Optimal Polynomial Coefficients Research Articles

Design Ultra-Compact Aspherical Lenses for Extended Sources Using Particle Swarm Optical Optimization Algorithm

This paper presents a global optimization algorithm specifically tailored for ultra-compact aspherical lens design problems for extended LED sources. The main purpose is to obtain prescribed illumination patterns, particularly the uniform illuminance distribution. This method begins by calculating the initial aspherical lens with two surfaces based on point source approximation. Then a system of polynomials is employed to fit the meridian curves of the two surfaces. In the optimization process, we use the particle swarm optimization (PSO) algorithm to automatically find the optimal polynomial coefficients when the ray tracing simulation is being performed. A series of ultra-compact aspherical lenses with dimension ratio of h / d ranging from 0.95 to 1.25 are presented, where h denotes the center height of the lens and d represents the diameter of the extended source. The results show the high efficiency and versatility of the proposed method in prescribed illumination design for extended LED sources in three-dimensional rotational symmetry geometry. Additionally, an aspherical lens is fabricated and tested, and its practical performance approaches the design.

A Pansharpening Approach Based on Multiple Linear Regression Estimation of Injection Coefficients

Pansharpening techniques allow a detailed reproduction of the Earth surface by fusing a multispectral (MS) and a panchromatic (PAN) image acquired over the same area. Classical pansharpening methods consist in the extraction of the details from the PAN image and their subsequent injection into the MS image through a linear function. In this letter, we propose to apply a nonlinear injection procedure that implements the detail injection through a polynomial function. Optimal polynomial coefficients in the least squares sense can be easily obtained in the closed form, and the consequent pansharpening algorithm is shown to obtain superior performance with respect to the existing linear approaches, especially for MS bands with a reduced wavelength overlap with the PAN channel.

Beam Hardening Correction Using Cone Beam Consistency Conditions.

The polychromatic X-ray spectrum and the energy-dependent attenuation coefficient of materials cause beam hardening artifacts in CT reconstructed volumes. These artifacts appear as cupping and streak artifacts depending on the material composition and the geometry of the imaged object. CT scanners employ projection linearization to transform polychromatic attenuation to monochromatic attenuation using a polynomial model. Polynomial coefficients are computed during calibration or using prior information such as X-ray spectrum and attenuation properties of the materials. In this paper, we are presenting a novel method to correct beam hardening artifacts by enforcing cone beam consistency conditions on the projection data. We used consistency conditions derived from Grangeat's fundamental relation between cone beam projection data and 3-D Radon transform. The optimal polynomial coefficients for artifact reduction are iteratively estimated by minimizing the inconsistency of a set of projection pairs. The results from simulated and real datasets show the visible reduction of artifacts. Our studies also demonstrate the robustness of the algorithm when the projections are perturbed with other physical measurement and geometrical errors. The proposed method requires neither calibration nor prior information like X-ray spectrum, attenuation properties of the materials and detector response. The algorithm can be used for beam hardening correction in clinical, pre-clinical, and industrial CT systems.

Dimensionality reduction for production optimization using polynomial approximations

The objective of this paper is to introduce a novel paradigm to reduce the computational effort in waterflooding global optimization problems while realizing smooth well control trajectories amenable for practical deployments in the field. In order to overcome the problems of slow convergence and non-smooth impractical control strategies, often associated with gradient-free optimization (GFO) methods, we introduce a generalized approach which represent the controls by smooth polynomial approximations either by a polynomial function or by a piecewise polynomial interpolation, which we denote as function control method (FCM) and interpolation control method (ICM), respectively. Using these approaches, we aim to optimize the coefficients of the selected functions or the interpolation points in order to represent the well-control trajectories along a time horizon. Our results demonstrate significant computational savings, due to a substantial reduction in the number of control parameters, as we seek the optimal polynomial coefficients or the interpolation points to describe the control trajectories as opposed to directly searching for the optimal control values (bottom hole pressure) at each time interval. We demonstrate the efficiency of the method on two and three-dimensional models, where we found the optimal variables using a parallel dynamic-neighborhood particle swarm optimization (PSO). We compared our FCM-PSO and ICM-PSO to the traditional formulation solved by both gradient-free and gradient-based methods. In all comparisons, both FCM and ICM show very good to superior performances.

An analytical representation of conformal mapping for genus-zero implicit surfaces and its application to surface shape similarity assessment

This paper develops an analytical representation of conformal mapping for genus-zero implicit surfaces based on algebraic polynomial functions, and its application to surface shape similarity assessment. Generally, the conformal mapping often works as a tool of planar or spherical parameterization for triangle mesh surfaces. It is further exploited for implicit surface matching in this study. The method begins with discretizing one implicit surface by triangle mesh, where a discrete harmonic energy model related to both the mesh and the other implicit surface is established based on a polynomial-function mapping. Then both the zero-center constraint and the landmark constraints are added to the model to ensure the uniqueness of mapping result with the Möbius transformation. By searching optimal polynomial coefficients with the Lagrange–Newton method, the analytical representation of conformal mapping is obtained, which reveals all global and continuous one-to-one correspondent point pairs between two implicit surfaces. Finally, a shape similarity assessment index for (two) implicit surfaces is proposed through calculating the differences of all the shape index values among those corresponding points. The proposed analytical representation method of conformal mapping and the shape assessment index are both verified by the simulation cases for the closed genus-zero implicit surfaces. Experimental results show that the method is effective for genus-zero implicit surfaces, which will offer a new way for object retrieval and manufactured surface inspection.

