Weighted Least-squares Algorithm Research Articles

Generalized Algorithms for Direct Reconstruction of Parametric Images From Dynamic PET Data

Indirect and direct methods have been developed for reconstructing parametric images from dynamic positron emission tomography (PET) data. Indirect methods are simple and easy to implement because reconstruction and kinetic modeling are performed in two separate steps. Direct methods estimate parametric images directly from dynamic PET sinograms and, in theory, can be statistically more efficient, but the algorithms are often difficult to implement and are very specific to the kinetic model being used. This paper presents a class of generalized algorithms for direct reconstruction of parametric images that are relatively easy to implement and can be adapted to different kinetic models. The proposed algorithms use optimization transfer principle to convert the maximization of a penalized likelihood into a pixel-wise weighted least squares (WLS) kinetic fitting problem at each iteration. Thus, it can employ existing WLS algorithms developed for kinetic models. The proposed algorithms resemble the empirical iterative implementation of the indirect approach, but converge to a solution of the direct formulation. Computer simulations showed that the proposed direct reconstruction algorithms are flexible and achieve a better bias-variance tradeoff than indirect reconstruction methods.

Two-dimensional phase unwrapping to help characterize an electromagnetic beam for quasi-optical mode converter design

An improved two-dimensional phase unwrapping procedure is discussed that uses a weighted least-squares algorithm, a congruence operation, and a filter to unwrap the phase distribution of an electromagnetic beam. These improvements make possible several advances for mirror designs used in gyrotron quasi-optical mode converters. The improved phase unwrapping procedure is demonstrated by applying it to a measured beam and a simulated beam that are used to design mirrors. The unwrapping procedure produces a smooth unwrapped phase that does not change the characteristics of the beam. The smooth unwrapped phase distribution is also used to find an estimate for the wavenumber vector distribution that is needed to design the mirrors.

Performance comparison of target localization for active sonar systems

The performance comparison of target localization for active sonar and effective fusion algorithms for target localization of multistatic sonar is investigated. Active sonar can be categorized into monostatic, bistatic, and multistatic, depending on the number of receiver elements. Target localization performance depends on the system configuration. The target localization performance of the monostatic, bistatic, and multistatic sonar systems is compared assuming that each element can receive both range and azimuth information of the target. In addition, we propose the weighted least square (WLS) algorithm, which incorporates judicial weighting to the conventional least square (LS) method, and an efficient sensor arrangement rule for target localization in the multistatic sonar system. The representative experimental results demonstrate that the target localization performance of multistatic sonar configuration is superior in terms of root-mean-square error (RMSE), to monostatic sonar and bistatic sonar by 35.98% and 37.45%, respectively, while the proposed WLS algorithm showed an improvement of 2.27% compared with the LS method.

Optimal and suboptimal smoothing algorithms for identification of time-varying systems with randomly drifting parameters

Noncausal estimation algorithms, which involve smoothing, can be used for off-line identification of nonstationary systems. Since smoothing is based on both past and future data, it offers increased accuracy compared to causal (tracking) estimation schemes, incorporating past data only. It is shown that efficient smoothing variants of the popular exponentially weighted least squares and Kalman filter-based parameter trackers can be obtained by means of backward-time filtering of the estimates yielded by both algorithms. When system parameters drift according to the random walk model and the adaptation gain is sufficiently small, the properly tuned two-stage Kalman filtering/smoothing algorithm, derived in the paper, achieves the Cramér–Rao type lower smoothing bound, i.e. it is the optimal noncausal estimation scheme. Under the same circumstances performance of the modified exponentially weighted least-squares algorithm is often only slightly inferior to that of the Kalman filter-based smoother.

QUASI-POISSON VS. NEGATIVE BINOMIAL REGRESSION: HOW SHOULD WE MODEL OVERDISPERSED COUNT DATA?

