5-1, voxels are relatively uncorrelated, and compared to smaller values of r have lower noise with only slight loss of spatialresolution. Iterative algorithms not using a specific metric or rationale for stopping iterations may produce images with an unknown and arbitrary level of convergence or smoothing. We resolve this issue by stopping iterations of a least-squares iterative algorithm when r reaches the range of 0.5-1. This defines a pCT image reconstruction method with consistent statistical properties optimal for clinical use, including for treatment planning with pCTimages.

A novel robot hand-eye calibration method to enhance calibration accuracy based on the POE model

Hand-eye calibration is crucial for vision-based robots using open-loop visual control. Based on the product of exponentials model, this study introduces a novel hand-eye calibration algorithm to improve calibration accuracy. As opposed to the traditional method of solving the AX = XB type in the hand-eye calibration using the marker, we transform the solution into an AX = B type without requiring accurate camera parameters. Firstly, the nominal marker pose in the robot base is described by the nominal kinematic parameters of the robot, camera, and end-effector-to-camera. The actual marker pose is obtained by physically contacting the marker with the probe. Subsequently, the error parameter of the hand-eye pose is defined as linear to the error between the nominal and actual marker poses and can be solved by an iterative least-squares algorithm. Finally, the calibrated hand-eye pose is acquired by multiplying its nominal pose and the exponential mapping of the error parameter. The simulations and real experiments illustrate the effectiveness and stability of the proposed algorithm. Noteworthy, the mean position accuracy of the vision-based robot is improved nearly 20 times after calibration. Furthermore, the comparison experiment demonstrates that the proposed algorithm is preferable for improving hand-eye calibration accuracy.

An Efficient Iterative Least Square Method for Indoor Visible Light Positioning Under Shot Noise

In this paper, we develop a set of effective algorithms for performing efficient and accurate visible light positioning (VLP) in the presence of shot noise, which is an important component in the received optical signal yet has been largely neglected in prior works. In particular, we formulate the positioning problem as a maximum log-likelihood optimization problem, which is nonconvex so that the standard numerical algorithm such as gradient descent (GD) and stochastic gradient descent (SGD) may not be able to find the global solution. To address this, we propose a novel least-square (LS) solver that can find a sub-optimal solution to the aforementioned non-convex optimization problem. Based on the LS solver, a set of more effective algorithms can be developed to further enhance the optimality of the solution. Specifically, we consider (1) combining the LS solver with GD, giving rise to the GD-LS algorithm; and (2) applying the LS solver in an iterative manner, giving rise to the iterative LS algorithm, which is a novel and efficient positioning algorithm. Moreover, we also provide a closed-form lower bound on the positioning error based on the Cramér-Rao lower bounds (CRLB). Numerical simulation shows that the proposed GD-LS and iterative LS algorithms cannot only achieve high positioning accuracy, but also enjoy low computation complexity: the average positioning accuracy of LS-GD is 0.009 m using computation time 0.046 s, and the iterative LS algorithm can achieve average positioning accuracy 0.023 m with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$1.94\times 10^{-4}$</tex-math></inline-formula> s computation time, which outperform GD and SGD method.

Erratum to Alternate Iterative Least-Squares Algorithm Based on Nonuniform Phase Shifting for Suppressing Nonlinearity Errors in Fringe Projection Profilometry

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Alternate Iterative Least-Squares Algorithm Based on Nonuniform Phase Shifting for Suppressing Nonlinearity Errors in Fringe Projection Profilometry

In fringe projection technique, nonlinearities of used devices are recognized as major factors inducing phase errors, especially when the used camera and projector simultaneously have nonlinear features. In this case, ripple artifacts appear to be unevenly distributed over the reconstructed surface depending on the local reflectivity. For suppressing these errors, this paper suggests a phase-measuring technique which is characterized by two critical steps including anti-aliasing phase-shifting and iterative least-squares phase estimation. Anti-aliasing phase-shifting requires the selected phase shifts to be nonuniform and to have no common divisors. It provides a possibility of accurately recovering phases from few to four fringe patterns and eliminating influence of high order harmonics on the measurement results. By expanding the fringe signals from projector into harmonics and modeling the camera response as a polynomial, we derive a least-squares algorithm based on alternate iterations to calculate the fringe phase map and the related nonlinearity coefficients simultaneously. Through practical experiments, it has been demonstrated that this technique allows us to decrease artifacts caused by the simultaneous nonlinearities of the camera and the projector, thus improving measurement accuracy significantly.

