Thesis (PhD)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: Multi-resolution analysis (MRA) has become a very popular eld of mathematical study in the past two decades, being not only an area rich in applications but one that remains lled with open problems. Building on the foundation of re nability of functions, MRA seeks to lter through levels of ever-increasing detail components in data sets { a concept enticing to an age where development of digital equipment (to name but one example) needs to capture more and more information and then store this information in di erent levels of detail. Except for designing digital objects such as animation movies, one of the most recent popular research areas in which MRA is applied, is inpainting, where \lost" data (in example, a photograph) is repaired by using boundary values of the data set and \smudging" these values into the empty entries. Two main branches of application in MRA are subdivision and wavelet analysis. The former uses re nable functions to develop algorithms with which digital curves are created from a nite set of initial points as input, the resulting curves (or drawings) of which possess certain levels of smoothness (or, mathematically speaking, continuous derivatives). Wavelets on the other hand, yield lters with which certain levels of detail components (or noise) can be edited out of a data set. One of the greatest advantages when using wavelets, is that the detail data is never lost, and the user can re-insert it to the original data set by merely applying the wavelet algorithm in reverse. This opens up a wonderful application for wavelets, namely that an existent data set can be edited by inserting detail components into it that were never there, by also using such a wavelet algorithm. In the recent book by Chui and De Villiers (see [2]), algorithms for both subdivision and wavelet applications were developed without using Fourier analysis as foundation, as have been done by researchers in earlier years and which have left such algorithms unaccessible to end users such as computer programmers. The fundamental result of Chapter 9 on wavelets of [2] was that feasibility of wavelet decomposition is equivalent to the solvability of a certain set of identities consisting of Laurent polynomials, referred to as Bezout identities, and it was shown how such a system of identities can be solved in a systematic way. The work in [2] was done in the univariate case only, and it will be the purpose of this thesis to develop similar results in the bivariate case, where such a generalization is entirely non-trivial. After introducing MRA in Chapter 1, as well as discussing the re nability of functions and introducing box splines as prototype examples of functions that are re nable in the bivariate setting, our fundamental result will also be that wavelet decomposition is equivalent to solving a set of Bezout identities; this will be shown rigorously in Chapter 2. In Chapter 3, we give a set of Laurent polynomials of shortest possible length satisfying the system of Bezout identities in Chapter 2, for the particular case of the Courant hat function, which will have been introduced as a linear box spline in Chapter 1. In Chapter 4, we investigate an application of our result in Chapter 3 to bivariate interpolatory subdivision. With the view to establish a general class of wavelets corresponding to the Courant hat function, we proceed in the subsequent Chapters 5 { 8 to develop a general theory for solving the Bezout identities of Chapter 2 separately, before suggesting strategies for reconciling these solution classes in order to be a simultaneous solution of the system. / AFRIKAAANSE OPSOMMING: Multi-resolusie analise (MRA) het in die afgelope twee dekades toenemende gewildheid geniet as 'n veld in wiskundige wetenskappe. Nie net is dit 'n area wat ryklik toepaslik is nie, maar dit bevat ook steeds vele oop vraagstukke. MRA bou op die grondleggings van verfynbare funksies en poog om deur vlakke van data-komponente te sorteer, of te lter, 'n konsep wat aanloklik is in 'n era waar die ontwikkeling van digitale toestelle (om maar 'n enkele voorbeeld te noem) sodanig moet wees dat meer en meer inligting vasgel^e en gestoor moet word. Behalwe vir die ontwerp van digitale voorwerpe, soos animasie- lms, word MRA ook toegepas in 'n mees vername navorsingsgebied genaamd inverwing, waar \verlore" data (soos byvoorbeeld in 'n foto) herwin word deur data te neem uit aangrensende gebiede en dit dan oor die le e data-dele te \smeer." Twee hooftakke in toepassing van MRA is subdivisie en gol e-analise. Die eerste gebruik verfynbare funksies om algoritmes te ontwikkel waarmee digitale krommes ontwerp kan word vanuit 'n eindige aantal aanvanklike gegewe punte. Die verkrygde krommes (of sketse) kan voldoen aan verlangde vlakke van gladheid (of verlangde grade van kontinue afgeleides, wiskundig gesproke). Gol es word op hul beurt gebruik om lters te bou waarmee gewensde dataof geraas-komponente verwyder kan word uit datastelle. Een van die grootste voordeel van die gebruik van gol es bo ander soortgelyke instrumente om data lters mee te bou, is dat die geraas-komponente wat uitgetrek word nooit verlore gaan nie, sodat die proses omkeerbaar is deurdat die gebruiker die sodanige geraas-komponente in die groter datastel kan terugbou deur die gol e-algoritme in trurat toe te pas. Hierdie eienskap van gol fies