Quasi-Poisson and negative binomial regression models have equal numbers of parameters, and either could be used for overdispersed count data. While they often give similar results, there can be striking differences in estimating the effects of covariates. We explain when and why such differences occur. The variance of a quasi-Poisson model is a linear function of the mean while the variance of a negative binomial model is a quadratic function of the mean. These variance relationships affect the weights in the iteratively weighted least-squares algorithm of fitting models to data. Because the variance is a function of the mean, large and small counts get weighted differently in quasi-Poisson and negative binomial regression. We provide an example using harbor seal counts from aerial surveys. These counts are affected by date, time of day, and time relative to low tide. We present results on a data set that showed a dramatic difference on estimating abundance of harbor seals when using quasi-Poisson vs. negative binomial regression. This difference is described and explained in light of the different weighting used in each regression method. A general understanding of weighting can help ecologists choose between these two methods.

Robust smooth segmentation approach for array CGH data analysis

Array comparative genomic hybridization (aCGH) provides a genome-wide technique to screen for copy number alteration. The existing segmentation approaches for analyzing aCGH data are based on modeling data as a series of discrete segments with unknown boundaries and unknown heights. Although the biological process of copy number alteration is discrete, in reality a variety of biological and experimental factors can cause the signal to deviate from a stepwise function. To take this into account, we propose a smooth segmentation (smoothseg) approach. To achieve a robust segmentation, we use a doubly heavy-tailed random-effect model. The first heavy-tailed structure on the errors deals with outliers in the observations, and the second deals with possible jumps in the underlying pattern associated with different segments. We develop a fast and reliable computational procedure based on the iterative weighted least-squares algorithm with band-limited matrix inversion. Using simulated and real data sets, we demonstrate how smoothseg can aid in identification of regions with genomic alteration and in classification of samples. For the real data sets, smoothseg leads to smaller false discovery rate and classification error rate than the circular binary segmentation (CBS) algorithm. In a realistic simulation setting, smoothseg is better than wavelet smoothing and CBS in identification of regions with genomic alterations and better than CBS in classification of samples. For comparative analyses, we demonstrate that segmenting the t-statistics performs better than segmenting the data. The R package smoothseg to perform smooth segmentation is available from http://www.meb.ki.se/~yudpaw.

Discrete Time Optimal Adaptive Control for Linear Stochastic Systems

The least-squares (LS) algorithm has been used for system modeling for a long time. Without any excitation conditions, only the convergence rate of the common LS algorithm can be obtained. This paper analyzed the weighted least-squares (WLS) algorithm and described the good properties of the WLS algorithm. The WLS algorithm was then used for adaptive control of linear stochastic systems to show that the linear closed-loop system was globally stable and that the system identification was consistent. Compared to the past optimal adaptive controller, this controller does not impose restricted conditions on the coefficients of the system, such as knowing the first coefficient before the controller. Without any persistent excitation conditions, the analysis shows that, with the regulation of the adaptive control, the closed-loop system was globally stable and the adaptive controller converged to the one-step-ahead optimal controller in some sense.

Kinematic modeling of wheeled mobile robots with slip

This work presents a kinematic modeling method for wheeled mobile robots with slip based on physical principles. First, we present the kinematic modeling of a mobile robot with no-slip considering four types of wheels: fixed, centered orientable, off-centered orientable (castor) and Swedish (also called Mecanum, Ilon or universal). Then, the dynamics of a wheeled mobile robot based on Lagrange formulation are derived and discussed. Next, a quasi-static motion is considered to obtain the kinematic conditions that provide the slip modeling equations. Several types of traction models for the slip between the wheel and the floor are indicated. In particular, for a frictional force linearly dependent on the sliding velocity, the no-slip kinematic equation of the wheeled mobile robot is related, through the weighted least-squares algorithm, with the slip modeling equations. To illustrate the applications of the proposed approach a tricycle vehicle is considered in a real situation. The experimental results obtained for the slip kinematic model are compared with the ones obtained for the well-known Kalman filter.

Sequential weighted least squares algorithm for PET image reconstruction

The weighted least squares (WLS) algorithm has proven useful for modern positron emission tomography (PET) scanners to approach reconstructions with non-Poisson precorrected measurement data. In this paper, we propose a new time recursive sequential WLS algorithm whose derivation uses the time-varying property of data acquisition of PET scanning. It ties close relationship with the time-varying Kalman filtering and can be extended appropriately to an iteration fashion as the absence of proper a priori initializations. The performance of sequential WLS is evaluated experimentally. The results show its fast convergence over both the multiplicative and coordinate-based iterative WLS methods. It also produces relative uniform estimate variances that makes it more suitable for routine applications.