Phase unwrapping of digital holographic microscopy using adaptive region segmentation and phase derivative calibration with respect to fringe density

Accurate phase unwrapping of the wrapped phase maps with high noise and uneven fringe densities is an essential problem for phase restoration in digital holographic microscopy. In this paper, a phase unwrapping algorithm using adaptive region segmentation and phase derivative calibration with respect to fringe density is proposed. The dense and sparse fringe regions of the wrapped phase map are adaptively segmented by using the sine/cosine spatial filtering and calculating the second-order phase gradient. Then, the first-order spatial wrapped phase derivative maps are divided into four phase derivative maps, and the phase anomalies on them are calibrated respectively. Finally, the four maps were recombined into two new phase derivative maps, and the unweighted least-squares iterative algorithm is implemented to obtain the continuous phase map. The advantage of our algorithm is that it can detect, calibrate and restore the phase anomalies based on fringe density automatically. Simulation and experimental results demonstrated that the proposed algorithm has better accuracy and effectiveness in dealing with the wrapped phase maps with high fringe density and high noise, which indicates that our algorithm is able to process the wrapped phase maps corrupted by coherent noise and phase aberration in the digital holographic microscopy.

Scanning error detection and compensation algorithm for white-light interferometry

As the primary error source in white-light interferometry (WLI), the error of scanning steps directly affects coherence peak sensing and greatly lowers the measurement accuracy. Based on the least-squares iterative algorithm, we present an algorithm to detect and compensate for scanning error in WLI. The actual scanning step is calculated from the continuous moving fringes with an iterative algorithm, and the surface is reconstructed with the calculated scanning steps to make compensation. Both simulations and experiments indicate that the high-accuracy measurements can be implemented under disturbance over a wide frequency band. The proposed algorithm could relax the environment requirement of white-light interferometer application.

Parametric identification of output error model for sampled systems with integer‐type time delay subject to load disturbance with unknown dynamics

In this paper, a bias-eliminated output error (OE) model identification method is proposed for single-input-single-output sampled systems with integer-type time delay subject to load disturbance with unknown dynamics. By viewing the load disturbance response as a time-variant parameter, an iterative least-squares identification algorithm is established to estimate the rational model parameters together with an integer-type delay parameter, while the disturbance response could be simultaneously estimated. To overcome the adverse effect of stochastic noise involved with output measurement, an auxiliary model is constructed to predict the noise-free system response. Moreover, a set of adaptive forgetting factors is introduced to expedite the convergence rate of model parameter estimation and the tracking performance of load disturbance response, respectively. In addition, a monotonically rising profile for evaluating the delay parameter is proposed for implementing the above iterative least-squares algorithm, in order to avoid the occurrence of multiple local minima of the loss function for model fitting. The asymptotic convergence on estimating the rational model parameters together with the delay parameter is analysed with a proof. An illustrative example is given to demonstrate the effectiveness and advantage of the proposed method.

Generating-function approach for double freeform lens design.

Many LED lighting applications involve the design of multiple optical surfaces. A prime example is a single lens with two refractive surfaces. In this paper, we consider an LED light source approximated as a point and a far-field target intensity. Using Hamilton's characteristic functions, the design problem is converted into two generalized Monge-Ampère equations by deriving a generating function for each optical surface. The generating function is a generalization of the cost function in optimal transport theory. The generalized Monge-Ampère equations are solved using an iterative least-squares algorithm. To compute the first optical surface, we choose an intermediate far-field target intensity. By choosing different intermediate target intensities based on the source and target intensity, we develop a "knob" to distribute the refractive power over both surfaces of the lens. We apply the algorithm on two example problems and show it is capable of producing complicated target distributions.