open 'n wonderlike toepassingsmoontlikheid daarvoor, naamlik dat 'n bestaande datastel verander kan word deur data-komponente daartoe te voeg wat nooit daarin was nie, deur so 'n gol e-algoritme te gebruik. In die onlangse boek deur Chui and De Villiers (sien [2]) is algoritmes ontwikkel vir die toepassing van subdivisie sowel as gol es, sonder om staat te maak op die grondlegging van Fourier-analise, soos wat die gebruik was in vroe ere navorsing en waardeur algoritmes wat ontwikkel is minder e ektief was vir eindgebruikers. Die fundamentele resultaat oor gol es in Hoofstuk 9 in [2], verduidelik hoe suksesvolle gol e-ontbinding ekwivalent is aan die oplosbaarheid van 'n sekere versameling van identiteite bestaande uit Laurent-polinome, bekend as Bezout-identiteite, en dit is bewys hoedat sodanige stelsels van identiteite opgelos kan word in 'n sistematiese proses. Die werk in [2] is gedoen in die eenveranderlike geval, en dit is die doelwit van hierdie tesis om soortgelyke resultate te ontwikkel in die tweeveranderlike geval, waar sodanige veralgemening absoluut nie-triviaal is. Nadat 'n inleiding tot MRA in Hoofstuk 1 aangebied word, terwyl die verfynbaarheid van funksies, met boks-latfunksies as prototipes van verfynbare funksies in die tweeveranderlike geval, bespreek word, word ons fundamentele resultaat gegee en bewys in Hoofstuk 2, naamlik dat gol e-ontbinding in die tweeveranderlike geval ook ekwivalent is aan die oplos van 'n sekere stelsel van Bezout-identiteite. In Hoofstuk 3 word 'n versameling van Laurent-polinome van korste moontlike lengte gegee as illustrasie van 'n oplossing van 'n sodanige stelsel van Bezout-identiteite in Hoofstuk 2, vir die besondere geval van die Courant hoedfunksie, wat in Hoofstuk 1 gede nieer word. In Hoofstuk 4 ondersoek ons 'n toepassing van die resultaat in Hoofstuk 3 tot tweeveranderlike interpolerende subdivisie. Met die oog op die ontwikkeling van 'n algemene klas van gol es verwant aan die Courant hoedfunksie, brei ons vervolglik in Hoofstukke 5 { 8 'n algemene teorie uit om die oplossing van die stelsel van Bezout-identiteite te ondersoek, elke identiteit apart, waarna ons moontlike strategie e voorstel vir die versoening van hierdie klasse van gelyktydige oplossings van die Bezout stelsel.

O conceito de limite é a base para qualquer aluno de um curso superior que faça a disciplina de cálculo, portanto deveria haver uma ênfase especial na sua aprendizagem por parte de professores e alunos. Na sala de aula a ênfase é dada sobre a definição formal e sobre a forma que ela é apresentada nos livros de cálculo. Neste trabalho foi feita uma pesquisa sobre a natureza do conceito de limite junto com uma reflexão sobre o que é um conceito. Houve uma pesquisa para entender quais os mecanismos cognitivos que estão envolvidos quando um sujeito atua sobre um problema que envolve tal conceito. Buscaram-se elementos de apoio para a investigação no processo histórico de desenvolvimento deste conceito matemático e é preciso destacar a importância deste processo para desenvolver objetos digitais interativos que pudessem ser utilizados em experimentos desenvolvidos para entender os mecanismos cognitivos deste conceito. De modo paralelo ao estudo da história do conceito e seus antecedentes houve um outro estudo sobre objetos digitais interativos e as possibilidades de utilizá-los na investigação dos mecanismos cognitivos. A teoria escolhida para a análise epistemológica do conceito foi a Epistemologia Genética de Jean Piaget, especificamente a teoria relativa a relações e operações infralógicas, lógicas e ao pensamento formal. Na investigação realizada foi utilizado um problema como desequilíbrio inicial. Este forneceu o ponto de partida para questionar as certezas e dúvidas do sujeito de forma a investigar os mecanismos cognitivos que formam o conceito de limite e do contínuo numérico. / The concept of limit is the basis for every student accessing the college and needs study calculus, therefore it should be an emphasis about its learning from students and teachers. At the classroom the emphasis is on the formal definition and over the way it is presented at the books of calculus. On this work a research was carried on and it was about the nature of concept of limit together with a reflexion about what is a concept. There was a research for understanding the cognitive mechanisms related when um individual acts on a problem with that concept. There was a search for elements, in the historic development, that supported the investigation about that mathematic concept and is necessary to regard the importance of that process for the design of interactive digital objects that could be used in experiments designed for understanding the cognitive mechanism related to the concept of limits. In a parallel with the study about the history of limit concept and his ancestors there was another study about interactive digital objects and the possibilities for using that in the investigation of cognitive mechanism. The theory to be used in the analysis was The Genetic Epistemology of Jean Piaget, especially the section about infralogic and logic relation and operations and the formal thinking. In this work it was used a start problem that supplied the beginning point to argue the certainties and doubts of the subject in a way to investigate the cognitive mechanisms that form the limit concept and the numeric continuum, its basis.

