Thesis (MSc)--University of Stellenbosch, 2011. / ENGLISH ABSTRACT: The nonlinear LULU smoothers excel at removing impulsive noise from sequences and possess a variety of theoretical properties that make it possible to perform a so-called Discrete Pulse Transform, which is a novel multiresolution analysis technique that decomposes a sequence into resolution levels with a large amount of structure, analogous to a Discrete Wavelet Transform. We explore the use of a one-dimensional Discrete Pulse Transform as the central element in a digital image compressor. We depend crucially on the ability of space-filling scanning orders to map the two-dimensional image data to one dimension, sacrificing as little image structure as possible. Both lossless and lossy image compression are considered, leading to five new image compression schemes that give promising results when compared to state-of-the-art image compressors. / AFRIKAANSE OPSOMMING: Die nielineêre LULU gladstrykers verwyder impulsiewe geraas baie goed uit rye en besit verskeie teoretiese eienskappe wat dit moontlik maak om ’n sogenoemde Diskrete Puls Transform uit te voer; ’n nuwe multiresolusie analise tegniek wat ’n ry opbreek in ’n versameling resolusie vlakke wat ’n groot hoeveelheid struktuur bevat, soortgelyk tot ’n Diskrete Golfie Transform. Ons ondersoek of ’n eendimensionele Diskrete Puls Transform as die sentrale element in ’n digitale beeld kompressor gebruik kan word. Ons is afhanklik van ruimtevullende skandeer ordes om die tweedimensionele beelddata om te skakel na een dimensie, sonder om te veel beeld struktuur te verloor. Vyf nuwe beeld kompressie skemas word bespreek. Hierdie skemas lewer belowende resultate wanneer dit met die beste hedendaagse beeld kompressors vergelyk word.

Thesis (PhD (Electronic Engineering))--University of Stellenbosch, 2010. / Contains bibliography and index. / ENGLISH ABSTRACT: Super-resolution imaging is the process whereby several low-resolution photographs of an object are combined to form a single high-resolution estimation. We investigate each component of this process: image acquisition, registration and reconstruction. A new feature detector, based on the discrete pulse transform, is developed. We show how to implement and store the transform efficiently, and how to match the features using a statistical comparison that improves upon correlation under mild geometric transformation. To simplify reconstruction, the imaging model is linearised, whereafter a polygon-based interpolation operator is introduced to model the underlying camera sensor. Finally, a large, sparse, over-determined system of linear equations is solved, using regularisation. The software developed to perform these computations is made available under an open source license, and may be used to verify the results. / AFRIKAANSE OPSOMMING: In super-resolusie beeldvorming word verskeie lae-resolusie foto's van 'n onderwerp gekombineer in 'n enkele, hoë-resolusie afskatting. Ons ondersoek elke stap van hierdie proses: beeldvorming, -belyning en hoë-resolusie samestelling. 'n Nuwe metode wat staatmaak op die diskrete pulstransform word ontwikkel om belangrike beeldkenmerke te vind. Ons wys hoe om die transform e ektief te bereken en hoe om resultate kompak te stoor. Die kenmerke word vergelyk deur middel van 'n statistiese model wat bestand is teen klein lineêre beeldvervormings. Met die oog op 'n vereenvoudigde samestellingsberekening word die beeldvormingsmodel gelineariseer. In die nuwe model word die kamerasensor gemodelleer met behulp van veelhoek-interpolasie. Uiteindelik word 'n groot, yl, oorbepaalde stelsel lineêre vergelykings opgelos met behulp van regularisering. Die sagteware wat vir hierdie berekeninge ontwikkel is, is beskikbaar onderhewig aan 'n oopbron-lisensie en kan gebruik word om die gegewe resultate te veri eer.

Thesis (MSc (Mathematics))--Stellenbosch University, 2008. / LULU smoothers is a class of nonlinear smoothers and they are compositions of the maximum and minimum operators. By analogy to the discrete Fourier transform and the discrete wavelet transform, one can use LULU smoothers to create a nonlinear multiresolution analysis of a sequence with pulses. This tool is known as the Discrete Pulse Transform (DPT). Some research have been done into the distributional properties of the LULU smoothers. There exist results on the distribution transfers of the basic LULU smoothers, which are the building blocks of the discrete pulse transform. The output distributions of further smoothers used in the DPT, in terms of input distributions, has been a challenging problem. We motivate the use of these smoothers by first considering linear filters as well as the median smoother, which has been very popular in signal and image processing. We give an overview of the attractive properties of the LULU smoothers after which we tackle their output distributions. The main result is the proof of a recursive formula for the output distribution of compositions of LULU smoothers in terms of a given input distribution.