TVL₁ Planarity Regularization for 3D Shape Approximation

The modern emergence of automation in many industries has given impetus to extensive research into mobile robotics. Novel perception technologies now enable cars to drive autonomously, tractors to till a field automatically and underwater robots to construct pipelines. An essential requirement to facilitate both perception and autonomous navigation is the analysis of the 3D environment using sensors like laser scanners or stereo cameras. 3D sensors generate a very large number of 3D data points when sampling object shapes within an environment, but crucially do not provide any intrinsic information about the environment which the robots operate within. This work focuses on the fundamental task of 3D shape reconstruction and modelling from 3D point clouds. The novelty lies in the representation of surfaces by algebraic functions having limited support, which enables the extraction of smooth consistent implicit shapes from noisy samples with a heterogeneous density. The minimization of total variation of second differential degree makes it possible to enforce planar surfaces which often occur in man-made environments. Applying the new technique means that less accurate, low-cost 3D sensors can be employed without sacrificing the 3D shape reconstruction accuracy

TVL₁shape approximation from scattered 3D data

With the emergence in 3D sensors such as laser scanners and 3D reconstruction from cameras, large 3D point clouds can now be sampled from physical objects within a scene. The raw 3D samples delivered by these sensors however, contain only a limited degree of information about the environment the objects exist in, which means that further geometrical high-level modelling is essential. In addition, issues like sparse data measurements, noise, missing samples due to occlusion, and the inherently huge datasets involved in such representations makes this task extremely challenging. This paper addresses these issues by presenting a new 3D shape modelling framework for samples acquired from 3D sensor. Motivated by the success of nonlinear kernel-based approximation techniques in the statistics domain, existing methods using radial basis functions are applied to 3D object shape approximation. The task is framed as an optimization problem and is extended using non-smooth L1 total variation regularization. Appropriate convex energy functionals are constructed and solved by applying the Alternating Direction Method of Multipliers approach, which is then extended using Gauss-Seidel iterations. This significantly lowers the computational complexity involved in generating 3D shape from 3D samples, while both numerical and qualitative analysis confirms the superior shape modelling performance of this new framework compared with existing 3D shape reconstruction techniques

Knot selection by boosting techniques

A novel concept for estimating smooth functions by selection techniques based on boosting is developed. It is suggested to put radial basis functions with different spreads at each knot and to do selection and estimation simultaneously by a componentwise boosting algorithm. The methodology of various other smoothing and knot selection procedures (e.g. stepwise selection) is summarized. They are compared to the proposed approach by extensive simulations for various unidimensional settings, including varying spatial variation and heteroskedasticity, as well as on a real world data example. Finally, an extension of the proposed method to surface fitting is evaluated numerically on both, simulation and real data. The proposed knot selection technique is shown to be a strong competitor to existing methods for knot selection

Emulation of multivariate simulators using thin-plate splines with application to atmospheric dispersion

It is often desirable to build a statistical emulator of a complex computer simulator in order to perform analysis which would otherwise be computationally infeasible. We propose methodology to model multivariate output from a computer simulator taking into account output structure in the responses. The utility of this approach is demonstrated by applying it to a chemical and biological hazard prediction model. Predicting the hazard area which results from an accidental or deliberate chemical or biological release is imperative in civil and military planning and also in emergency response. The hazard area resulting from such a release is highly structured in space and we therefore propose the use of a thin-plate spline to capture the spatial structure and fit a Gaussian process emulator to the coefficients of the resultant basis functions. We compare and contrast four different techniques for emulating multivariate output: dimension-reduction using (i) a fully Bayesian approach with a principal component basis, (ii) a fully Bayesian approach with a thin-plate spline basis, assuming that the basis coefficients are independent, and (iii) a “plug-in” Bayesian approach with a thin-plate spline basis and a separable covariance structure; and (iv) a functional data modeling approach using a tensor-product (separable) Gaussian process. We develop methodology for the two thin-plate spline emulators and demonstrate that these emulators significantly outperform the principal component emulator. Further, the separable thin-plate spline emulator, which accounts for the dependence between basis coefficients, provides substantially more realistic quantification of uncertainty, and is also computationally more tractable, allowing fast emulation. For high resolution output data, it also offers substantial predictive and computational ad- vantages over the tensor-product Gaussian process emulator