Data Reduction of Horizontal Load Full-Scale Tests on Bored Concrete Piles and Pile Groups

This paper proposes a new approach for data reduction of horizontal load full-scale tests on piles and pile groups. This approach has been developed on results from tests run on bored concrete piles embedded in homogeneous and nonhomogeneous ground. Due to nonlinear response of pile material and also to nonhomogeneous embedding ground, the problem of fitting reliable curves for representing strains along shafts is increased. It is suggested that B-splines fixed by a weighted least-squares algorithm should be used to overcome that problem. Taking advantage of the mathematical properties of B-splines, an algorithm for computing the internal force distribution amongst pile heads direct from test results is also proposed for pile groups. It is shown that the integration of the curvatures to compute pile movements should be done using natural boundary conditions instead of pile head measurements whenever possible. Despite the concrete crack, the distribution of bending moments can be computed from curvatures provided a reliable reinforced concrete model is used. Finally, it is proposed to compute the soil reactions by the integration of bending moments, solving an integral equation by again using B-spline functions.

A novel weighted least squares PET image reconstruction method using adaptive variable index sets

In this paper, we present a novel image reconstruction method based on weighted least squares (WLS) objective function for positron emission tomography (PET). Unlike a usual WLS algorithm, the proposed method, which we call it SA-WLS, combines the SAGE algorithm with WLS algorithm. It minimized the WLS objective function using single coordinate descent (SCD) method in a sequence of small ''hidden'' data spaces (HDS). Although SA-WLS used a strategy to update parameter sequentially just like common SCD method, the use of these small HDS makes it converge much faster and produce the reconstructed images with greater contrast and detail than the usual WLS method. In order to decrease further the actual CPU time per iteration, the adaptive variable index sets were introduced to modify SA-WLS (MSA-WLS). Instead of optimizing each pixel, this MSA-WLS method sequentially optimizes many pixels located in an index set at one time. The index sets were automatically modified during each iteration step. MSA-WLS gathers the virtue of simultaneously and sequentially updating the parameters so that it achieves a good compromise between the convergence rate and the computational cost in PET reconstruction problem. Details of these algorithms were presented and the performances were evaluated by a simulated head phantom.

Joint Weighted Least-Squares Estimation of Carrier-Frequency Offset and Timing Offset for OFDM Systems Over Multipath Fading Channels

This work presents an algorithm for joint estimation of carrier-frequency offset and timing offset for orthogonal frequency-division multiplexing (OFDM) systems in the tracking mode. The proposed weighted least-squares algorithm derives its estimates based on phase differences in the received pilot subcarrier signals between two symbols. Moreover, the optimal weights in two different channel conditions are derived. Both analysis and simulation show that the weighted least-squares algorithm can effectively and accurately estimate the carrier-frequency offset as well as the timing offset of OFDM signals in multipath fading channels.

Genetic programming for model selection of TSK-fuzzy systems

This paper compares a genetic programming (GP) approach with a greedy partition algorithm (LOLIMOT) for structure identification of local linear neuro-fuzzy models. The crisp linear conclusion part of a Takagi-Sugeno-Kang (TSK) fuzzy rule describes the underlying model in the local region specified in the premise. The objective of structure identification is to identify an optimal partition of the input space into Gaussian, axis-orthogonal fuzzy sets. The linear parameters in the rule consequent are then estimated by means of a local weighted least-squares algorithm. LOLIMOT is an incremental tree-construction algorithm that partitions the input space by axis-orthogonal splits. In each iteration it greedily adds the new model that minimizes the classification error. GP performs a global search for the optimal partition tree and is therefore able to backtrack in case of sub-optimal intermediate split decisions. We compare the performance of both methods for function approximation of a highly non-linear two-dimensional test function and an engine characteristic map.

A global energy approach to facet model and its minimization using weighted least-squares algorithm

A global energy approach and a weighted least-squares facet algorithm have been developed using the facet model. The global energy method gives a near-optimal estimation of the ideal image as predicted by the facet model using a regularization framework. The weighted least-squares facet algorithm improves the estimation of the facet parameters by taking into account the variances of estimations from different resolution cells. Experimental results show the validity of the global energy approach and the superiority of the WLS facet algorithms in both synthetic images and real range images.

