A Newton method for pose refinement of 3D models

Different optimization methods can be employed to optimize a numerical estimate for the match between an instantiated object model and an image. In order to take advantage of gradient-based optimization methods, perspective inversion must be used in this context. We show that convergence can be very fast by extrapolating to maximum goodness-of-fit with Newton's method. This approach is related to methods which either maximize a similar goodness-of-fit measure without use of gradient information, or else minimize distances between projected model lines and image features. Newton's method combines the accuracy of the former approach with the speed of convergence of the latter.

Approximate Gauss-Newton methods for nonlinear least squares problems

The Gauss–Newton algorithm is an iterative method regularly used for solving nonlinear least squares problems. It is particularly well suited to the treatment of very large scale variational data assimilation problems that arise in atmosphere and ocean forecasting. The procedure consists of a sequence of linear least squares approximations to the nonlinear problem, each of which is solved by an “inner” direct or iterative process. In comparison with Newton’s method and its variants, the algorithm is attractive because it does not require the evaluation of second-order derivatives in the Hessian of the objective function. In practice the exact Gauss–Newton method is too expensive to apply operationally in meteorological forecasting, and various approximations are made in order to reduce computational costs and to solve the problems in real time. Here we investigate the effects on the convergence of the Gauss–Newton method of two types of approximation used commonly in data assimilation. First, we examine “truncated” Gauss–Newton methods where the inner linear least squares problem is not solved exactly, and second, we examine “perturbed” Gauss–Newton methods where the true linearized inner problem is approximated by a simplified, or perturbed, linear least squares problem. We give conditions ensuring that the truncated and perturbed Gauss–Newton methods converge and also derive rates of convergence for the iterations. The results are illustrated by a simple numerical example. A practical application to the problem of data assimilation in a typical meteorological system is presented.

The Limnology of Slapton Ley

Slapton Ley, a freshwater lake, located in south Devon (National Grid Reference SX 825 439), has been the focus of a wide range of research studies since the foundation of the Field Studies Council Centre in Slapton village in 1959, and the creation of the Slapton Ley Nature Reserve. Early concerns over eutrophication of the Lower Ley led to a range of studies focused on the impacts of land use change in the catchment, on nutrient delivery to the Ley, and on interpreting the impact of long-term nutrient enrichment of the Ley from palaeolimnological studies. What has been missing to date, however, is a focused study of the impacts of nutrient enrichment on the chemical and ecological structure and function of the combined Lower and Higher Ley systems. This paper attempts to draw together the various areas of study on the Ley to date in order to provide a review of current understanding of the limnology of Slapton Ley and to identify gaps in our knowledge. The past, present and future trophic status of the Ley is re-interpreted in the light of current understanding of the eutrophication process in the wider scientific community. Recommendations for future research are then made, with a view to the monitoring and management of Slapton Ley and its catchment.

The Natural History of Slapton Ley Nature Reserve XVIII: nitrogen and phosphorus losses from the catchment - an export coefficient modelling approach

Nitrogen and phosphorus losses from the catchment of Slapton Ley, a small coastal lake in SW England, were calculated using an adaptation of a model developed by Jorgensen (1980). A detailed survey of the catchment revealed that its land use is dominated by both permanent and temporary grassland (respectively 38 and 32% of its total area), and that the remainder is made up of the cultivation of cereals and field vegetables, and market gardening. Livestock numbers in the catchment constitute ca. 6600 head of cattle, 10,000 sheep, 590 pigs, 1700 poultry and 58 horses. The permanent human population of the area is ca. 2000, served by two small gravity-fed sewage treatment works (STWs). Inputs to, and losses from, farmland in the catchment were computed using Jorgensen’s model, and coefficients derived from the data of Cooke (1976), Gostick (1982), Rast and Lee (1983) and Vollenweider (1968). Allowing for outputs from STWs, the total annual external load of N and P upon Slapton Ley is 160 t (35 kg ha-1) a-1 N, and 4.8 t (1.05 kg ha-1) a-1 P. Accordingly to Vollenweider (1968, 1975), such loadings exceed OECD permissible level by a factor of ca. 50 in the case of N, and ca. 5 in that of P. In order to reduce nutrient loads, attention would need to be paid to both STW and agricultural sources.