Two-step blind phase-shifting interferometry based on the extreme value of interference and least-squares iterative algorithm

To improve the measuring accuracy in two-step phase-shifting interferometry (PSI), a new approach combining the extreme value of interference (EVI) and the least-squares iterative algorithm (LSIA) is proposed to extract the phase from two-frame blind phase-shifting interferograms. This method first evaluates the phase shift between two interferograms by the EVI algorithm, and then constructs the fitted interferogram by the addition of two interferograms after filtering the corresponding background intensities, so the phase with high precision can be retrieved by combining two real interferograms and this fitted interferogram using the LSIA method. The proposed algorithm expands the flexibility of the LSIA method and has the high-precision performance compared with the existing algorithms in two-step PSI. Simulation and experiment are performed to verify the feasibility of the proposed algorithm.

Solving Two Coupled Fuzzy Sylvester Matrix Equations Using Iterative Least-squares Solutions

ABSTRACT In this paper, five iterative methods for solving two coupled fuzzy Sylvester matrix equations are considered. The two coupled fuzzy Sylvester matrix equations are expressed by using the generalized inverse of the coefficient matrix, then iterative solutions are constructed by applying the hierarchical identification principle and by using the block-matrix inner product (the star product for short). A proposed modification to this algorithm to solve the first coupled fuzzy Sylvester matrix equations is suggested. This proposed modification is compared with the first algorithm where our modification exhibits fast convergence behavior. Also, we suggested two least-squares iterative algorithm by applying a hierarchical identification principle to solve the two coupled fuzzy Sylvester matrix equations. The proposed methods are illustrated by numerical examples.

Modal parameter identification of time-varying systems via wavelet-based frequency response function

Modal parameter identification of systems, structures and machines with variable physical parameters is an important task for their damage diagnosis, maintenance and repair, and life cycle management, especially when time-varying vibration modes are involved. The paper proposes a new combined, two-step, i.e., estimated wavelet-based frequency response function (FRF) and least-squares iterative algorithm, modeling approach in order to determine time-varying vibration modes based on Gaussian white noise input excitation. The time-varying wavelet-based FRF is estimated based on output noise estimation model $$H_1$$ ; the second step identifies time-varying modal parameters based on estimated wavelet-based FRF with the use of least-squares iterative algorithm. Estimated wavelet-based FRF can reveal the dynamic characteristics of the system correctly similar to the theoretical FRF by single degree of freedom (SDOF). The combined method is demonstrated using system identification analysis based on the simulated stiffness-varying and mass-varying multiple-degree-of-freedom (MDOF) system subjected to random Gaussian white noise excitation. The results show that the proposed combined method accurately identifies modal parameter of the time-invariant analyzed structure, and correctly captures modal parameter of the time-varying analyzed structure. The modal frequency and shape agree with theoretical results well. The analysis results also indicate that the proposed combined method is sensitive to the position of input excitation. The estimation accuracy wavelet-based FRF can be improved by selecting the average number of times and wavelet parameters.

Chordal Based Error Function for 3-D Pose-Graph Optimization

Pose-graph optimization (PGO) is a well-known problem in the robotics community. Optimizing a graph means finding the configuration of the nodes that best satisfies the edges. This is generally achieved using iterative approaches that refine a current solution until convergence. Nowadays, iterative least-squares (ILS) algorithms such as Gauss–Newton (GN) or Levenberg–Marquardt (LM) are dominant. Common to all these implementations is the influence of the error function used to measure the difference between prediction and observation. The smoother the error function is, the better the convergence properties of the system become, resulting in an increased convergence basin and more stable behavior. In this letter, we propose an alternative error function based on a variant of the Frobenious norm between transformation matrices. The proposed approach leads to a larger convergence basin and to numerical properties in the Jacobian computation that can potentially speedup the system. In contrast with some existing approximations, our formulation allows isotropic and anistropic noise covariances to be modeled. To validate our conjectures, we present an extensive comparative analysis between our approach and one of the most used error functions that computes the distance in the unit-quaternion space.

