Multiplicative Iterative Algorithm Research Articles

A Dynamic Traffic Zone Division Scheme Based on Game Theory

In order to overcome the defects such as complex computation, poor flexibility and hardness to reflect internal characteristics of traffic flow which exist in the traditional static traffic zone division methods, this article proposes a dynamic traffic zone division scheme based on game theory. It introduces the conception of traffic zone core based on traffic similarity and takes each section node in the road network as a rational game participant, who seeks the most similarity payoff between itself and the traffic zone. Automatic partition and dynamic adjustment of traffic zones are realized through a multiple game iterative algorithm. Taking part of the road network in Jinan as an example, a simulation is given to validate the proposed method, and results show that it is simple, flexible, and with high reliability.

Independent and arbitrary generation of spots in the 3D space domain with computer generated holograms written on a phase-only liquid crystal spatial light modulator

An improved multiple independent iterative plane algorithm, based on a projection optimization idea, is proposed for the independent and arbitrary generation of one spot or multiple spots in a speckle-suppressed 3D work-area. Details of the mathematical expressions of the algorithm are given to theoretically show how it is improved for 3D spot generation. Both simulations and experiments are conducted to investigate the performance of the algorithm for independent and arbitrary 3D spot generation in several different cases. Simulation results agree well with experimental results, which validates the effectiveness of the algorithm proposed. Several additional experiments are demonstrated for fast and independent generation of four or more spots in the 3D space domain, which confirms the capabilities and practicalities of the algorithm further.

A 2D non-invasive ultrasonic method for simultaneous measurement of arterial strain and flow pattern

Many cardiovascular diseases are closely associated with the mechanical properties of arterial wall and hemodynamic parameters. Simultaneous measurements of the arterial strain and flow pattern may aid diagnosis of cardiovascular diseases and may be useful to study fluid-structure interaction between blood and vessel. This paper proposes a 2D non-invasive ultrasonic method to simultaneously measure arterial strain and flow pattern with sub-pixel accuracy. The method uses a multiple iterative algorithm to estimate the geometrical transformations of arterial wall and high-velocity gradient flows simultaneously. The accuracy of the method was validated by an in vitro arterial phantom and in vivo common carotid arteries (CCAs) of 12 mice using a Sonix RP (10 MHz) and a VisualSonics Vevo 2100 (30 MHz) ultrasound imaging system, respectively. For the arterial phantom, the calculated elasticity modulus from the strain profile shows good agreement with the mechanical testing value, deviating no more than 9.3%. The calculated flow velocity agrees well with the value obtained from the rotameter, deviating only 4.3%. For the CCAs of mice, good agreement is found between the calculated flow velocity and the measured value by ultrasound Doppler. The mean elasticity modulus of CCAs is 134.62 ± 54.3 kPa, which is in accordance with published data. The proposed method is capable of measuring the arterial wall strain and flow velocity pattern. This may be clinically useful for early detecting and monitoring cardiovascular diseases and may provide an essential tool in modelling the fluid-structure interaction between the blood and blood vessel.

POSITIVELY CONSTRAINED TOTAL VARIATION PENALIZED IMAGE RESTORATION

The total variation (TV) minimization models are widely used in image processing, mainly due to their remarkable ability in preserving edges. There are many methods for solving the TV model. These methods, however, seldom consider the positivity constraint one should impose on image-processing problems. In this paper we develop and implement a new approach for TV image restoration. Our method is based on the multiplicative iterative algorithm originally developed for tomographic image reconstruction. The advantages of our algorithm are that it is very easy to derive and implement under different image noise models and it respects the positivity constraint. Our method can be applied to various noise models commonly used in image restoration, such as the Gaussian noise model, the Poisson noise model, and the impulsive noise model. In the numerical tests, we apply our algorithm to deblur images corrupted by Gaussian noise. The results show that our method give better restored images than the forward–backward splitting algorithm.

Comparative study on several blind deconvolution algorithms applied to underwater image restoration

Although the use of blind deconvolution of image restoration is a widely known concept, only few reports have discussed in detail its application to solving problem of restoration of underwater range-gated laser images. A comparative study of underwater image restoration using the Richardson-Lucy algorithm, the least-squares algorithm, and the multiplicative iterative algorithm for blind deconvolution is presented. All the deconvolution approaches use denoised underwater images and Wells’ small angle approximation theory of derived point spread function as the initial object and degradation guess, respectively. Owing the underwater no-reference imaging environment, image quality judgment based on the blur metric method is incorporated in our comparison to determine the appropriate deconvolution iteration number for each algorithm, which objectively evaluates the image restoration results. The performance of the three algorithms applied to underwater image restoration is discussed and reported.

