In this paper, we present a unified framework for the rigid and nonrigid point set registration problem in the presence of significant amounts of noise and outliers. The key idea of this registration framework is to represent the input point sets using Gaussian mixture models. Then, the problem of point set registration is reformulated as the problem of aligning two Gaussian mixtures such that a statistical discrepancy measure between the two corresponding mixtures is minimized. We show that the popular iterative closest point (ICP) method and several existing point set registration methods in the field are closely related and can be reinterpreted meaningfully in our general framework. Our instantiation of this general framework is based on the the L2 distance between two Gaussian mixtures, which has the closed-form expression and in turn leads to a computationally efficient registration algorithm. The resulting registration algorithm exhibits inherent statistical robustness, has an intuitive interpretation, and is simple to implement. We also provide theoretical and experimental comparisons with other robust methods for point set registration.

Robust Point Set Registration Using Gaussian Mixture Models

In his seminal book, Shewhart (1931) makes no demand on the distribution of the characteristic to be plotted on a control chart. How then can we explain the idea that normality is, if not required, at least highly desirable? I believe that it has come about through the many statistical studies of control-chart behavior. If one is to study how a control chart behaves, it is necessary to relate it to some distribution. The obvious choice is the normal distribution because of its ubiquity as a satisfactory model. This is bolstered by the existence of the Central Limit Theorem.

Sampling: Design and Analysis

This letter describes algorithms for nonnegative matrix factorization (NMF) with the β-divergence (β-NMF). The β-divergence is a family of cost functions parameterized by a single shape parameter β that takes the Euclidean distance, the Kullback-Leibler divergence, and the Itakura-Saito divergence as special cases (β = 2, 1, 0 respectively). The proposed algorithms are based on a surrogate auxiliary function (a local majorization of the criterion function). We first describe a majorization-minimization algorithm that leads to multiplicative updates, which differ from standard heuristic multiplicative updates by a β-dependent power exponent. The monotonicity of the heuristic algorithm can, however, be proven for β ∈ (0, 1) using the proposed auxiliary function. Then we introduce the concept of the majorization-equalization (ME) algorithm, which produces updates that move along constant level sets of the auxiliary function and lead to larger steps than MM. Simulations on synthetic and real data illustrate the faster convergence of the ME approach. The letter also describes how the proposed algorithms can be adapted to two common variants of NMF: penalized NMF (when a penalty function of the factors is added to the criterion function) and convex NMF (when the dictionary is assumed to belong to a known subspace).

https://hal.archives-ouvertes.fr/hal-00522289/document

Algorithms for nonnegative matrix factorization with the β-divergence

We consider hierarchical generalized linear models which allow extra error components in the linear predictors of generalized linear models. The distribution of these components is not restricted to be normal; this allows a broader class of models, which includes generalized linear mixed models. We use a generalization of Henderson's joint likelihood, called a hierarchical or h-likelihood, for inferences from hierarchical generalized linear models. This avoids the integration that is necessary when marginal likelihood is used. Under appropriate conditions maximizing the h-likelihood gives fixed effect estimators that are asymptotically equivalent to those obtained from the use of marginal likelihood; at the same time we obtain the random effect estimates that are asymptotically best unbiased predictors. An adjusted profile h-likelihood is shown to give the required generalization of restricted maximum likelihood for the estimation of dispersion components. A scaled deviance test for the goodness of fit, a model selection criterion for choosing between various dispersion models and a graphical method for checking the distributional assumption of random effects are proposed. The ideas of quasi-likelihood and extended quasi-likelihood are generalized to the new class. We give examples of the Poisson-gamma, binomial-beta and gamma-inverse gamma hierarchical generalized linear models. A resolution is proposed for the apparent difference between population-averaged and subject-specific models. A unified framework is provided for viewing and extending many existing methods.

https://rss.onlinelibrary.wiley.com/doi/pdf/10.1111/j.2517-6161.1996.tb02105.x

Hierarchical Generalized Linear Models

We provide a compact survey and brief interpretations and comments on some 170 publications on process capability indices which appeared in widely scattered sources during the years 1992 to 2000. A...

Process capability indices- a review. 1992–2000

A minimum divergence estimation method is developed for robust parameter estimation. The proposed approach uses new density-based divergences which, unlike existing methods of this type such as minimum Hellinger distance estimation, avoid the use of nonparametric density estimation and associated complications such as bandwidth selection. The proposed class of ‘density power divergences’ is indexed by a single parameter α which controls the trade-off between robustness and efficiency. The methodology affords a robust extension of maximum likelihood estimation for which α = 0. Choices of α near zero afford considerable robustness while retaining efficiency close to that of maximum likelihood.

/pdf/robust-and-efficient-estimation-by-minimising-a-density-49vbgv47gs.pdf

Robust and efficient estimation by minimising a density power divergence

This paper compares the minimum divergence estimator of Basu et al. (1998) to a competing minimum divergence estimator which turns out to be equivalent to a method proposed from a different perspective by Windham (1995). Both methods can be applied for any parametric model and contain maximum likelihood as a special case. Efficiencies are compared under model conditions, and robustness properties are studied. Overall the two methods are found to perform quite similarly. Some relatively small advantages of the former method over the latter are identified.

/pdf/a-comparison-of-related-density-based-minimum-divergence-5es5q9lqxn.pdf

A Comparison of related density-based minimum divergence estimators

SUMMARY The paper examines predictive distributions, concentrating on measuring their fit to the true distribution by average Kullback-Leibler divergence. The notion of an 'averaged bootstrap' predictive distribution is introduced. This predictive distribution is shown to be asymptotically superior to the estimative distribution, in terms of average KullbackLeibler divergence, when the true distribution is in a natural exponential family. Smallsample results are presented for the Poisson and binomial distributions which suggest that the bootstrap distribution performs well in these cases.

Predictive fit for natural exponential families

Absence of adaptive learning from the oviposition foraging behaviour of a checkerspot butterfly

The power divergence (PD) and the density power divergence (DPD) families have proven to be useful tools in the area of robust inference. In this paper, we consider a superfamily of divergences which contains both of these families as special cases. The role of this superfamily is studied in several statistical applications, and desirable properties are identified and discussed. In many cases, it is observed that the most preferred minimum divergence estimator within the above collection lies outside the class of minimum PD or minimum DPD estimators, indicating that this superfamily has real utility, rather than just being a routine generalization. The limitation of the usual first order influence function as an effective descriptor of the robustness of the estimator is also demonstrated in this connection.

/pdf/a-generalized-divergence-for-statistical-inference-1ae8trxrcr.pdf

Ian R. Harris

Papers

Robust and efficient estimation by minimising a density power divergence

A Comparison of related density-based minimum divergence estimators

Predictive fit for natural exponential families

Absence of adaptive learning from the oviposition foraging behaviour of a checkerspot butterfly

A generalized divergence for statistical inference