Two-dimensional Affine Transformations Research Articles

Non-negative least-squares variance and covariance component estimation using the positive-valued function for errors-in-variables models

Abstract Although (co)variance component estimation has been widely applied in the errors-in-variables (EIV) model, the occurrence of negative variance components is still a major issue in the estimated variance components. This problem may be due to the following unfavorable factors: 1) unreasonable selection of initial variance values; 2) low redundancy in the EIV functional model; 3) improper design in the EIV stochastic model, and 4) other data quality problems. Many attempts have been made to prevent the appearance of negative variance components. In this contribution, a novel and efficient non-negative least-squares variance component estimation using the PVF (PVF-NLS-VCE) is introduced, which can simultaneously estimate the unknown (co)variance components in the EIV stochastic model and the parameters in the EIV functional model. Its principle is to implicitly impose a non-negative constraint by replacing the variance component with the positive-valued function (PVF) whose range is the set of positive real numbers. Two numerical examples using real and simulated data are presented. The numerical results of linear regression are identical to those obtained based on least-squares variance component estimation (LS-VCE) with positive initial values of variance components. The numerical results of two-dimensional affine transformation are shown to prevent negative variance components and precede those obtained by LS-VCE with a negative initial value of variance component. Both numerical examples verify the effectiveness of the PVF-NLS-VCE method whether the initial values of variance components are positive or negative. The proposed PVF-NLS-VCE is a simple, convenient and flexible method to achieve the non-negative estimates of variance components, which can reduce sensitivity to initial value selection and automatically guarantee a non-negative definite covariance matrix.

Assessment of ICESat-2’s Horizontal Accuracy Using an Iterative Matching Method Based on High-Accuracy Terrains

The Ice, Cloud, and Land Elevation Satellite-2 (ICESat-2), launched in September 2018, has been widely used in forestry and surveying. A high-accuracy digital elevation model (DEM)/digital surface model (DSM) for terrain matching can effectively evaluate the ICESat-2 design requirements and provide essential data support for further study. The conventional terrain-matching methods regard the laser ground track as a whole, ignoring the individual differences caused by the interaction of photons during flight. Therefore, a novel terrain-matching method using a two-dimensional affine transformation model was proposed to describe the deformation of laser tracks. The least-square optimizes the model parameters with the high-accuracy terrain data to obtain the best matching result. The results in McMurdo Dry Valley (MDV), Antarctica, and Zhengzhou (ZZ), China, demonstrate that the proposed method can verify geolocation accuracy and indicate that the average horizontal accuracy of ICESat-2 V5 data is about 3.86 m in MDV and 4.67 m in ZZ. It shows that ICESat-2 has good positioning accuracy, even in mountainous areas with complex terrain. Additionally, the random forest (RF) model was calculated to analyze the influence of four factors on geographic location accuracy. The slope and signal-to-noise ratio (SNR) are considered the crucial factors affecting the accuracy of ICESat-2 data.

Total Msplit estimation

Msplit estimation is a method that enables the estimation of mutually competing versions of parameters in functional observation models. In the presented study, the classical functional models found in it are replaced by errors-in-variables (EIV) models. Similar to the weighted total least-squares (WTLS) method, the random components of these models were assigned covariance matrix models. Thus, the proposed method, named Total Msplit (TMsplit) estimation, corresponds to the basic rules of WTLS. TMsplit estimation objective function is constructed using the components of squared Msplit and WTLS estimation objective functions. The TMsplit estimation algorithm is based on the Gauss–Newton method that is applied using a linear approximation of EIV models. The basic properties of the method are presented using examples of the estimation of regression line parameters and the estimation of parameters in a two-dimensional affine transformation.

Virtual-real fusion with geometric consistency based on two-dimensional affine transformation

Virtual-real fusion with geometric consistency based on two-dimensional affine transformation

Iteratively reweighted total least squares for PEIV model

The weighted total least-squares (WTLS) estimate for the partial EIV (PEIV) model is very susceptible to outliers. In view of outliers in observations and coefficient matrix, a robust estimate of WTLS which makes using of iteratively reweighted technique (IRTLS) for the PEIV model is developed by combining a new two-step iterated algorithm of the WTLS estimate with the sensitivity-analysis based on the robust M-estimation. The uniformly most powerful test statistics are constructed to determine the down-weighting factor and the variance of unit weight is estimated by least median squares (LMS) method possessing high break-down point. Depended on the PEIV model, two different down-weighting schemes are presented. In the first scheme down-weighting is only implemented for the coefficient matrix and not for observations when the elements of the coefficient matrix are estimated, and the second scheme is contrary. The most attractive aspect is that the suggested computational formulae are the same with the traditional robust least-squares (LS) methods. The two-dimensional affine transformation and liner fitting example are analyzed, and some comparisons are performed with different available weight functions. The proposed approach (Scheme 1) proves to be a powerful tool in detecting outliers for the PEIV model.