Two modified least-squares iterative algorithms for the Lyapunov matrix equations

In this paper two modified least-squares iterative algorithms are presented for solving the Lyapunov matrix equations. The first algorithm is based on the hierarchical identification principle, which can be viewed as a surrogate of the least-squares iterative algorithm proposed by Ding et al., whose convergence has not been proved until now. The second one is motivated by a new form of fixed point iterative scheme. With the tool of a new matrix norm, the proof of both algorithms’ global convergence is offered. Furthermore, the feasible sets of their convergence factors are analyzed. Finally, a numerical example is presented to illustrate the rationality of theoretical results.

An iterative damped least-squares algorithm for simultaneously monitoring the development of hemorrhagic and secondary ischemic lesions in brain injuries.

Electrical impedance tomography (EIT) is a non-invasive and real-time imaging method that has the potential to be used for monitoring intracerebral hemorrhage (ICH). Recent studies have proposed that ischemia secondary to ICH occurs simultaneously in the brain. Real-time monitoring of the development of hemorrhage and risk of secondary ischemia is crucial for clinical intervention. However, few studies have explored the performance of EIT monitoring in cases where hemorrhage and secondary ischemia exist. When these lesions get close to each other, or their conductivity and volume changes differ greatly, it becomes challenging for dynamic EIT algorithms to simultaneously reconstruct subtle injuries. To address this, an iterative damped least-squares (IDLS) algorithm is proposed in this study. The quality of the IDLS algorithm was assessed using blur radius and temporal response during computer simulation and a phantom 3D head-shaped model where bidirectional disturbance targets were simulated. The results showed that the IDLS algorithm enhanced contrast and concurrently reconstructed bidirectional disturbance targets in images. Moreover, it showed superior performance in decreasing the blur radius and was time cost-effective. With further improvement, the IDLS algorithm has the potential to be used for monitoring the development of hemorrhage and risk of ischemia secondary to ICH. Graphical abstract (a) and (b) are simulation images of bidirectional disturbance targets with different change ratios of volume (Vr) and conductivity (σr) based on the damped least-squares (DLS) algorithm and iterative damped least-squared (IDLS) algorithm, respectively. (c) shows the performance metrics of blur radius and temporal response with different volume ratio (corresponding to Vr). (d) shows the performance metrics of blur radius and temporal response with different conductivity change percentage (corresponding to σr).

Estimation of fracture weaknesses and integrated attenuation factors from azimuthal variations in seismic amplitudes

Seismic wave propagation in fractured reservoirs exhibits anisotropy and attenuation, which are in turn related to fracture properties (e.g., fracture density) and fluid parameters (e.g., moduli and viscosity). Based on the linear slip theory, stiffness parameters can be determined for fractured and dissipative rocks, from which integrated attenuation factors involving host-rock intrinsic attenuation and fracture-induced attenuation emerge. Using a simplified mathematical form for these stiffness parameters, a linearized mathematical relationship directly relating the reflection coefficient to fracture weaknesses and integrated attenuation factors is available. A two-step inversion approach, involving (1) an iterative damped least-squares algorithm to predict P- and S-wave moduli using seismic angle gathers along the fracture orientation azimuth and (2) an iterative inversion method to estimate fracture weaknesses and integrated attenuation factors from azimuthal amplitude differences, is examined. The objective function for the second step is constructed based on a Bayesian framework. Synthetic testing confirms that fracture weaknesses and integrated attenuation factors are stably determined from seismic amplitudes exhibiting a moderate signal-to-noise ratio. The approach is applied to a field data set from a fractured carbonate reservoir. We observe that geologically reasonable results of fracture weaknesses and integrated attenuation factors are obtained. We conclude that this estimation procedure provides a reliable tool in fracture prediction and inverted attenuation factors appear as additional proofs to identify fluid type in fractured reservoirs.

