Finite precision error analysis of Zernike moments computation schemes and a new, efficient, robust recursive algorithm

As an orthogonal moment, Zernike moment (ZM) is an attractive image feature in a number of application scenarios due to its distinguishing properties. However, we find that for digital images, the commonly used Cartesian method for ZM computation has compromised the advantages of ZMs because of their non-ideal accuracy stemming from two inherent sources of errors, i.e., the geometric error and the integral error. There exists considerable errors in image reconstruction using ZMs calculated with the Cartesian method. In this paper, we propose a polar coordinate based algorithm for the computation of ZMs, which avoids the two kinds of errors and greatly improves the accuracy of ZM computation. We present solutions to the key issues in ZM computation under polar coordinate system, including the derivation of computation formulas, the polar pixel arrangement scheme, and the interpolation-based image conversion etc. As a result, ZM-based image reconstruction can be performed much more accurately.

We present a way to construct a complete set of scaling rotation and translation invariants extract directly from Zernike moments. Zernike moment can be constructed by Radial moment. In our method in order to construct invariant Zernike moment is to achieve invariant Radial moment which is component of Zernike moment. We use matrix form to denote relationship between Radial and Zernike moment, which makes derivation more comprehensible. The translation invariant Radial moment is first introduced, for it is most complicated part of all the three invariant. Rotation and scaling invariant Radial moment is achieved by normalizing the factor caused by rotation and scaling. The form of invariant radial moment is to combine three parts of invariant. Some experiment has done to test the performance of invariance. In this experiment we take an image library containing 23,329 files which are built by translation rotation and zoom in out of one origin Latin character image. Most of the value of standard deviation ratio by mean of proposed moments is nearly 1%. In addition, retrieval experiment is to test the discrimination ability. MPEG-7 CE shape1 - Part A library is taken in this experiment. The recall rate in part A1 is 96.6% and is 100% in part A2.

Analysis of noise sensitivity of Tchebichef and Zernike moments with application to image watermarking

Chapter Two - Direct numerical and large–eddy simulation of complex turbulent flows

Discrete Chebyshev moments (due to discrete polynomial basis) do not have the discretization errors that continuous-domain Legendre and Zernike moments contain. Calculation of polynomial basis coefficients of discrete moments is generally performed using recurrence relationships. Such recurrence equations cause numerical error accumulation especially for calculation of higher-order moments and for larger image sizes, causing significant degradation of image reconstruction from these moments. A method for better image reconstruction from high orders of discrete Chebyshev moments is demonstrated. This is accomplished by calculating Chebyshev polynomial coefficients directly from their definition formulas using arbitrary precision arithmetic and by forming lookup tables from these coefficients.

The 3-D reconstruction is of great importance for measuring, inspecting, monitoring, or understanding surface characteristics. Previous active target-based vision methods provide the reconstruction with more than three images for monocular vision, usually using extra external monocular, stereo, or multiview vision to expand the measurement range. This article aims to achieve the reconstruction from the initial image of monocular vision without precalibration. The subpixel centers on the target with groups of three collinear concentric circles in the radial pattern are obtained in the region of interest (ROI) by histogram of oriented gradient (HOG)–support vector machine (SVM), grouped by electromagnetic (EM), and extracted by the Zernike moment (ZM). The intrinsic parameters of the projection of the single image are modeled by the two projections of the general-direction infinity points generated from the collinear centers of the radial pattern. Moreover, the ratio of the cross ratios of CSI and CST features with infinity points is proposed to determine moderate images with small errors for reconstruction. The mutual benchmark model is presented to enlarge the effective measurement range without extra external observation instruments. Moreover, the reconstruction is modeled and further refined in SE(3) to achieve the orthogonality, accuracy, and large measurement range, which are proved by simulation and experimental verification.