O conceito de limite é a base para qualquer aluno de um curso superior que faça a disciplina de cálculo, portanto deveria haver uma ênfase especial na sua aprendizagem por parte de professores e alunos. Na sala de aula a ênfase é dada sobre a definição formal e sobre a forma que ela é apresentada nos livros de cálculo. Neste trabalho foi feita uma pesquisa sobre a natureza do conceito de limite junto com uma reflexão sobre o que é um conceito. Houve uma pesquisa para entender quais os mecanismos cognitivos que estão envolvidos quando um sujeito atua sobre um problema que envolve tal conceito. Buscaram-se elementos de apoio para a investigação no processo histórico de desenvolvimento deste conceito matemático e é preciso destacar a importância deste processo para desenvolver objetos digitais interativos que pudessem ser utilizados em experimentos desenvolvidos para entender os mecanismos cognitivos deste conceito. De modo paralelo ao estudo da história do conceito e seus antecedentes houve um outro estudo sobre objetos digitais interativos e as possibilidades de utilizá-los na investigação dos mecanismos cognitivos. A teoria escolhida para a análise epistemológica do conceito foi a Epistemologia Genética de Jean Piaget, especificamente a teoria relativa a relações e operações infralógicas, lógicas e ao pensamento formal. Na investigação realizada foi utilizado um problema como desequilíbrio inicial. Este forneceu o ponto de partida para questionar as certezas e dúvidas do sujeito de forma a investigar os mecanismos cognitivos que formam o conceito de limite e do contínuo numérico. / The concept of limit is the basis for every student accessing the college and needs study calculus, therefore it should be an emphasis about its learning from students and teachers. At the classroom the emphasis is on the formal definition and over the way it is presented at the books of calculus. On this work a research was carried on and it was about the nature of concept of limit together with a reflexion about what is a concept. There was a research for understanding the cognitive mechanisms related when um individual acts on a problem with that concept. There was a search for elements, in the historic development, that supported the investigation about that mathematic concept and is necessary to regard the importance of that process for the design of interactive digital objects that could be used in experiments designed for understanding the cognitive mechanism related to the concept of limits. In a parallel with the study about the history of limit concept and his ancestors there was another study about interactive digital objects and the possibilities for using that in the investigation of cognitive mechanism. The theory to be used in the analysis was The Genetic Epistemology of Jean Piaget, especially the section about infralogic and logic relation and operations and the formal thinking. In this work it was used a start problem that supplied the beginning point to argue the certainties and doubts of the subject in a way to investigate the cognitive mechanisms that form the limit concept and the numeric continuum, its basis.