Iterative parameter identification for pseudo‐linear systems with ARMA noise using the filtering technique

This study considers the parameter identification problem of a pseudo-linear autoregressive moving average system (i.e. linear-in-parameter autoregressive output-error ARMA systems), whose disturbance is an ARMA process. By means of the filtering technique, a filtering-based gradient iterative (F-GI) algorithm and a filtering-based least squares iterative (LSI) algorithms are presented for enhancing the estimation accuracy. Furthermore, a filtering-based decomposition LSI algorithm is derived for improving the computational efficiency. The key is to use the hierarchical identification principle, to apply the data filtering technique for identification, and to replace the unknown terms in the information vectors with their estimates. Compared with the F-GI algorithm, the filtering-based LSI algorithm and the filtering-based decomposition LSI algorithm have faster convergence rates. The simulation results indicate that the proposed algorithms are effective.

Prestack AVA inversion of exact Zoeppritz equations based on modified Trivariate Cauchy distribution

Obtaining interlayer weak reflection information that helps identify properties and accurate density information from complex and elusive reservoirs is particularly important for reservoir characterization and detection. However, conventional prestack amplitude variation with incidence angle inversion method is strongly influenced by the accuracy of the approximate Zoeppritz equations, which suppresses weak reflections coming from the commonly used prior distribution. In this paper, we address these problems by using exact Zoeppritz equations. First, the objective function of the inverse problem was constructed and the modified Cauchy distribution was introduced as the prior information by utilizing Bayes' theorem. In the complicated objective function, the forward operators are the sophisticated and nonlinear Zoeppritz equations with respect to estimate parameters. We then combined the idea of generalized linear inversion with iterative reweighed least-squares algorithm in order to solve the problem. Generalized linear inversion was used to solve the objective function, from which a nonlinear solution of the model parameters' perturbations can be calculated. The iterative reweighed least-squares algorithm was applied to solve the nonlinear expression in an attempt to obtain an updated iterative formula of the model parameters. Therefore the prestack amplitude variation with incidence angle inversion was able to be performed in order to better characterize a reservoir. Both synthetic and field data examples show that the new method can not only directly inverse P-wave velocity, S-wave velocity and density, but also provides accurate estimation results, particularly for density. The introduction of the modified Trivariate Cauchy prior constraints effectively estimated and inverted elastic parameters of weak reflections. Both examples demonstrated the feasibility and effectiveness of the proposed method.

Fast Detection of Orthogonal Space-Time Block Codes With Unknown Channel

This letter investigates the problem of blind detection of orthogonal space-time block codes (OSTBC) over a quasi-static flat multiple-input multiple-output (MIMO) Rayleigh fading channel. We first introduce a core iterative least-squares (ILS) algorithm to blindly detect OSTBC signals without the knowledge of channel state information (SCI) at the receiver. This ILS algorithm has low computational complexity but may converge to local optimum which offers unreliable detection result. Then, in order to improve the detection performance, we propose an enhanced ILS (E-ILS) approach which is based on statistical analysis of repeated independent ILS procedures on received data. Extensive simulation studies prove the efficiency of the proposed E-ILS algorithm with blind detection performance approaching the optimal maximum-likelihood detector with known CSI.

Phase retrieval based on temporal and spatial hybrid matching in simultaneous phase-shifting dual-wavelength interferometry.

In simultaneous phase-shifting dual-wavelength interferometry, by matching both the phase-shifting period number and the fringe number in interferogram of two wavelengths to the integers, the phase with high accuracy can be retrieved through combining the principle component analysis (PCA) and least-squares iterative algorithm (LSIA). First, by using the approximate ratio of two wavelengths, we can match both the temporal phase-shifting period number and the spatial fringe number in interferogram of two wavelengths to the integers. Second, using above temporal and spatial hybrid matching condition, we can achieve accurate phase shifts of single-wavelength of phase-shifting interferograms through using PCA algorithm. Third, using above phase shifts to perform the iterative calculation with the LSIA method, the wrapped phases of single-wavelength can be determined. Both simulation calculation and experimental research demonstrate that by using the temporal and spatial hybrid matching condition, the PCA + LSIA based phase retrieval method possesses significant advantages in accuracy, stability and processing time.