The relationship between changes in the Economic Sentiment Indicator and real GDP growth: a time-varying coefficient approach

The aim of this paper is to capture the time-varying effects of the relationship between changes in the Economic Sentiment Indicator (ESI) and economic growth. We use penalized regression splines to estimate the different point effects over time. Evidence from six European countries supports the idea that the elasticity of the ESI is time-varying.Economic Sentiment Indicator; Real GDP Growth; Thin Plate Regression Splines; Time-Varying Coefficient Model

Semiparametric Construction of Spatial Generalized Hedonic Models for Private Properties

This paper analyzes the empirical hedonic prices for non-standard condominiums and single family houses using nonparametric estimates as well as a generalized additive model for the spatial generalization of the attractiveness of all Swiss communities. We find that the assumption of log-linearity for continuous variables does not hold but can be replaced by partwise log-linear or quadratic terms. Due to the topographical segmentation of Switzerland, driving times seem to be more adequate than geographical distances to explain the price level of a village with the price level of its neighbours. We show, that using metric multidimensional scaling, the driving times between the villages can be converted into artificial coordinates with three principal axis to serve as a basis for the prediction of the macro-locations of the Swiss villagesHedonic prices; private property; Switzerland; robust regression; splines; multidimensional scaling

Approximating data with weighted smoothing splines

n.a. --Approximation,Residuals,Smoothing Splines,Thin Plate Splines

Generalized Functional Additive Mixed Models

We propose a comprehensive framework for additive regression models for non-Gaussian functional responses, allowing for multiple (partially) nested or crossed functional random effects with flexible correlation structures for, e.g., spatial, temporal, or longitudinal functional data as well as linear and nonlinear effects of functional and scalar covariates that may vary smoothly over the index of the functional response. Our implementation handles functional responses from any exponential family distribution as well as many others like Beta- or scaled non-central $t$-distributions. Development is motivated by and evaluated on an application to large-scale longitudinal feeding records of pigs. Results in extensive simulation studies as well as replications of two previously published simulation studies for generalized functional mixed models demonstrate the good performance of our proposal. The approach is implemented in well-documented open source software in the "pffr()" function in R-package "refund"

Reduced-rank spatio-temporal modeling of air pollution concentrations in the Multi-Ethnic Study of Atherosclerosis and Air Pollution

There is growing evidence in the epidemiologic literature of the relationship between air pollution and adverse health outcomes. Prediction of individual air pollution exposure in the Environmental Protection Agency (EPA) funded Multi-Ethnic Study of Atheroscelerosis and Air Pollution (MESA Air) study relies on a flexible spatio-temporal prediction model that integrates land-use regression with kriging to account for spatial dependence in pollutant concentrations. Temporal variability is captured using temporal trends estimated via modified singular value decomposition and temporally varying spatial residuals. This model utilizes monitoring data from existing regulatory networks and supplementary MESA Air monitoring data to predict concentrations for individual cohort members. In general, spatio-temporal models are limited in their efficacy for large data sets due to computational intractability. We develop reduced-rank versions of the MESA Air spatio-temporal model. To do so, we apply low-rank kriging to account for spatial variation in the mean process and discuss the limitations of this approach. As an alternative, we represent spatial variation using thin plate regression splines. We compare the performance of the outlined models using EPA and MESA Air monitoring data for predicting concentrations of oxides of nitrogen (NO$_x$)-a pollutant of primary interest in MESA Air-in the Los Angeles metropolitan area via cross-validated $R^2$. Our findings suggest that use of reduced-rank models can improve computational efficiency in certain cases. Low-rank kriging and thin plate regression splines were competitive across the formulations considered, although TPRS appeared to be more robust in some settings.Comment: Published in at http://dx.doi.org/10.1214/14-AOAS786 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org