Performance Bounds for Parameter Estimation under Misspecified Models: Fundamental findings and applications

Inferring information from a set of acquired data is the main objective of any signal processing (SP) method. In particular, the common problem of estimating the value of a vector of parameters from a set of noisy measurements is at the core of a plethora of scientific and technological advances in the last decades; for example, wireless communications, radar and sonar, biomedicine, image processing, and seismology, just to name a few. Developing an estimation algorithm often begins by assuming a statistical model for the measured data, i.e. a probability density function (pdf) which if correct, fully characterizes the behaviour of the collected data/measurements. Experience with real data, however, often exposes the limitations of any assumed data model since modelling errors at some level are always present. Consequently, the true data model and the model assumed to derive the estimation algorithm could differ. When this happens, the model is said to be mismatched or misspecified. Therefore, understanding the possible performance loss or regret that an estimation algorithm could experience under model misspecification is of crucial importance for any SP practitioner. Further, understanding the limits on the performance of any estimator subject to model misspecification is of practical interest. Motivated by the widespread and practical need to assess the performance of a mismatched estimator, the goal of this paper is to help to bring attention to the main theoretical findings on estimation theory, and in particular on lower bounds under model misspecification, that have been published in the statistical and econometrical literature in the last fifty years. Secondly, some applications are discussed to illustrate the broad range of areas and problems to which this framework extends, and consequently the numerous opportunities available for SP researchers.Comment: To appear in the IEEE Signal Processing Magazin

Estimation of longitudinal stability and control derivatives for an icing research aircraft from flight data

The results of applying a modified stepwise regression algorithm and a maximum likelihood algorithm to flight data from a twin-engine commuter-class icing research aircraft are presented. The results are in the form of body-axis stability and control derivatives related to the short-period, longitudinal motion of the aircraft. Data were analyzed for the baseline (uniced) and for the airplane with an artificial glaze ice shape attached to the leading edge of the horizontal tail. The results are discussed as to the accuracy of the derivative estimates and the difference between the derivative values found for the baseline and the iced airplane. Additional comparisons were made between the maximum likelihood results and the modified stepwise regression results with causes for any discrepancies postulated

Cramer-Rao bounds for nonparametric surface reconstruction from range data

technical reportThe Cramer-Rao error bound provides a fundamental limit on the expected performance of a statistical estimator. The error bound depends on the general properties of the system, but not on the specific properties of the estimator or the solution. The Cramer-Rao error bound has been applied to scalar- and vector-valued estimators and recently to parametric shape estimators. However, nonparametric, low-level surface representations are an important important tool in 3D reconstruction, and are particularly useful for representing complex scenes with arbitrary shapes and topologies. This paper presents a generalization of the Cramer-Rao error bound to nonparametric shape estimators. Specifically, we derive the error bound for the full 3D reconstruction of scenes from multiple range images

On the accuracy of solving confluent Prony systems

In this paper we consider several nonlinear systems of algebraic equations which can be called "Prony-type". These systems arise in various reconstruction problems in several branches of theoretical and applied mathematics, such as frequency estimation and nonlinear Fourier inversion. Consequently, the question of stability of solution with respect to errors in the right-hand side becomes critical for the success of any particular application. We investigate the question of "maximal possible accuracy" of solving Prony-type systems, putting stress on the "local" behavior which approximates situations with low absolute measurement error. The accuracy estimates are formulated in very simple geometric terms, shedding some light on the structure of the problem. Numerical tests suggest that "global" solution techniques such as Prony's algorithm and ESPRIT method are suboptimal when compared to this theoretical "best local" behavior

Range Precision of LADAR Systems

A key application of Laser Detection and Ranging (LADAR) systems is measurement of range to a target. Many modern LADAR systems are capable of transmitting laser pulses that are less than a few nanoseconds in duration. These short-duration pulses provide excellent range precision. However, randomness in the detected laser signals places limits on the precision. The goal of this dissertation is to quantify the range precision limits of LADAR systems. The randomness in the time between photon arrivals, which is called shot noise, is discussed in depth. System-dependent noise sources such as dark current and detector gain variation are considered. The effect of scene-dependent parameters including background light, target obscuration, and target orientation is also discussed. Finally, noise mitigation strategies such as pulse averaging and gain equalization are described and tested on simulated and real LADAR data

Semiparametric estimation of shifts on compact Lie groups for image registration

In this paper we focus on estimating the deformations that may exist between similar images in the presence of additive noise when a reference template is unknown. The deformations aremodeled as parameters lying in a finite dimensional compact Lie group. A generalmatching criterion based on the Fourier transformand itswell known shift property on compact Lie groups is introduced. M-estimation and semiparametric theory are then used to study the consistency and asymptotic normality of the resulting estimators. As Lie groups are typically nonlinear spaces, our tools rely on statistical estimation for parameters lying in a manifold and take into account the geometrical aspects of the problem. Some simulations are used to illustrate the usefulness of our approach and applications to various areas in image processing are discussed

Régularisation de problèmes inverses linéaires avec opérateur inconnu

Dans cette thèse, nous étudions des méthodes de résolution pour différents types de problèmes inverses linéaires. L'objectif est d'estimer un paramètre de dimension infinie (typiquement une fonction ou une mesure) à partir de l'observation bruitée de son image par un opérateur linéaire. Nous nous intéressons plus précisément à des problèmes inverses dits discrets, pour lesquels l'opérateur est à valeurs dans un espace de dimension finie. Pour ce genre de problème, la non­injectivité de l'opérateur rend impossible l'identification du paramètre à partir de l'observation. Un aspect de la régularisation consiste alors à déterminer un critère de sélection d'une solution parmi un ensemble de valeurs possibles. Nous étudions en particulier des applications de la méthode du maximum d'entropie sur la moyenne, qui est une méthode Bayésienne de régularisation permettant de définir un critère de sélection à partir d'information a priori. Nous traitons également des questions de stabilité en problèmes inverses sous des hypothèses de compacité de l'opérateur, dans un problème de régression non-paramétrique avec observations indirectes.We study regularization methods for different kinds of linear inverse problems. The objective is to estimate an infinite dimensional parameter (typically a function or a measure) from the noisy observation of its image through a linear operator. We are interested more specifically to discret inverse problems, for which the operator takes values in a finite dimensional space. For this kind of problems, the non-injectivity of the operator makes impossible the identification of the parameter from the observation. An aspect of the regularization is then to determine a criterion to select a solution among a set of possible values. We study in particular some applications of the maximum entropy on the mean method, which is a Bayesian regularization method that allows to choose a solution from prior informations. We also treat stability issues in inverse problems under compacity assumptions on the operator, in a general nonparametric regression framework with indirect observations