Minimal Solver Research Articles

Aero-engine bearing fault detection: A clustering low-rank approach

Different from classic exponential-decaying feature waveform, the fault information of high-speed aero-engine bearing presents an overlapping distortion or even approximately-harmonic morphology, which brings a new challenge for popular bearing diagnosis techniques. From the nonlocal similarity perspective of vibration waveforms, this study first reveals the singular value coupled pattern among strong similarity structure, weak similarity structure and noises, which consequently yields the overlapping coherent pathology in the SVD-based low-rank domain. Leveraging an adaptive clustering into the low-rank regularization theory, a tailored diagnostic framework (dubbed AMS-CluLR) is then proposed to address the challenging problem. The main highlight of AMS-CluLR is to adaptively cluster local feature waveforms into multiple isolated groups to guarantee that different similarity structures are reliably concentrated into their matched low-rank domain, which effectively eliminates the singular value overlapping coherent pathology while keeping the structural completeness of relatively weak similarity features. Moreover, an alternative minimization solver is developed for the AMS-CluLR model to rapidly achieve a satisfying stationary solution. The embedded clustering operation’s necessity and AMS-CluLR’s superiority are profoundly investigated through a set of comprehensive numerical studies. An aero-engine bearing experiment under high-speed conditions with rotational speed up to 25,000 rev/min is further conducted to corroborate AMS-CluLR’s superiority. Experimental results show that the proposed framework substantially outperforms state-of-the-art bearing fault detection methods in terms of both information volumes and visual quality.

Deblending using the focal transformation with an efficient greedy inversion solver

Nowadays, using simultaneous source seismic acquisition is becoming popular in order to achieve a denser source spacing in an efficient manner. For further processing these data need to be deblended, meaning that they have to be separated in their non-overlapping constituents. In this work, we adopt a novel greedy inversion solver to design a faster version of the double focal transform, which we can use to eliminate blending noise in simultaneous source acquisition. The greedy inversion introduces a coherence-oriented mechanism to enhance focusing of the significant model space, leading to a sparse model space and fast convergence rate. Although the convergence rate of greedy inversion is larger than that of the Spectral Projected Gradient for ℓ1 minimization (SPGL1) solver, the greedy solver is still not fast enough. We propose to use a relative tolerance strategy to speed up the greedy solver: the LSQR (Sparse Equations and Least Squares) algorithm in the inner loop of the greedy solver is stopped when the misfit change between inner iterations is less than a pre-defined relative tolerance value or when it reaches a maximum number of iterations. Synthetics and numerically blended field data examples demonstrate the validity of its application for deblending. We also investigate the effect of random noise on the deblending process, which shows that it is better to apply a denoising process before deblending in order to get an optimum result.

Subspace Clustering by Block Diagonal Representation.

This paper studies the subspace clustering problem. Given some data points approximately drawn from a union of subspaces, the goal is to group these data points into their underlying subspaces. Many subspace clustering methods have been proposed and among which sparse subspace clustering and low-rank representation are two representative ones. Despite the different motivations, we observe that many existing methods own the common block diagonal property, which possibly leads to correct clustering, yet with their proofs given case by case. In this work, we consider a general formulation and provide a unified theoretical guarantee of the block diagonal property. The block diagonal property of many existing methods falls into our special case. Second, we observe that many existing methods approximate the block diagonal representation matrix by using different structure priors, e.g., sparsity and low-rankness, which are indirect. We propose the first block diagonal matrix induced regularizer for directly pursuing the block diagonal matrix. With this regularizer, we solve the subspace clustering problem by Block Diagonal Representation (BDR), which uses the block diagonal structure prior. The BDR model is nonconvex and we propose an alternating minimization solver and prove its convergence. Experiments on real datasets demonstrate the effectiveness of BDR.

