The output distribution of important LULU-operators

Two procedures to compute the output distribution phi_S of certain stack filters S (so called erosion-dilation cascades) are given. One rests on the disjunctive normal form of S and also yields the rank selection probabilities. The other is based on inclusion-exclusion and e.g. yields phi_S for some important LULU-operators S. Properties of phi_S can be used to characterize smoothing properties of S. One of the methods discussed also allows for the calculation of the reliability polynomial of any positive Boolean function (e.g. one derived from a connected graph).Comment: 20 pages, up to trivial differences this is the final version to be published in Quaestiones Mathematicae 201

Quasi-inverses and approximation with min-max operators in the ℓ 1-norm

The semi-group of min-max operators, as used for nonlinear smoothing or multiresolution analysis, has no nontrivial inverses. Having chosen a smoother for a specific purpose, the secondary approximation problem of minimising damage was considered by showing that quasi-inverses exist. This was done with respect to the total variation as norm in ℓ 1, as this is natural for these operators. We show that these quasi-inverses also minimise the residual in the more usual 1-norm. © 2006 NISC Pty Ltd.Articl

Fully trend preserving operators

The concept of fully trend preserving (ftp) operators on sequences was introduced to prove variation preservation on the outputs of LULU-separators. It turns out that this property is central in further analysis of compositions of LULU-smoothers, and other smoothers. Several important results follow, amongst them the co-idempotence of a large class of smoothers. This has been an unresolved problem up to now. © 2004 NISC Pty Ltd.Articl

On the consistency of a separator

A smoother can be called a separator if it is idempotent and co-idempotent, with the motivation coming from a physical analogy. This is a rudementary consistency, and type of "linearity", which has been found to occur in practice. All the basic LULU-separator turn out to act linearly on any non-negative combination of the two outputs of such separators. This is arguably a very consistent behaviour. We prove this, and investigate how far this follows for general separators. We also prove that this type of linearity is extendable to composite "separators" that are used in a nonlinear multiresolution decomposition with LULU-operator. This has important implications for image processing. © 2010 NISC Pty Ltd.Articl

Calculating the output distribution of stack filters that are erosion-dilation cascades, in particular LULU-filters

Two procedures to compute the output distribution ϕs of certain stack filters S (so called erosion-dilation cascades) are given. One rests on the disjunctive normal form of S and also yields the rank selection probabilities. The other is based on inclusion-exclusion and e.g. yields ϕs for some important LULU-operators S. Properties of ϕs can be used to characterize smoothing properties of S. Also, in the same way as our polynomials ϕs are computed one could compute the reliability polynomial of a connected graph, or more generally the reliability polynomial w.r.t. any positive Boolean function.Keywords: Stack filter, output distribution, LULU filter, reliability polynomial, inclusion-exclusion

ORNL long-range environmental and waste management plan

This report, the ORNL Long-Range Environmental and Waste Management Plan, is the annual update in a series begun in fiscal year 1985. Its primary purpose is to provide a thorough and systematic planning document to reflect the continuing process of site assessment, strategy development, and planning for the current and long-term control of environmental issues, waste management practices, and remedial action requirements. The document also provides an estimate of the resources required to implement the current plan. This document is not intended to be a budget document; it is, however, intended to provide guidance to both Martin Marietta Energy Systems, Inc., and the US Department of Energy (DOE) management as to the near order of magnitude of the resources (primarily funding requirements) and the time frame required to execute the strategy in the present revision of the plan. As with any document of this nature, the near-term (one to three years) part of the plan is a pragmatic assessment of the current program and ongoing capital projects and reflects the efforts perceived to be necessary to comply with all current state and federal regulations and DOE orders. It also should be in general agreement with current budget (funding) requests and obligations for these immediate years. 55 figs., 72 tabs