Estimation of an unknown cartographic projection and its parameters from the map

This article presents a new off-line method for the detection, analysis and estimation of an unknown cartographic projection and its parameters from a map. Several invariants are used to construct the objective function ? that describes the relationship between the 0D, 1D, and 2D entities on the analyzed and reference maps. It is minimized using the Nelder-Mead downhill simplex algorithm. A simplified and computationally cheaper version of the objective function ? involving only 0D elements is also presented. The following parameters are estimated: a map projection type, a map projection aspect given by the meta pole K coordinates [φ _k , λ _k ], a true parallel latitude φ ₀, central meridian longitude λ ₀, a map scale, and a map rotation. Before the analysis, incorrectly drawn elements on the map can be detected and removed using the IRLS. Also introduced is a new method for computing the L ₂ distance between the turning functions Θ₁, Θ₂ of the corresponding faces using dynamic programming. Our approach may be used to improve early map georeferencing; it can also be utilized in studies of national cartographic heritage or land use applications. The results are presented both for the real cartographic data, representing early maps from the David Rumsay Map Collection, and for the synthetic tests.     

Defocus map estimation from a single image

In this paper, we address the challenging problem of recovering the defocus map from a single image. We present a simple yet effective approach to estimate the amount of spatially varying defocus blur at edge locations. The input defocused image is re-blurred using a Gaussian kernel and the defocus blur amount can be obtained from the ratio between the gradients of input and re-blurred images. By propagating the blur amount at edge locations to the entire image, a full defocus map can be obtained. Experimental results on synthetic and real images demonstrate the effectiveness of our method in providing a reliable estimation of the defocus map.     

Optimum control of an unknown linear plant using Bayesian estimation of the error

In this paper, the theory of the optimum control of an unknown linear plant is discussed. Rather than try to estimate the coeffcients of the plant, the future error as defined by a quadratic measure is estimated using a Bayesian estimator. In this manner, better system performance can be expected since the effect of estimation errors on the estimator is obtained. The optimum control signal is then obtained which minimizes the estimated future error. It is shown that it is a linear function of the present output of the system. In the final section, a necessary and sufficient condition is obtained for the convergence of this procedure to the optimum system obtained with known coefficients.     

Comparison of adaptive methods for function estimation from samples

The problem of estimating an unknown function from a finite number of noisy data points has fundamental importance for many applications. This problem has been studied in statistics, applied mathematics, engineering, artificial intelligence, and, more recently, in the fields of artificial neural networks, fuzzy systems, and genetic optimization. In spite of many papers describing individual methods, very little is known about the comparative predictive (generalization) performance of various methods. We discuss subjective and objective factors contributing to the difficult problem of meaningful comparisons. We also describe a pragmatic framework for comparisons between various methods, and present a detailed comparison study comprising several thousand individual experiments. Our approach to comparisons is biased toward general (nonexpert) users. Our study uses six representative methods described using a common taxonomy. Comparisons performed on artificial data sets provide some insights on applicability of various methods. No single method proved to be the best, since a method's performance depends significantly on the type of the target function, and on the properties of training data.     

The optimal projection equations for reduced-order state estimation

First-order necessary conditions for optimal, steady-state, reduced-order state estimation for a linear, time-invariant plant in the presence of correlated disturbance and nonsingular measurement noise are derived in a new and highly simplified form. In contrast to the lone matrix Riccati equation arising in the full-order (Kalman filter) case, the optimal steady-state reduced-order estimator is characterized by three matrix equations (one modified Riccati equation and two modified Lyapunov equations) coupled by a projection whose rank is precisely equal to the order of the estimator and which determines the optimal estimator gains. This coupling is a graphic reminder of the suboptimality of proposed approaches involving either model reduction followed by "full-order" estimator design or full-order estimator design followed by estimator-reduction techniques. The results given here complement recently obtained results which characterize the optimal reduced-order model by means of a pair of coupled modified Lyapunov equations [7] and the optimal fixed-order dynamic compensator by means of a coupled system of two modified Riceati equations and two modified Lyapunov equations [6].     

A Hough Transform with projection for velocity estimation

An extended Hough Transform is proposed to estimate the velocity of Regions Of Interest (ROI). Each ROI is first characterized by a spatio-temporal point (centroids for example). The velocity of each point is then computed by searching the line that passes through this point and fits at best the other ones. In order to avoid high dimension problems (a three-dimensional (3D) spatio-temporal line is defined by four parameters), the data are projected on several planes. The velocities are deduced by combining the results obtained on each plane using weighted least squares. The robustness of this method called “Hough Transform with Projection (HTP)”, compared to the Standard Hough Transform (applied on 3D spatio-temporal data) is demonstrated. Finally, experimental results on synthetic and real sequences validate the approach.     

Global estimation of n unknown frequencies

Given a signal which is available for measurements and consists of the sum of a bias and n sinusoids with unknown amplitudes, frequencies and phases, a dynamic algorithm is designed to recover online the unknown values of the frequencies, for any initial condition and any value of frequencies, amplitudes and phases.     

A gradient projection algorithm for relaxation methods

We consider a particular problem which arises when apply-ing the method of gradient projection for solving constrained optimiza-tion and finite dimensional variational inequalities on the convex set formed by the convex hull of the standard basis unit vectors. The method is especially important for relaxation labeling techniques applied to problems in artificial intelligence. Zoutendijk's method for finding feasible directions, which is relatively complicated in general situations, yields a very simple finite algorithm for this problem. We present an extremely simple algorithm for performing the gradient projection and an independent verification of its correctness.     

Optimization of projection methods for solving ill-posed problems

In this paper we propose a modification of the projection scheme for solving ill-posed problems. We show that this modification allows to obtain the best possible order to accuracy of Tikhonov Regularization using an amount of information which is far less than for the standard projection technique.     