Fast and reliable minimal relative pose estimation under planar motion

This paper proposes fast, reliable, and minimal non-iterative relative pose solvers under planar motion constraint. Relative pose estimation is popularly utilized in many important problems such as visual odometry and SLAM, and planar motion is common for mobile robots and vehicles on floors and roads. We transform the original problem formulation of finding intersections of two ellipses into more accessible form of finding intersections of a line and unit circle. Such transformation leads to a non-iterative and closed-form solver, which enables significant speed-up compared to previous methods. The proposed algorithm is almost 9 times faster than the previous minimal solver with planar motion and around 90 times faster than the previous minimal solver with general motion. In addition, our algorithms provide reliable relative pose in degeneracy of the previous minimal planar solvers. Effectiveness of the proposed algorithms is demonstrated with two types of experiments: relative pose estimation with synthetic data and monocular visual odometry with real image sequences.

Algorithms for the Orthographic-n-Point Problem

We examine the orthographic-n-point problem (OnP), which extends the perspective-n-point problem to telecentric cameras. Given a set of 3D points and their corresponding 2D points under orthographic projection, the OnP problem is the determination of the pose of the 3D point cloud with respect to the telecentric camera. We show that the OnP problem is equivalent to the unbalanced orthogonal Procrustes problem for non-coplanar 3D points and to the sub-Stiefel Procrustes problem for coplanar 3D points. To solve the OnP problem, we apply existing algorithms for the respective Procrustes problems and also propose novel algorithms. Furthermore, we evaluate the algorithms to determine their robustness and speed and conclude which algorithms are preferable in real applications. Finally, we evaluate which algorithm is most suitable as a minimal solver in a RANSAC scheme.

Two-View Orthographic Epipolar Geometry: Minimal and Optimal Solvers

We will in this paper present methods and algorithms for estimating two-view geometry based on an orthographic camera model. We use a previously neglected nonlinear criterion on rigidity to estimate the calibrated essential matrix. We give efficient algorithms for estimating it minimally (using only three point correspondences), in a least squares sense (using four or more point correspondences), and optimally with respect to the number of inliers. The inlier-optimal algorithm is based on a three-point solver and gives a fourth-order polynomial time algorithm. These methods can be used as building blocks to robustly find inlier correspondences in the presence of high degrees of outliers. We show experimentally that our methods can be used in many instances, where the orthographic camera model isn’t generally used. A case of special interest is situations with repetitive structures, which give high amounts of outliers in the initial feature point matching.

A theory of point-wise homography estimation

We propose a method, called HAF, to estimate planar homography from an affine correspondence satisfying the epipolar constraint in an image pair. An affine correspondence consists of a point pair and the related local affine transformation mapping the pixels infinitely close to the point locations from the first to the second images. As a minimal solver, it estimates the homography from only one correspondence, however, it is generalized for the over-determined case as well. The required local affinities are obtained by affine-covariant feature detectors accurately. As a side-effect of the tests, the state-of-the-art affine-covariant detectors are compared to each other w.r.t. the accuracy of the estimated point-wise homographies. The proposed method is validated both on the publicly available AdelaideRMF dataset and in a synthetic testing environment.

Protein folding optimization based on 3D off-lattice model via an improved artificial bee colony algorithm

Protein folding is a fundamental topic in molecular biology. Conventional experimental techniques for protein structure identification or protein folding recognition require strict laboratory requirements and heavy operating burdens, which have largely limited their applications. Alternatively, computer-aided techniques have been developed to optimize protein structures or to predict the protein folding process. In this paper, we utilize a 3D off-lattice model to describe the original protein folding scheme as a simplified energy-optimal numerical problem, where all types of amino acid residues are binarized into hydrophobic and hydrophilic ones. We apply a balance-evolution artificial bee colony (BE-ABC) algorithm as the minimization solver, which is featured by the adaptive adjustment of search intensity to cater for the varying needs during the entire optimization process. In this work, we establish a benchmark case set with 13 real protein sequences from the Protein Data Bank database and evaluate the convergence performance of BE-ABC algorithm through strict comparisons with several state-of-the-art ABC variants in short-term numerical experiments. Besides that, our obtained best-so-far protein structures are compared to the ones in comprehensive previous literature. This study also provides preliminary insights into how artificial intelligence techniques can be applied to reveal the dynamics of protein folding. Graphical Abstract Protein folding optimization using 3D off-lattice model and advanced optimization techniques.