Ultrasonic particle image velocimetry for improved flow gradient imaging: algorithms, methodology and validation

This paper presents a new algorithm for ultrasonic particle image velocimetry (Echo PIV) for improving the flow velocity measurement accuracy and efficiency in regions with high velocity gradients. The conventional Echo PIV algorithm has been modified by incorporating a multiple iterative algorithm, sub-pixel method, filter and interpolation method, and spurious vector elimination algorithm. The new algorithms' performance is assessed by analyzing simulated images with known displacements, and ultrasonic B-mode images of in vitro laminar pipe flow, rotational flow and in vivo rat carotid arterial flow. Results of the simulated images show that the new algorithm produces much smaller bias from the known displacements. For laminar flow, the new algorithm results in 1.1% deviation from the analytically derived value, and 8.8% for the conventional algorithm. The vector quality evaluation for the rotational flow imaging shows that the new algorithm produces better velocity vectors. For in vivo rat carotid arterial flow imaging, the results from the new algorithm deviate 6.6% from the Doppler-measured peak velocities averagely compared to 15% of that from the conventional algorithm. The new Echo PIV algorithm is able to effectively improve the measurement accuracy in imaging flow fields with high velocity gradients.

Positively Constrained Multiplicative Iterative Algorithm for Maximum Penalized Likelihood Tomographic Reconstruction

This paper first develops a general multiplicative iterative (MI) algorithm for tomographic image reconstructions, where the objective function is only specified as a general function containing two components: a data mismatch component and a penalty component. This general algorithm is then applied to different objective functions deduced from different probability models for measurements in emission or transmission tomography, such as Poisson, Gaussian, or shifted Poisson models. Furthermore, an approximate line search step can be easily incorporated into the algorithm so that the objective function is guaranteed to increase during the iterations. This MI algorithm (with line search) is easy to implement, as it performs only one forward- and one or two back-projection in each iteration, and it respects the positivity constraint usually imposed on reconstructions.

Hybridization and Postprocessing Techniques for Mixed Eigenfunctions

We introduce hybridization and postprocessing techniques for the Raviart–Thomas approximation of second-order elliptic eigenvalue problems. Hybridization reduces the Raviart–Thomas approximation to a condensed eigenproblem. The condensed eigenproblem is nonlinear, but smaller than the original mixed approximation. We derive multiple iterative algorithms for solving the condensed eigenproblem and examine their interrelationships and convergence rates. An element-by-element postprocessing technique to improve accuracy of computed eigenfunctions is also presented. We prove that a projection of the error in the eigenspace approximation by the mixed method (of any order) superconverges and that the postprocessed eigenfunction approximations converge faster for smooth eigenfunctions. Numerical experiments using a square and an L-shaped domain illustrate the theoretical results.

Retrieval of sub-pixel snow cover information in the Himalayan region using medium and coarse resolution remote sensing data

Numerous constrained and unconstrained algorithms have been used to retrieve sub-pixel snow-cover information quantitatively using medium and coarse spatial resolution multispectral images from the Advanced Wide Field Sensor (AWiFS) and Moderate Resolution Imaging Spectrometer (MODIS) sensors over the Himalayan region. Both the methods give slow convergence rates and inaccurate estimation of sub-pixel components analysed using root mean square (RMS) error and image deviation. Multiplicative iterative algorithms such as the Expectation Maximization Maximum Likelihood Method (EMML) and the Image Space Reconstruction Algorithm (ISRA) based on the minimization of least squares and Kullback–Leibler distances have been attempted to compute the endmembers' abundances in unmixing of satellite data. In this paper we discuss the eigenvalues of minimum noise fraction (MNF) transformation bands, data noise removal using MNF transformation and selection of pure endmembers using satellite images. The normalized difference snow index (NDSI) is also estimated using field spectral reflectance results and satellite images in green and shortwave infrared (SWIR) wavelength regions in order to carry out a comparative analysis for its variations with sub-pixel snow cover fractions. The present analysis shows the advantage of iterative over direct (constrained and unconstrained) methods; constraints are easily handled and allow better regularization of the solution for the ill-conditioned cases. Iterative methods are found to be faster compared to those of direct methods and can be used operationally for all resolution data for accurate estimation of sub-pixel snow cover.

Blind deconvolution of a noisy degraded image

We develop a unified algorithm for performing blind deconvolution of a noisy degraded image. By incorporating a low-pass filter into the asymmetric multiplicative iterative algorithm and extending it to multiframe blind deconvolution, this algorithm accomplishes the blind deconvolution and noise removal concurrently. We report numerical experiments of applying the algorithm to the restoration of short-exposure atmosphere turbulence degraded images. These experiments evidently demonstrate that the unified algorithm has both good blind deconvolution performance and high-resolution image restoration.

Multiframe blind image restoration based on a multiplicative iterative algorithm

Based on a multiplicative iterative algorithm, a novel algorithm named multiframe multiplicative iterative algorithm (MMIA) has been developed for performing multiframe blind image restoration. By further extending the multiplicative iterative algorithm to multiframe blind deconvolution, the MMIA is derived. Experimental results well demonstrate that the MMIA not only has good performance and efficiency, but also achieves high resolution of the restored image.