FPGA-Based Implementation of a New Phase-to-Sine Amplitude Conversion Architecture

The classical structure of linear interpolation-based phase-to-sine mapper (PSM) consists of at least two ROMs for polynomial coefficient storage. Other architectures may include extra ROM for storing residual errors. However, ROMs dissipate high power and occupy a significant amount of the die area. This study presents a new technique that eliminates the ROM by including the computation of segment initial coefficients in the hardware. Therefore, it becomes possible to trim down noticeable hardware resources. The proposed direct digital frequency synthesizer (DDFS) architecture has been encoded in VHDL and synthesized with Quartus II software. Post simulation results show that the proposed design is capable of achieving the theoretical spurious-free dynamic range (SFDR) upper bound when optimal polynomial coefficients are considered. For 32 piecewise linear segments, the SFDR of the synthesized sinusoid is 84.15 dBc. A ROM compression ratio of 597.3:1 was also achieved. The performance of the DDFS is compared with previously presented DDFS techniques and the results show that the proposed design has advantages of high ROM compression ratio and low hardware complexity. DOI: http://dx.doi.org/10.5755/j01.eee.19.10.5905

Series Approach to Modeling Ion Size Effects for Asymmetric Electrolytes in the Diffuse Double Layer

The theory of the diffuse layer for asymmetric electrolytes is reconsidered with emphasis on the effects of ion size on the diffuse layer potential drop and differential capacity. For asymmetric 2:1 and 1:2 electrolytes, this potential drop is expressed in terms of a polynomial with a linear, quadratic, and cubic term in the corresponding estimate in the Gouy-Chapman theory. Optimal polynomial coefficients and model validation for 2:1 electrolytes are provided by least-squares regression of Monte Carlo data obtained for a restricted electrolyte in a primitive solvent. These coefficients are then expressed as simple functions of the parameters commonly associated with the mean spherical approximation. The series approach accurately describes potential drops and differential capacities of the diffuse layer for 2:1 and 1:2 electrolytes for the chosen assumptions.

Series Approach to Modeling Ion Size Effects for Symmetric Electrolytes in the Diffuse Double Layer

An analytical model is developed for the potential drop and differential capacity across the diffuse layer which considers the effects of ion size on these properties. For symmetric electrolytes, this potential drop is expressed in terms of a cubic polynomial in the corresponding estimate in the Gouy-Chapman theory. Optimal polynomial coefficients and model validation for 1:1 and 2:2 electrolytes are provided by fits of Monte Carlo data obtained for a restricted electrolyte in a primitive solvent. Simple relationships between these coefficients and parameters commonly associated with the mean spherical approximation are obtained. It is shown that the series approach accurately describes potential drops and differential capacities of the diffuse layer for 1:1 and 2:2 electrolytes for the chosen assumptions.

Noniterative WLS design of allpass variable fractional-delay digital filters

This paper presents a noniterative weighted-least-squares (WLS) method for designing allpass variable fractional-delay (VFD) digital filters. After expressing each coefficient of an allpass VFD filter as a polynomial of the VFD parameter p, we develop a noniterative technique for finding the optimal polynomial coefficients, and show that the allpass VFD filter design problem can be efficiently solved without using any iterative procedure while a closed-form solution can be easily obtained through solving a matrix equation. Compared with the existing iterative WLS method that solves a series of approximately linearized WLS minimization problems, the proposed noniterative one can yield much better design results with significantly reduced computational complexity. Moreover, the new WLS method does not involve any convergence issue.

Design of 1-D and 2-D variable fractional delay allpass filters using weighted least-squares method

In this paper, a weighted least-squares method is presented to design one-dimensional and two-dimensional variable fractional delay allpass filters. First, each coefficient of the variable allpass filter is expressed as the polynomial of the fractional delay parameter. Then, the nonlinear phase error is approximated by a weighted equation error such that the cost function can be converted into a quadratic form. Next, by minimizing the weighted equation error, the optimal polynomial coefficients can be obtained iteratively by solving a set of linear simultaneous equations at each iteration. Finally, the design examples are demonstrated to illustrate the effectiveness of the proposed approach.

Eigenfilter approach for the design of variable fractional delay FIR and all-pass filters

An eigenfilter approach is presented for designing 1-D and 2-D variable fractional delay FIR and all-pass filters. First, the coefficients of filters are expressed as a polynomial of the fractional delay parameter. Then, the optimal polynomial coefficients are obtained from the elements of the eigenvector corresponding to the minimum eigenvalue of a real, symmetric and positive definite matrix. Finally, several design examples of 1-D and 2-D variable fractional delay FIR and all-pass filters are used to illustrate the effectiveness of the eigenfilter approach.

Weighted least-squares method for designing arbitrarily variable 1-D FIR digital filters

Variable digital filters with variable cutoff frequencies can be designed by frequency transformation methods. Such methods are simple, but they are not applicable to the design of variable filters with arbitrarily variable frequency responses. This paper proposes an efficient method for designing variable one-dimensional (1-D) finite-impulse-response (FIR) digital filters with arbitrarily variable magnitude characteristics and specified linear or nonlinear phase responses. First, each coefficient of the variable FIR filter is assumed to be a multi-dimensional (M-D) polynomial of spectral parameters that specify different variable magnitude characteristics. Then we present a pair of least-squares algorithms for finding the optimal polynomial coefficients by minimizing the weighted squared error between the desired variable frequency response and the actual frequency response. The first one is for designing 1-D FIR filters with variable magnitude response and linear-phase, and the second one is for designing variable 1-D FIR filters with variable magnitude response and nonlinear-phase. Although the first algorithm can be regarded as a special case of the second one, using the first one in the linear case can simplify the design significantly. The proposed methods are very straightforward and efficient for designing variable digital filters with arbitrarily variable magnitude responses and specified linear or nonlinear phases.