Abstract. Hydrological models are usually systems of nonlinear differential equations for which no analytical solutions exist and thus rely on numerical solutions. While some studies have investigated the relationship between numerical method choice and model error, the extent to which extreme precipitation such as that observed during hurricanes Harvey and Katrina impacts numerical error of hydrological models is still unknown. This knowledge is relevant in light of climate change, where many regions will likely experience more intense precipitation. In this experiment, a large number of hydrographs are generated with the modular modeling framework FUSE (Framework for Understanding Structural Errors), using eight numerical techniques across a variety of forcing data sets. All constructed models are conceptual and lumped. Multiple model structures, parameter sets, and initial conditions are incorporated for generality. The computational cost and numerical error associated with each hydrograph were recorded. Numerical error is assessed via root mean square error and normalized root mean square error. It was found that the root mean square error usually increases with precipitation intensity and decreases with event duration. Some numerical methods constrain errors much more effectively than others, sometimes by many orders of magnitude. Of the tested numerical methods, a second-order adaptive explicit method is found to be the most efficient because it has both a small numerical error and a low computational cost. A small literature review indicates that many popular modeling codes use numerical techniques that were suggested by this experiment to be suboptimal. We conclude that relatively large numerical errors may be common in current models, highlighting the need for robust numerical techniques, in particular in the face of increasing precipitation extremes.

Identification and Kinematic Calculation of Laser Tracker Errors

Continuous orthogonal moments, for which continuous functions are used as kernel functions, are invariant to rotation and scaling, and they have been greatly developed over the recent years. Among continuous orthogonal moments, polar harmonic Fourier moments (PHFMs) have superior performance and strong image description ability. In order to improve the performance of PHFMs in noise resistance and image reconstruction, PHFMs, which can only take integer numbers, are extended to fractional-order polar harmonic Fourier moments (FrPHFMs) in this paper. Firstly, the radial polynomials of integer-order PHFMs are modified to obtain fractional-order radial polynomials, and FrPHFMs are constructed based on the fractional-order radial polynomials; subsequently, the strong reconstruction ability, orthogonality, and geometric invariance of the proposed FrPHFMs are proven; and, finally, the performance of the proposed FrPHFMs is compared with that of integer-order PHFMs, fractional-order radial harmonic Fourier moments (FrRHFMs), fractional-order polar harmonic transforms (FrPHTs), and fractional-order Zernike moments (FrZMs). The experimental results show that the FrPHFMs constructed in this paper are superior to integer-order PHFMs and other fractional-order continuous orthogonal moments in terms of performance in image reconstruction and object recognition, as well as that the proposed FrPHFMs have strong image description ability and good stability.

Numerical smearing, ray effect, and angular false scattering in radiation transfer computation

Virtual manufacturing systems can provide useful means for products to be manufactured without the need of physical testing on the shop floor. As a result, the time and cost of part production can be decreased. There are different error sources in machine tools such as tool deflection, geometrical deviations of moving axis and thermal distortions of machine tool structures. Some of these errors can be decreased by controlling the machining process and environmental parameters. However other errors like tool deflection and geometrical errors which have a big portion of the total error, need more attention. This paper presents a virtual machining system in order to enforce dimensional, geometrical and tool deflection errors in three-axis milling operations. The system receives 21 dimensional and geometrical errors of a machine tool and machining codes of a specific part as input. The output of the system is the modified codes which will produce actual machined part in the virtual environment.

Electric energy, as an economical and clean energy, plays a significant role in the development of science and technology and the economy. The motor is the core equipment of the power station; therefore, monitoring the motor vibration and predicting time series of the bearing vibration can effectively avoid hazards such as bearing heating and reduce energy consumption. Time series forecasting methods of motor bearing vibration based on sliding window forecasting, such as CNN, LSTM, etc., have the problem of error accumulation, and the longer the time-series forecasting, the larger the error. In order to solve the problem of error accumulation caused by the conventional methods of time series forecasting of motor bearing vibration, this paper innovatively introduces Informer into time series forecasting of motor bearing vibration. Based on Transformer, Informer introduces ProbSparse self-attention and self-attention distilling, and applies random search to optimize the model parameters to reduce the error accumulation in forecasting, achieve the optimization of time and space complexity and improve the model forecasting. Comparing the forecasting results of Informer and those of other forecasting models in three publicly available datasets, it is verified that Informer has excellent performance in time series forecasting of motor bearing vibration and the forecasting results reach ∼.