O conceito de limite é a base para qualquer aluno de um curso superior que faça a disciplina de cálculo, portanto deveria haver uma ênfase especial na sua aprendizagem por parte de professores e alunos. Na sala de aula a ênfase é dada sobre a definição formal e sobre a forma que ela é apresentada nos livros de cálculo. Neste trabalho foi feita uma pesquisa sobre a natureza do conceito de limite junto com uma reflexão sobre o que é um conceito. Houve uma pesquisa para entender quais os mecanismos cognitivos que estão envolvidos quando um sujeito atua sobre um problema que envolve tal conceito. Buscaram-se elementos de apoio para a investigação no processo histórico de desenvolvimento deste conceito matemático e é preciso destacar a importância deste processo para desenvolver objetos digitais interativos que pudessem ser utilizados em experimentos desenvolvidos para entender os mecanismos cognitivos deste conceito. De modo paralelo ao estudo da história do conceito e seus antecedentes houve um outro estudo sobre objetos digitais interativos e as possibilidades de utilizá-los na investigação dos mecanismos cognitivos. A teoria escolhida para a análise epistemológica do conceito foi a Epistemologia Genética de Jean Piaget, especificamente a teoria relativa a relações e operações infralógicas, lógicas e ao pensamento formal. Na investigação realizada foi utilizado um problema como desequilíbrio inicial. Este forneceu o ponto de partida para questionar as certezas e dúvidas do sujeito de forma a investigar os mecanismos cognitivos que formam o conceito de limite e do contínuo numérico. / The concept of limit is the basis for every student accessing the college and needs study calculus, therefore it should be an emphasis about its learning from students and teachers. At the classroom the emphasis is on the formal definition and over the way it is presented at the books of calculus. On this work a research was carried on and it was about the nature of concept of limit together with a reflexion about what is a concept. There was a research for understanding the cognitive mechanisms related when um individual acts on a problem with that concept. There was a search for elements, in the historic development, that supported the investigation about that mathematic concept and is necessary to regard the importance of that process for the design of interactive digital objects that could be used in experiments designed for understanding the cognitive mechanism related to the concept of limits. In a parallel with the study about the history of limit concept and his ancestors there was another study about interactive digital objects and the possibilities for using that in the investigation of cognitive mechanism. The theory to be used in the analysis was The Genetic Epistemology of Jean Piaget, especially the section about infralogic and logic relation and operations and the formal thinking. In this work it was used a start problem that supplied the beginning point to argue the certainties and doubts of the subject in a way to investigate the cognitive mechanisms that form the limit concept and the numeric continuum, its basis.

Complex 3D digital objects are used in many domains such as animation films, scientific visualization, medical imaging and computer vision. These objects are usually represented by triangular meshes with many triangles. The simplification of those objects in order to keep them as close as possible to the original has received a lot of attention in the recent years. In this context, we propose a simplification algorithm which is focused on the accuracy of the simplifications. The mesh simplification uses edges collapses with vertex relocation by minimizing an error metric. Accuracy is obtained with the two error metrics we use: the Accurate Measure of Quadratic Error (AMQE) and the Symmetric Measure of Quadratic Error (SMQE). AMQE is computed as the weighted sum of squared distances between the simplified mesh and the original one. Accuracy of the measure of the geometric deviation introduced in the mesh by an edge collapse is given by the distances between surfaces. The distances are computed in between sample points of the simplified mesh and the faces of the original one. SMQE is similar to the AMQE method but computed in the both, direct and reverse directions, i.e. simplified to original and original to simplified meshes. The SMQE approach is computationnaly more expensive than the AMQE but the advantage of computing the AMQE in a reverse fashion results in the preservation of boundaries, sharp features and isolated regions of the mesh. For both measures we obtain better results than methods proposed in the literature.

This study investigates the contributions that the use of information and communication technologies brings to the learning processes of students. Many actions have been developed for building a curriculum that integrates ICT into the daily life of the school, these actions are directed from teacher training to the production of teaching materials, including the Learning Objects. This research aims to investigate what are the contributions that the learning object "Kingdom hope – the disappearance of Graceful", built through the Project IVEN, brings to the processes of interaction between partners. In order to identify those elements adopted a qualitative approach through a case study. The theoretical contribution that allowed the analysis was based on Vygotsky's theory, according to which interaction promotes learning. 20 students participated in the research of 2nd year of primary school which, grouped in pairs, used this learning object, having its use videogravada. The research data were transcribed and analysed in vygotskyana perspective and showed that the use of the learning object allows an active interaction of studied students with their partners and that this interaction occurs through the processes of negotiation, motivation and discussion of contents present in all interactions analysed. This study brought contributions to a better understanding of the benefits of the use of ICT, especially the OA. / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Este estudo investiga as contribuições que a utilização das Tecnologias da Informação e Comunicação traz aos processos de aprendizagem de alunos. Muitas ações têm sido desenvolvidas para a construção de um currículo que integre as TIC ao cotidiano da escola, essas ações estão voltadas desde a formação de professores à produção de materiais pedagógicos, dentre eles, destacam-se os Objetos de Aprendizagem. Esta pesquisa tem como objetivo investigar quais as contribuições que o objeto de aprendizagem “Reino Esperança – O sumiço de Graciosa”, construído por intermédio do Projeto RIVED, traz aos processos de interação entre parceiros. A fim de identificar esses elementos adotou-se uma abordagem qualitativa através de um estudo de caso. O aporte teórico que permitiu as análises fundamentou-se na Teoria de Vygotsky, segundo a qual a interação promove a aprendizagem. Participaram da pesquisa 20 alunos do 2º ano do Ensino Fundamental que, agrupados em duplas, utilizaram o referido objeto de aprendizagem, tendo sua utilização videogravada. Os dados da pesquisa foram transcritos e analisados na perspectiva vygotskyana e revelaram que o uso do objeto de aprendizagem estudado permite uma ativa interação dos alunos com seus parceiros e que esta interação ocorre mediante os processos de negociação, motivação e discussão de conteúdos presentes em todas as interações analisadas. Este estudo trouxe contribuições para uma melhor compreensão dos benefícios da utilização das TIC, principalmente dos OA.