State and unknown input estimation for linear discrete-time systems

In this paper, a state and unknown input delayed estimator is designed for discrete-time linear systems even if some well-known matching condition does not hold. This result is obtained using a constructive algorithm that analyzes the state observability and the left invertibility of the systems with unknown inputs and that provides a suitable canonical transformation for the design of the estimator.     

Extension of minimum variance estimation for systems with unknown inputs

In this paper, we address the problem of minimum variance estimation for discrete-time time-varying stochastic systems with unknown inputs. The objective is to construct an optimal filter in the general case where the unknown inputs affect both the stochastic model and the outputs. It extends the results of Darouach and Zasadzinski (Automatica 33 (1997) 717) where the unknown inputs are only present in the model. The main difficulty in treating this problem lies in the fact that the estimation error is correlated with the systems noises, this fact leads generally to suboptimal filters. Necessary and sufficient conditions for the unbiasedness of this filter are established. Then conditions under which the estimation error and the system noises are uncorrelated are presented, and an optimal estimator and a predictor filters are derived. Sufficient conditions for the existence of these filters are given and sufficient conditions for their stability are obtained for the time-invariant case. A numerical example is given in order to illustrate the proposed method.     

Discrete gradient methods and linear projection methods for preserving a conserved quantity of stochastic differential equations

Numerical methods preserving a conserved quantity for stochastic differential equations are considered. A class of discrete gradient methods based on the skew-gradient form is constructed, and the sufficient condition of convergence order 1 in the mean-square sense is given. Then a class of linear projection methods is constructed. The relationship of the two classes of methods for preserving a conserved quantity is proved, which is, the constructed linear projection methods can be considered as a subset of the constructed discrete gradient methods. Numerical experiments verify our theory and show the efficiency of proposed numerical methods.     

Advanced point-mass method for nonlinear state estimation

State estimation of discrete-time nonlinear non-Gaussian stochastic systems by point-mass approach, which is based on discretization of state space by a regular grid and numerical solution of Bayesian recursive relations, is treated. The stress is laid to grid design which is crucial for estimator quality and significantly affects the computational demands of the estimator. Boundary-based grid design, thrifty convolution, and multigrid design with grid splitting and merging are proposed. The main advantages of these techniques are nonnegligible support delimitation, time-saving computation of convolution, and effective processing of multimodal probability density functions, respectively. The techniques are involved into the basic point-mass approach and a new general-purpose, more sophisticated point-mass algorithm is designed. Computational demands and estimation quality of the designed algorithm are presented and compared with the particle filter in a numerical example.     

New methods for self-organising map visual analysis

Self-organising maps (SOMs) have been used effectively in the visualisation and analysis of multidimensional data, with applications in exploratory data analysis (EDA) and data mining. We present three new techniques for performing visual analysis of SOMs. The first is a computationally light contraction method, closely related to the SOMs training algorithm, designed to facilitate cluster and trajectory analysis. The second is an enhanced geometric interpolation method, related to multidimensional scaling, which forms a mapping from the input space onto the map. Finally, we propose the explicit representation of graphs like the SOMs induced Delaunay triangulation for topology preservation and cluster analysis. The new methods provide an enhanced interpretation of the information contained in an SOM, leading to a better understanding of the data distributions with which they are trained, as well as providing insight into the maps formation.     

Numerical integration software for projection and projection-grid methods

Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 181–184, January–February, 1994.     

Confidence estimation methods for neural networks: a practicalcomparison

Feedforward neural networks, particularly multilayer perceptrons, are widely used in regression and classification tasks. A reliable and practical measure of prediction confidence is essential. In this work three alternative approaches to prediction confidence estimation are presented and compared. The three methods are the maximum likelihood, approximate Bayesian, and the bootstrap technique. We consider prediction uncertainty owing to both data noise and model parameter misspecification. The methods are tested on a number of controlled artificial problems and a real, industrial regression application, the prediction of paper "curl". Confidence estimation performance is assessed by calculating the mean and standard deviation of the prediction interval coverage probability. We show that treating data noise variance as a function of the inputs is appropriate for the curl prediction task. Moreover, we show that the mean coverage probability can only gauge confidence estimation performance as an average over the input space, i.e., global performance and that the standard deviation of the coverage is unreliable as a measure of local performance. The approximate Bayesian approach is found to perform better in terms of global performance.     

Unbiased minimum-variance state estimation for linear systems with unknown input

The problem of state estimation for a linear system with unknown input, which affects both the system and the output, is discussed in this paper. A recursive optimal filter with global optimality in the sense of unbiased minimum variance over all linear unbiased estimators, is provided. The necessary and sufficient condition for the convergence and stability is also given, which is milder than existing approaches.     

Rapid estimation and detection scheme for unknown discretized rectangular inputs

An algorithm to detect unknown discretized rectangular inputs and to estimate the state simultaneously, has been described in this note. The discretized rectangular input is represented by a sequence of equal magnitude inputs to the state. Four alternate hypotheses are formed at every stage of observation and Bayes risks are compared to determine the correct one. A minimum variance estimate of the inputs is used to improve the estimate of the state. Computer results from a numerical example are shown to demonstrate that a possible divergence of the Kalman filter can be prevented by incorporating the detection scheme.     

Residuals of refined projection methods for large matrix eigenproblems

In terms of the deviation of the eigenvector ϕ from a projection subspace E, a priori error bounds are established for the residual norm of an approximate eigenpair obtained by a refined projection method. It is shown that the residual converges to zero as the deviation tends to zero. Finally, how to efficiently compute refined Ritz vectors is discussed in the refined symmetric Lanczos method.