Link-Based Performance Optimization of Spatial Mechanisms

In the design of spatial linkages, the finite-position kinematics is fully specified by the position of the joint axes, i.e., a set of lines in space. However, most of the tasks have additional requirements regarding motion smoothness, obstacle avoidance, force transmission, or physical dimensions, to name a few. Many of these additional performance requirements are fully or partially independent of the kinematic task and can be fulfilled using a link-based optimization after the set of joint axes has been defined. This work presents a methodology to optimize the links of spatial mechanisms that have been synthesized for a kinematic task, so that additional requirements can be satisfied. It is based on considering the links as anchored to sliding points on the set of joint axes, and making the additional requirements a function of the location of the link relative to the two joints that it connects. The optimization of this function is performed using a hybrid algorithm, including a genetic algorithm (GA) and a gradient-based minimization solver. The combination of the kinematic synthesis together with the link optimization developed here allows the designer to interactively monitor, control, and adjust objectives and constraints, to yield practical solutions to realistic spatial mechanism design problems.

TOA sensor network self-calibration for receiver and transmitter spaces with difference in dimension

We study and solve the previously unstudied problem of finding both transmitter and receiver positions using only time of arrival (TOA) measurements when there is a difference in dimensionality between the affine subspaces spanned by receivers and transmitters. Anchor-free TOA network calibration has uses both in radio, radio strength and sound applications, such as calibrating ad hoc microphone arrays. Using linear techniques and requiring only minimal number of receivers and transmitters, an algorithm is constructed for general dimension p for the lower dimensional subspace. Degenerate cases are determined and partially characterized as when receivers or transmitters inhabit a lower dimensional affine subspace than was given as input. The algorithm is further extended to overdetermined cases in a straightforward manner. Utilizing the minimal solver, an algorithm using the Random Sample Consensus (RANSAC) paradigm has been constructed to simultaneously solve the calibration problem and remove severe outliers, a common problem in TOA applications. Simulated experiments show good performance for the minimal solver and the RANSAC-like algorithm under noisy measurements. Two indoor environment experiments using microphones and speakers give a RMSE of 2.35cm and 3.95cm on receiver and transmitter positions compared to computer vision reconstructions.

SU‐E‐I‐52: Validation of Multi‐Frequency Electrical Impedance Tomography Using Computed Tomography

Purpose:Multi‐frequency EIT has been reported to be a potential tool in distinguishing a tissue anomaly from background. In this study, we investigate the feasibility of acquiring functional information by comparing multi‐frequency EIT images in reference to the structural information from the CT image through fusion.Methods:EIT data was acquired from a slice of winter melon using sixteen electrodes around the phantom, injecting a current of 0.4mA at 100, 66, 24.8 and 9.9 kHz. Differential EIT images were generated by considering different combinations of pair frequencies, one serving as reference data and the other as test data. The experiment was repeated after creating an anomaly in the form of an off‐centered cavity of diameter 4.5 cm inside the melon. All EIT images were reconstructed using Electrical Impedance Tomography and Diffuse Optical Tomography Reconstruction Software (EIDORS) package in 2‐D differential imaging mode using one‐step Gaussian Newton minimization solver. CT image of the melon was obtained using a Phillips CT Scanner. A segmented binary mask image was generated based on the reference electrode position and the CT image to define the regions of interest. The region selected by the user was fused with the CT image through logical indexing.Results:Differential images based on the reference and test signal frequencies were reconstructed from EIT data. Result illustrated distinct structural inhomogeneity in seeded region compared to fruit flesh. The seeded region was seen as a higherimpedance region if the test frequency was lower than the base frequency in the differential EIT reconstruction. When the test frequency was higher than the base frequency, the signal experienced less electrical impedance in the seeded region during the EIT data acquisition.Conclusion:Frequency‐based differential EIT imaging can be explored to provide additional functional information along with structural information from CT for identifying different tissues.