Trajectory control and image encryption using affine transformation of lorenz system

This paper presents a generalization of chaotic systems using two-dimensional affine transformations with six introduced parameters to achieve scaling, reflection, rotation, translation and/or shearing. Hence, the location of the strange attractor in space can be controlled without changing its chaotic dynamics. In addition, the embedded parameters enhance the randomness and sensitivity of the system and control its response. This approach overpasses performing the transformations as post-processing stages by applying them on the resulting time series. Trajectory control through dynamic parameters is demonstrated. Simulation results validate the proposed analysis for both the simplest and Lorenz chaotic systems. An image encryption scheme is implemented using transformed Lorenz system resulting in a more secure encryption scheme in comparison to Lorenz and other recent related works. The scheme exhibits good performance when assessed using the PRNG properties, encrypted image histogram and its uniformity through chi square test, pixel correlation, Mean Squared Error (MSE), entropy, Peak Signal-to-Noise Ratio (PSNR), the National Institute of Standards & Technology (NIST) test, key space, key sensitivity, resistance to differential, ciphertext-only, known plaintext, and chosen plaintext attacks, robustness against noise and computation time.

Korišćenje afine transformacija za generisanje meta u simulatoru za obuku u rukovanju vatrenim oružjem

The paper proposes a conversion of plane shooting targets using twodimensional affine transformations. It is assumed that affine transformations will be useful for target generation in firearms training simulators (FATS). The structure and the matrix formation algorithm of target vertex matrices are described as well as transformation matrices and output matrices. Vertex matrices were formed for twelve plane shooting targets. Transformation matrices 3×3 in size were used for conversions. Mathematical modelling and computer simulation of software-configurable plane shooting targets for FATS were performed in the Mathcad v.15 software.

OUTLIER DETECTION IN PARTIAL ERRORS-IN-VARIABLES MODEL

The weighed total least square (WTLS) estimate is very sensitive to the outliers in the partial EIV model. A new procedure for detecting outliers based on the data-snooping is presented in this paper. Firstly, a two-step iterated method of computing the WTLS estimates for the partial EIV model based on the standard LS theory is proposed. Secondly, the corresponding w-test statistics are constructed to detect outliers while the observations and coefficient matrix are contaminated with outliers, and a specific algorithm for detecting outliers is suggested. When the variance factor is unknown, it may be estimated by the least median squares (LMS) method. At last, the simulated data and real data about two-dimensional affine transformation are analyzed. The numerical results show that the new test procedure is able to judge that the outliers locate in x component, y component or both components in coordinates while the observations and coefficient matrix are contaminated with outliers

On Falconer's formula for the generalised Rényi dimension of a self-affine measure

We investigate a formula of K. Falconer which describes the typical value of the generalised R´enyi dimension, or generalised q-dimension, of a selfaffine measure in terms of the linear components of the affinities. We show that in contrast to a related formula for the Hausdorff dimension of a typical self-affine set, the value of the generalised q-dimension predicted by Falconer’s formula varies discontinuously as the linear parts of the affinities are changed. Conditionally on a conjecture of J. Bochi and B. Fayad, we show that the value predicted by this formula for pairs of two-dimensional affine transformations is discontinuous on a set of positive Lebesgue measure. These discontinuities derive from discontinuities of the lower spectral radius which were previously observed by the author and J. Bochi.