Global stabilization of a class of discrete-time nonlinear stochastic systems

For a class of nonlinear stochastic systems with parametric-strict-feedback form, global stabilization of closed-loop system is established by designing an adaptive controller based on the weighted least-squares algorithm and by using a backstepping method where nonlineacity vectors appearing in the last state equation are permitted to have the growth rate faster than linear.

Computer vision assisted virtual reality calibration

A computer vision assisted semi-automatic virtual reality (VR) calibration technology has been developed that can accurately match a virtual environment of graphically simulated 3D models to the video images of the real task environment. We have developed a new 20-variable weighted least-squares algorithm that updates both camera and object models simultaneously for given two camera views of two mating objects. This simultaneous update enables accurate model matching even with rough, approximate initial camera calibrations. The developed semi-automatic VR calibration supports automated intermediate updates, eliminating nearly all operator interaction except for initial coarse matching. In our quasi-static supervisory telerobotic applications, intermediate VR calibrations are performed intermittently at a few robot stopping poses only, as a cost-effective and safer approach compared to real-time visual servoing. Extensive experimental results comparing alignment errors under various viewing conditions are presented.

Tailoring of EOFs for Ionospheric Tomography and the Weighted, Partitioned, Least-Squares Algorithm

Among the linear-algebraic class of inversion techniques used for ionospheric tomography is the Weighted, Damped, Least-Squares (WDLS) algorithm. One application of it uses empirical orthogonal functions (EOFs) as a vertical basis set and Fourier harmonics horizontally. To date, the EOFs employed have been generated from a large database of electron-density profiles with the intent of spanning all ionospheric states. In this work, we explore the utility of tailoring the EOFs for specific ionospheric conditions by limiting their span in latitude, longitude, local time, season, solar epoch, and geomagnetic disturbance level. In this paper, we demonstrate early results, showing some improvement by tailoring in latitude. We also describe a successor to the WDLS method, the Weighted, Partitioned, Least-Squares (WPLS) algorithm, which combines weighting with the partitioning capability of singular-value decomposition (SVD). In particular, we introduce a simple criterion for truncating the SVD series of singular values.

Weighted adaptive minimum variance controller with unknown input coefficient

For a standard autoregressive moving average system with extraneous input this paper designs a weighted adaptive minimum variance controller based on a weighted least-squares algorithm. It is shown that the closed-loop system is stable, the general adaptive tracking is asymptotically optimal and the closed-loop transfer function is invariant, with its denominator differing from a fixed value. It is found that there is an equivalent relation between the consistent estimation and the system stability. Simulation demonstrates that the results developed in the paper are efficient. No a priori information for the unknown input coefficient and no modification for its estimate are made.

Linear-phase quadrature mirror filters with coefficients −1, 0 and + 1

This paper considers the linear-phase quadrature mirror filters (LP-QMFs) with coefficients constrained to the values of −1, 0 and + 1. A new filter structure and a design technique based on a recently developed weighted least-squares (WLS) algorithm are presented. First, we design an LP-QMF with continuous coefficients using the WLS algorithm. This design process provides two favourable design results that the prototype LP analysis filter has least-squares stopband response and the resulting QMF bank shows quasi-equiripple reconstruction error behavior. In conjunction with the new proposed filter structure, we then present an efficient method to obtain a design solution with coefficients restricted to −1, 0 and + 1 in the minimax sense. Several design examples demonstrating the effectiveness of the proposed technique are provided.

An integral approach to free-form object modeling

Presents an approach to free-form object modeling from multiple range images. In most conventional approaches, successive views are registered sequentially. In contrast to the sequential approaches, we propose an integral approach which reconstructs statistically optimal object models by simultaneously aggregating all data from multiple views into a weighted least-squares (WLS) formulation. The integral approach has two components. First, a global resampling algorithm constructs partial representations of the object from individual views, so that correspondence can be established among different views. Second, a weighted least-squares algorithm integrates resampled partial representations of multiple views, using the techniques of principal component analysis with missing data (PCAMD). Experiments show that our approach is robust against noise and mismatch.