Evaluation of the Least-Squares Method for the Solution of the Population Balance Equation

Spectral methods are evaluated for the solution of population balance problems. Both a simplified population balance equation (PBE) with an analytical solution available and a rigorous PBE are considered in this numerical analysis. In comparison with the orthogonal collocation, tau, and Galerkin methods, the least-squares method does not produce the same favorable results. On the other hand, the least-squares method with a direct minimization solver is capable of producing equally accurate results of the model equations as the orthogonal collocation, tau, and Galerkin methods. Compared to the conventional least-squares method, the direct minimization formulation is better conditioned but does not produce a symmetric positive-definite system matrix.

SU‐E‐T‐195; An Algorithm for the Reconstruction of An Entrance Beam Fluence Using a Simulated Exit Phantom Fluence

Purpose: To design and test an algorithm for reconstructing entrance fluence from an exit fluence through a cylindrical phantom using fluence kernels generated in the BEAMnrc monte carlo code. This work may be extended to replace the simulated fluence with direct measurements on an EPID where dosimetry data from treatment can be acquired and compared with direct measures of fluence from devices used commonly for IMRT verification. Method and Materials: A parameter space array of monte carlo fluence kernels is calculated for each voxel in a 256x256 pixel grid measuring across 16.384cm2 at 60cm SPD, which corresponds to a 40.96cm2 imager at 150cm SPD. An array of coefficients of parameter space elements is then used to calculate a fluence at 150cm SPD, and an iterative minimization solver is configured to minimize the sum of the square of the differences between the calculated fluence and the measured fluence. Results: Comparison between two IMRT beam sequences simulated using the same BEAMnrc model used to generate the parameter space array were performed. The iterative solver was allowed to run for 15 minutes on a 128 core cluster on each phase space generated. A comparison of directly simulated entrance fluence and entrance fluence calculated by the proposed algorithm was performed, with a comparison criterion of the percentage of points in the fluence array with a threshold of 10% of maximum intensity, within 1% of the simulated fluence intensity; passing rates of 94.5% and 97.9% were achieved. Conclusion: These results demonstrate that the algorithm is capable of reconstructing entrance fluence from simulated exit fluence within a reasonable tolerance for verification of delivery. Additionally, patient and beam specific parameter space arrays may be generated, allowing live measurements during treatment to be used for the reconstruction of entrance fluence with a high degree of accuracy.

Estimating the fundamental matrix under pure translation and radial distortion

Four algorithms for estimating the fundamental matrix and radial distortion for the case that the images are related by a pure camera translation are proposed. This restricted class of motions is important in robotics and industrial applications. It is assumed that the radial distortions can be modeled by the division model. First, a linear 3-point minimal solver is derived, which is essential to drive a RANSAC algorithm. The minimal solver is then extended to an overdetermined linear solver, which is essential to compute starting values for the third algorithm, a Gold Standard solver that returns the maximum likelihood estimate fundamental matrix and radial distortion. The three algorithms are combined into a RANSAC algorithm that determines the fundamental matrix and radial distortion automatically. An experimental evaluation determines the characteristics of the algorithms and shows the high accuracy, precision, and robustness that can be achieved. Furthermore, the evaluation shows that using algorithms that determine the fundamental matrix and radial distortion for general camera motions are not a viable alternative for the case of pure camera translations because they are significantly slower and less robust than the algorithms proposed here.

Evaluating bound-constrained minimization software

Bound-constrained minimization is a subject of active research. To assess the performance of existent solvers, numerical evaluations and comparisons are carried on. Arbitrary decisions that may have a crucial effect on the conclusions of numerical experiments are highlighted in the present work. As a result, a detailed evaluation based on performance profiles is applied to the comparison of bound-constrained minimization solvers. Extensive numerical results are presented and analyzed.