Blind image deconvolution by means of asymmetric multiplicative iterative algorithm

A novel scheme for blind deconvolution is presented. Adapted from our previous multiplicative iterative algorithm, a more general scheme--the asymmetric multiplicative iterative algorithm--is derived. Additionally, in order to obtain a meaningful estimate, a penalized asymmetric multiplicative iterative algorithm is used to further improve the performance of this scheme. Results of a numerical experiment with the asymmetric multiplicative iterative algorithm are presented.

Nonnegative matrix factorization with constrained second-order optimization

Nonnegative matrix factorization (NMF) solves the following problem: find nonnegative matrices A ∈ R + M × R and X ∈ R + R × T such that Y ≅ AX , given only Y ∈ R M × T and the assigned index R. This method has found a wide spectrum of applications in signal and image processing, such as blind source separation (BSS), spectra recovering, pattern recognition, segmentation or clustering. Such a factorization is usually performed with an alternating gradient descent technique that is applied to the squared Euclidean distance or Kullback–Leibler divergence. This approach has been used in the widely known Lee–Seung NMF algorithms that belong to a class of multiplicative iterative algorithms. It is well known that these algorithms, in spite of their low complexity, are slowly convergent, give only a strictly positive solution, and can easily fall into local minima of a nonconvex cost function. In this paper, we propose to take advantage of the second-order terms of a cost function to overcome the disadvantages of gradient (multiplicative) algorithms. First, a projected quasi-Newton method is presented, where a regularized Hessian with the Levenberg–Marquardt approach is inverted with the Q-less QR decomposition. Since the matrices A and/or X are usually sparse, a more sophisticated hybrid approach based on the gradient projection conjugate gradient (GPCG) algorithm, which was invented by More and Toraldo, is adapted for NMF. The gradient projection (GP) method is exploited to find zero-value components (active), and then the Newton steps are taken only to compute positive components (inactive) with the conjugate gradient (CG) method. As a cost function, we used the α -divergence that unifies many well-known cost functions. We applied our new NMF method to a BSS problem with mixed signals and images. The results demonstrate the high robustness of our method.

Multiplicative Algorithms for Maximum Penalized Likelihood Inversion with Non Negative Constraints and Generalized Error Distributions

In many linear inverse problems the unknown function f (or its discrete approximation Θ p×1), which needs to be reconstructed, is subject to the non negative constraint(s); we call these problems the non negative linear inverse problems (NNLIPs). This article considers NNLIPs. However, the error distribution is not confined to the traditional Gaussian or Poisson distributions. We adopt the exponential family of distributions where Gaussian and Poisson are special cases. We search for the non negative maximum penalized likelihood (NNMPL) estimate of Θ. The size of Θ often prohibits direct implementation of the traditional methods for constrained optimization. Given that the measurements and point-spread-function (PSF) values are all non negative, we propose a simple multiplicative iterative algorithm. We show that if there is no penalty, then this algorithm is almost sure to converge; otherwise a relaxation or line search is necessitated to assure its convergence.

An accelerated multiplicative iterative algorithm in image reconstruction

AbstractBased on the ML‐EM (maximum likelihood expectation maximization) algorithm and AWLS (one kind of multiplicative weighted least square) reconstruction, a new algorithm named RMITC (rapid multiplicative iteration with total‐count conservation) is proposed. The new method assumes a higher order correction factor and incorporates a total‐count conservation constraint to obtain better images reconstructed while achieving a higher speed of convergence. Computer simulated phantom data and real positron emission tomography (PET) transmission data were used to compare the new method with other reconstruction algorithms, such as ML‐EM and AWLS. Results demonstrated that the new method is faster and better quantitatively than both ML‐EM and AWLS. © 2002 Wiley Periodicals, Inc. Int J Imaging Syst Technol 12, 97–100, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ima.10016

On the convergence of multiplicative iterative algorithms with inexact line search

This paper studies the convergence properties of multiplicative iterative algorithms with inexact line search. We prove that the convergence can be guaranteed for a general form of line search rule, under the assumption of convexity of objective function or the assumption of convergence of the sequence generated by the algorithm. This answers an open problem put forward by Iusem.

Discussion of the Paper by Vardi and Lee

Discussion of the Paper by Vardi and Lee

Multiplicative iterative algorithms for convex programming

We study multiplicative iterative algorithms for the minimization of a differentiable, convex function defined on the positive orthant of R N. If the objective function has compact level sets and has a locally Lipschitz continuous gradient, then these algorithms converge to a solution of the minimization problem. Moreover, the convergence is nearly monotone in the sense of Kullback-Leibler divergence. Special cases of these algorithms have been applied in position emission tomography and are formally related to the EM algorithm for position emission tomography.

Iterative reconstruction of thyroidal SPECT images.

First SPECT results using a multiplicative iterative reconstruction algorithm are presented. The superiority of the iterative technique over filtered backprojection is demonstrated in two thyroid SPECT studies. Obvious benefits of the new reconstruction technique are better defined outlines of the imaged organ and patient body as well as negligible artificial image amplitudes outside the patient.