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

Discrete pulse transform of images and applications

The LULU operators Ln and Un operate on neighbourhoods of size n. The Discrete Pulse Transform (DPT) of images is obtained via recursive peeling of so-called local maximum and minimum sets with the LULU operators as n increases from 1 to the maximum number of elements in the array. The DPT provides a new nonlinear decomposition of a multidimensional array. This thesis investigates the theoretical and practical soundness of the decomposition for image analysis. Properties for the theoretical justification of the DPT are provided as consistency of the decomposition (a pseudo-linear property), and its setting as a nonlinear scale-space, namely the LULU scalespace. A formal axiomatic theory for scale-space operators and scale-spaces is also presented. The practical soundness of the DPT is investigated in image sharpening, best approximation of an image, noise removal in signals and images, feature point detection with ideas to extending work to object tracking in videos, and image segmentation. LULU theory on multidimensional arrays and the DPT is now at a point where concrete signal, image and video analysis algorithms can be developed for a wide variety of applications.Thesis (PhD)--University of Pretoria, 2013.Mathematics and Applied Mathematicsunrestricte

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

Original article available at http://arxiv.org/ENGLISH ABSTRACT: Two procedures to compute the output distribution 0S 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 0S for some important LULU-operators S. Properties of 0S can be used to characterize smoothing properties.Preprin

SHAPE FROM FOCUS USING LULU OPERATORS AND DISCRETE PULSE TRANSFORM IN THE PRESENCE OF NOISE

A study of three dimension (3D) shape recovery is an interesting and challenging area of research. Recovering the depth information of an object from normal two dimensional (2D) images has been studied for a long time with different techniques. One technique for 3D shape recovery is known as Shape from Focus (SFF). SFF is a method that depends on different focused values in reconstructing the shape, surface, and depth of an object. The different focus values are captured by taking different images for the same object by varying the focus length or varying the distance between object and camera. This single view imaging makes the data gathering simpler in SFF compared to other shape recovery techniques. Calculating the shape of the object using different images with different focused values can be done by applying sharpness detection methods to maximize and detect the focused values. However, noise destroys many information in an image and the result of noise corruption can change the focus values in the images. This thesis presents a new 3D shape recovery technique based on focus values in the presence of noise. The proposed technique is based on LULU operators and Discrete Pulse Transform (DPT). LULU operators are nonlinear rank selector operators that hold consistent separation, total variation and shape preservation properties. The proposed techniques show better and more accurate performance in comparison with the existing SFF techniques in noisy environment

Integrable ODEs on Associative Algebras

In this paper we give definitions of basic concepts such as symmetries, first integrals, Hamiltonian and recursion operators suitable for ordinary differential equations on associative algebras, and in particular for matrix differential equations. We choose existence of hierarchies of first integrals and/or symmetries as a criterion for integrability and justify it by examples. Using our componentless approach we have solved a number of classification problems for integrable equations on free associative algebras. Also, in the simplest case, we have listed all possible Hamiltonian operators of low order.Comment: 19 pages, LaTe

FUSION OF HYPERSPECTRAL AND MULTISPECTRAL IMAGERY WITH REGRESSION KRIGING AND THE LULU OPERATORS; A COMPARISON

In this digital world, there is a large requirement of high resolution satellite image. Images at a low resolution may contain relevant information that has to be integrated with the high resolution image to obtain the required information. This is being fulfilled by image fusion. Image fusion is merging of different resolution images into a single image. The output image contains more information, as the information is integrated from both the images Image fusion was conducted with two different algorithms: regression kriging and the LULU operators. First, regression Kriging estimates the value of a dependent variable at unsampled location with the help of auxiliary variables. Here we used regression Kriging with the Hyperion image band as the response variables and the LISS III image bands are the explanatory variables. The fused image thus has the spectral variables from Hyperion image and the spatial variables from the LISS III image. Second, the LULU operator is an image processing methods that can be used as well in image fusion technique. Here we explored to fuse the Hyperion and LISS III image. The LULU operators work in three stages of the process, viz the decomposition stage, the fusion and the reconstruction stage. Quality aspects of the fused image for both techniques have been compared for spectral quality (correlation) and spatial quality (entropy). The study concludes that the quality of the fused image obtained with regression kriging is better than that obtained with the LULU operator

Proving Environmental Inequity in Siting Locally Unwanted Land Uses

This paper advances a process for determining whether, e.g., waste-to-energy facilities are disproportionately located in minority and poor communities, and the author asks others to join in searching for a scientifically sound and fair process of resolving conflicting interests in locating LULUs. He also discusses some difficult issues and argues that they need to be addressed by a representative panel