Levente Hunyadi

Constrained quadratic errors-in-variables fitting

We propose an estimation method to fit conics and quadrics to data in the context of errors-in-variables where the fit is subject to constraints. The proposed algorithm is based on algebraic distance minimization and consists of solving a few generalized eigenvalue (or singular value) problems and is not iterative. Nonetheless, the algorithm produces accurate estimates, close to those obtained with maximum likelihood, while the constraints are also guaranteed to be satisfied. Important special cases, fitting ellipses, hyperbolas, parabolas, and ellipsoids to noisy data are discussed.

Identifying Unstructured Systems in the Errors-in-Variables Context

Abstract We present an approach to identifying static systems from noisy data where the system model is a composition of simple functions each of which can be recast into a form linear both in data and in parameters, e.g. a set of straight lines and quadratic curves in two dimensions. The proposed algorithm is an optimization method that alternates between a parametric estimation step that estimates curve or surface parameters based on a set of points and a data reassignment step that maps points to a most feasible curve or surface.

Fitting a Model to Noisy Data Using Low-Order Implicit Curves and Surfaces

Fitting a compact model to measured data that captures the underlying relationship is a fundamental task in computer graphics and computer-aided design. Low-order implicit curves and surfaces are a practical choice in grasping this relationship since they are closed under several geometric operations (e.g. intersection, union, offset) while they offer a higher degree of smoothness than their parametric counterparts, and may be preferred especially if the object under study itself is a composition of geometric shapes. We present a method based on a blend of iterative maximum likelihood approximation of linear and quadratic curves and surfaces (with constraints), and of an alternating optimization scheme in the flavor of the standard algorithm for k-means. The algorithm alternates between two steps: (1) fitting a set of linear and quadratic curves and surfaces to previously identified groups of noisy data points, and (2) identifying new groups by assignment to the most feasible shape. Non-iterative direct methods are proposed to seed the maximum likelihood estimator with initial parameter values.

Implicit model fitting to an unorganized set of points

Constructing a computer model from a mass of unorganized coordinate data acquired of a physical object is a frequent problem in engineering. Unfortunately, data are usually observed with noise due to surface attributes of the physical object, impact of the environment and uncertainty of the measuring device. The aim is thus to reconstruct the physical model in a way as to minimize the misfit between the reconstructed model and the true object. We present an approach that alloys clustering and generalized total least squares regression to detect groups in observations and estimate local parameters over naturally delineated domains in a noisy context. The global model is an aggregate of local models, each of which is described by a primitive, which evolves during an iterative refinement process. A resampling step with random initial assignment is added to minimize the probability that the method converges to a suboptimal solution.

Identification of errors-in-variables systems using data clustering

The fact that simultaneous estimation of process and noise parameters using second-order properties is not possible under fairly general conditions is a well-known result in literature in the context of dynamic errors-in-variables systems. In order to make systems identifiable, additional restrictions have to be imposed. One possibility is that data are separable into two distinct clusters, which can be independently identified and the estimated parameters compared. This paper outlines an approach to system identification using principal component analysis to cluster data and the generalized Koopmans-Levin method to derive parameter estimates.

Prosper: a framework for extending prolog applications with a web interface

Clear separation of presentation and code-behind, declarative use of visual control elements and a supportive background framework to automate recurring tasks are fundamental to rapid web application development. This poster presents a framework that facilitates extending Prolog applications with a web front-end. The framework relies on Prolog to the greatest possible extent, supports code re-use, and integrates easily into existing web server solutions.

MODELING BY FITTING A UNION OF POLYNOMIAL FUNCTIONS TO DATA IN AN ERRORS-IN-VARIABLES CONTEXT

We present a model construction method based on a local fitting of polynomial functions to noisy data and building the entire model as a union of regions explained by such polynomial functions. Local fitting is shown to reduce to solving a polynomial eigenvalue problem where the matrix coefficients are data covariance and approximated noise covariance matrices that capture distortion effects by noise. By defining the asymmetric distance between two points as the projection of one onto the function fitted to the neighborhood of the other, we use a best weighted cut method to find a proper partitioning of the entire set of data into feasible regions. Finally, the partitions are refined using a modified version of a k-planes algorithm.