Iterative algorithm for weighted total least squares adjustment

In this contribution, an iterative algorithm is developed for parameter estimation in a nonlinear measurement error model y−e = (A−EA)x, which is based on the complete description of the variance–covariance matrices of the observation errors e and of the coefficient matrix errors EA without any restriction, e.g. in the case that there are correlations among observations. This paper derives the weighted total least squares solution without applying Lagrange multipliers in a straightforward manner. The algorithm is simple in the concept, easy in the implementation, and fast in the convergence. The final exact solution can be achieved through iteration. Based on the similarity between the proposed algorithm and the ordinary least squares method, the estimate for the covariance matrix of the unknown parameters can be analogously computed by using the error propagation law. The efficacy of the proposed WTLS algorithm is demonstrated by solving three WTLS problems, i.e. a linear regression model, a planar similarity transformation and two-dimensional affine transformation in the case of diagonal and fully populated covariance matrices in both start and transformed coordinate systems.

Vision-Based Sensing in Dynamic Tests

The availability of a suitable data acquisition sensor network is a key implementation issue to link models with real world structures. Non-contact displacement sensors should be preferred since they do not change the system properties. A two-dimensional vision-based displacement measurement sensor is the focus of this contribution. In particular, the perspective distortion introduced by the angle between the optic axis of the camera and the normal to the plane in which the structural system deforms is considered. A two-dimensional affine transformation is utilized to eliminate the distortion from the recorded to the distortion-free image. The results of a laboratory experiment show the potential of the proposed approach.

On undecidability bounds for matrix decision problems

In this paper we consider several reachability problems such as vector reachability, membership in matrix semigroups and reachability problems in piecewise linear maps. Since all of these questions are undecidable in general, we work on lowering the bounds for undecidability. In particular, we show an elementary proof of undecidability of the reachability problem for a set of 5 two-dimensional affine transformations. Then, using a modified version of a standard technique, we also prove that the vector reachability problem is undecidable for two (rational) matrices in dimension 11. The above result can be used to show that the system of piecewise linear functions of dimension 12 with only two intervals has an undecidable set-to-point reachability problem. We also show that the “zero in the upper right corner” problem is undecidable for two integral matrices of dimension 18 lowering the bound from 23.

Estimation of Two-Dimensional Affine Transformations Through Polar Curve Matching and Its Application to Image Mosaicking and Remote-Sensing Data Registration

This paper presents a new and effective method for estimating two-dimensional affine transformations and its application to image registration. The method is based on matching polar curves obtained from the radial projections of the image energies, defined as the squared magnitudes of their Fourier transforms. Such matching is formulated as a simple minimization problem whose optimal solution is found with the Levenberg-Marquardt algorithm. The analysis of affine transformations in the frequency domain exploits the well-known property whereby the translational displacement in this domain can be factored out and separately estimated through phase correlation after the four remaining degrees of freedom of the affine warping have been determined. Another important contribution of this paper, emphasized through one example of image mosaicking and one example of remote sensing image registration, consists in showing that affine motion can be accurately estimated by applying our algorithm to the shapes of macrofeatures extracted from the images to register. The excellent performance of the algorithm is also shown through a synthetic example of motion estimation and its comparison with another standard registration technique.

Effect of the perturbed correlation method for optical character recognition

In order to enhance the ability of correlation method in character recognition, we introduce the perturbation where all six parameters of a two-dimensional affine transformation are changed slightly from the original value, the correlation for all cases of the parameter set are tried, and the best correlation is taken as the final result. We used reciprocal directional features as the feature representation for the correlation, and it was applied to the handwritten Kanji character database ETL8b2. As a result of the experiment, the recognition accuracy was improved by 5%, and the maximum correct recognition rate was 97.4%. The problem to be solved is the enormous amount of computation that the procedure requires.

Expanders obtained from affine transformations

A bipartite graphG=(U, V, E) is an (n, k, δ, α) expander if |U|=|V|=n, |E|≦kn, and for anyX⊆U with |X|≦αn, |Γ G (X)|≧(1+δ(1−|X|/n)) |X|, whereΓ G (X) is the set of nodes inV connected to nodes inX with edges inE. We show, using relatively elementary analysis in linear algebra, that the problem of estimating the coefficientδ of a bipartite graph is reduced to that of estimating the second largest eigenvalue of a matrix related to the graph. In particular, we consider the case where the bipartite graphs are defined from affine transformations, and obtain some general results on estimating the eigenvalues of the matrix by using the discrete Fourier transform. These results are then used to estimate the expanding coefficients of bipartite graphs obtained from two-dimensional affine transformations and those obtained from one-dimensional ones.