An Inexact Restoration Package for Bilevel Programming Problems

Bilevel programming problems are a class of optimization problems with hierarchical structure where one of the con-straints is also an optimization problem. Inexact restoration methods were introduced for solving nonlinear programming problems a few years ago. They generate a sequence of, generally, infeasible iterates with intermediate iterations that consist of inexactly restored points. In this paper we present a software environment for solving bilevel program-ming problems using an inexact restoration technique without replacing the lower level problem by its KKT optimality conditions. With this strategy we maintain the minimization structure of the lower level problem and avoid spurious solutions. The environment is a user-friendly set of Fortran 90 modules which is easily and highly configurable. It is prepared to use two well-tested minimization solvers and different formulations in one of the minimization subproblems. We validate our implementation using a set of test problems from the literature, comparing different formulations and the use of the minimization solvers.

Egomotion Estimation Using Assorted Features

We propose a novel minimal solver for recovering camera motion across two views of a calibrated stereo rig. The algorithm can handle any assorted combination of point and line features across the four images and facilitates a visual odometry pipeline that is enhanced by well-localized and reliably-tracked line features while retaining the well-known advantages of point features. The mathematical framework of our method is based on trifocal tensor geometry and a quaternion representation of rotation matrices. A simple polynomial system is developed from which camera motion parameters may be extracted more robustly in the presence of severe noise, as compared to the conventionally employed direct linear/subspace solutions. This is demonstrated with extensive experiments and comparisons against the 3-point and line-sfm algorithms.

Evolution strategies based adaptive Lp LS-SVM

Not only different databases but two classes of data within a database can also have different data structures. SVM and LS-SVM typically minimize the empirical ϕ-risk; regularized versions subject to fixed penalty ( L 2 or L 1 penalty) are non-adaptive since their penalty forms are pre-determined. They often perform well only for certain types of situations. For example, LS-SVM with L 2 penalty is not preferred if the underlying model is sparse. This paper proposes an adaptive penalty learning procedure called evolution strategies (ES) based adaptive L p least squares support vector machine (ES-based L p LS-SVM) to address the above issue. By introducing multiple kernels, a L p penalty based nonlinear objective function is derived. The iterative re-weighted minimal solver (IRMS) algorithm is used to solve the nonlinear function. Then evolution strategies (ES) is used to solve the multi-parameters optimization problem. Penalty parameter p, kernel and regularized parameters are adaptively selected by the proposed ES-based algorithm in the process of training the data, which makes it easier to achieve the optimal solution. Numerical experiments are conducted on two artificial data sets and six real world data sets. The experiment results show that the proposed procedure offer better generalization performance than the standard SVM, the LS-SVM and other improved algorithms.

Fast electromagnetic simulation algorithm based on hierarchical and curvilinear finite elements

In this article, hierarchical finite element method (FEM) based on curvilinear elements is used to study three-dimensional (3D) electromagnetic problems. The incomplete Cholesky preconditioned loose generalized minimal residual solver (LGMRES) based on decomposition algorithm (DA) is applied to solve the FEM equations. The efficiency of the proposed approach is studied on several numerical problems. Numerical results demonstrate that the DA-based LGMRES is especially effective when the hierarchical FEM is used to solve 3D electromagnetic problems. © 2010 Wiley Periodicals, Inc. Microwave Opt Technol Lett 53:324–331, 2011; View this article online at wileyonlinelibrary.com. DOI 10.1002/mop.25714

A Preconditioned Minimal Residual Solver for a Class of Linear Operator Equations

Abstract We consider the class of linear operator equations with operators admitting self-adjoint positive definite and m-accretive splitting (SAS). This splitting leads to an ADI-like iterative method which is equivalent to a fixed point problem where the operator is a 2 by 2 matrix of operators. An infinite dimensional adaptation of a minimal residual algorithm with Symmetric Gauss-Seidel and polynomial preconditioning is then applied to solve the resulting matrix operator equation. Theoretical analysis shows the convergence of the methods, and upper bounds for the decrease rate of the residual are derived. The convergence of the methods is numerically illustrated with the example of the neutron transport problem in 2-D geometry.