To address the problems of the low accuracy of geometric error identification and incomplete identification results of the linear axis detection of computer numerical control (CNC) machine tools, a new 21-item geometric error identification method based on double ball-bar measurement was proposed. The model between the double ball-bar reading and the geometric error term in each plane was obtained according to the three-plane arc trajectory measurement. The mathematical model of geometric error components of CNC machine tools is established, and the error fitting coefficients are solved through the beetle antennae search particle swarm optimization (BAS–PSO) algorithm, in which 21 geometric errors, including roll angle errors, were identified. Experiments were performed to compare the optimization effect of the BAS–PSO and PSO and BAS and genetic particle swarm optimization (GA–PSO) algorithms. Experimental results show that the PSO algorithm is trapped in the local optimum, and the BAS–PSO is superior to the other three algorithms in terms of convergence speed and stability, has higher identification accuracy, has better optimization performance, and is suitable for identifying the geometric error coefficient of CNC machine tools. The accuracy and validity of the identification results are verified by the comparison with the results of the individual geometric errors detected through laser interferometer experiments. The identification accuracy of the double ball-bar is below 2.7 µm. The proposed identification method is inexpensive, has a short processing time, is easy to operate, and possesses a reference value for the identification and compensation of the linear axes of machine tools.

Recently, unsupervised methods for monocular visual odometry (VO), with no need for quantities of expensive labeled ground truth, have attracted much attention. However, these methods are inadequate for long-term odometry task, due to the inherent limitation of only using monocular visual data and the inability to handle the error accumulation problem. By utilizing supplemental low-cost inertial measurements, and exploiting the multi-view geometric constraint and sequential constraint, an unsupervised visual-inertial odometry framework (UnVIO) is proposed in this paper. Our method is able to predict the per-frame depth map, as well as extracting and self-adaptively fusing visual-inertial motion features from image-IMU stream to achieve long-term odometry task. A novel sliding window optimization strategy, which consists of an intra-window and an inter-window optimization, is introduced for overcoming the error accumulation and scale ambiguity problem. The intra-window optimization restrains the geometric inferences within the window through checking the photometric consistency. And the inter-window optimization checks the 3D geometric consistency and trajectory consistency among predictions of separate windows. Extensive experiments have been conducted on KITTI and Malaga datasets to demonstrate the superiority of UnVIO over other state-of-the-art VO / VIO methods. The codes are open-source.

CNC machine tools play an extremely important role in the equipment manufacturing industry. Because a machine tool has multiple geometric errors which are highly coupled, the error identification process is difficult. Most direct and indirect measurement methods do not consider the dynamic errors in the actual cutting process. This study aims to optimize the error identification process of three-axis CNC machine tools, and proposes an efficient identification and separation method for the dynamic error and the quasi-static error based on feature workpiece cutting. In this study, a stepped feature workpiece that can reflect geometric errors is designed, and the mapping relationship between the geometric error components and the features is established. After the feature workpiece is cut, it performs two tests of on-machine measurement and CMM calibration sequentially. It can realize the identification and separation of 7 geometric errors and corresponding dynamic errors based on the two sets of test data. This method takes the actual cutting workpiece as the error source and can identify and separate the dynamic error and the quasi-static error simultaneously. Thus, the measurement efficiency of this method is higher than direct measurements. The geometric error identification result is highly consistent with the laser interferometer measurement, which proves the method in this study is feasible, efficient, and accurate.