While it is generally accepted that environmental data are critical to understanding environmental phenomena, there are yet improvements to be made in their consistent collection, curation, and sharing. This thesis describes two research efforts to improve two different aspects of hydrologic data collection and management. First described is a recipe for the design, development, and deployment of a low-cost environmental data logging and transmission system for environmental sensors and its connection to an open source data-sharing network. The hardware is built using several low-cost, open-source, mass-produced components. The system automatically ingests data into HydroServer, a standards-based server in the open source Hydrologic Information System (HIS) created by the Consortium of Universities for the Advancement of Hydrologic Sciences Inc (CUAHSI). A recipe for building the system is provided along with several test deployment results. Second, a connection between HydroServer and HydroShare is described. While the CUAHSI HIS system is intended to empower the hydrologic sciences community with better data storage and distribution, it lacks support for the kind of “Web 2.0” collaboration and social-networking capabilities that are increasing scientific discovery in other fields. The design, development, and testing of a software system that integrates CUAHSI HIS with the HydroShare social hydrology architecture is presented. The resulting system supports efficient archive, discovery, and retrieval of data, extensive creator and science metadata, assignment of a persistent digital identifier such as a Digital Object Identifier (DOI), scientific discussion and collaboration around the data and other basic social-networking features. In this system, HydroShare provides functionality for social interaction and collaboration while the existing HIS provides the distributed data management and web services framework. The system is expected to enable scientists, for the first time, to access and share both national- and research lab-scale hydrologic time series in a standards-based web services architecture combined with a social network developed specifically for the hydrologic sciences.These two research projects address and provide a solution for significant challenges in the automatic collection, curation, and feature-rich sharing of hydrologic data.

The writings of Yuk Hui, bringing together technical and philosophical insights, provides new theory for the digital, something desired by many in archival research. The aim of this study is to assess the use of Hui’s theories on digital objects and web ontologies as a unifying theory for archival studies, looking at the concept of digital records in particular. Two types of ontology are needed in order to understand digital objects according to Hui: (1) ontologies which relates to the technical language and hierarchies of knowledge representation, like metadata; (2) Ontology, which is the question of Being in the tradition of Heidegger’s phenomenology, the understanding of what it is to be in the world. Both of these ontologies are needed to account for the nature of the digital object. Further, digital objects are produced and constituted through their relations, which can be divided into discursive (technical and logical) and existential (the understanding of being in the world) relations. Digital objects must be approached both as technical formal entities, and as beings in the world. The first register is common, the second – more existential approach – is rarely applied. Using a metatheoretical and hermeneutic method, different conceptions of digital records are discussed, in order to find commonalities and possibilities in relation to Hui’s theory. The digital record as: a process of becoming, information as affordance, context, temporality and evidence are all related to and structured through the application of Hui’s perspective.   The thesis ends with the contribution of a possible redefinition of digital records: A digital record is constituted and concretised as material by discursive and existential relations, that taken together function as a persistent representation. Further, Hui’s digital object theory is suggested as a structuring and unifying theory. In relation to digital records Hui’s theory on digital objects can function as a metatheoretical measure, and a method of critical reading. / Yuk Hui för samman tekniska och filosofiska perspektiv och erbjuder ny teori för det digitala, något som ofta efterfrågas inom arkivvetenskapen. Syftet med denna studie är att applicera Huis teori om digitala objekt och webontologier som en förenande teori för arkivstudier, specifikt i förhållandet till konceptet digitala records. Två typer av ontologi behövs för att förstå digitala objekt enligt Hui: (1) ontologier som relaterar till tekniskt språk och hierarkiska kunskapsrepresentationer, så som metadata; (2) Ontologi som frågan om varat i fenomenologen Martin Heideggers bemärkelse; förståelsen av vad det är att vara i världen. Båda dessa ontologier behövs för att redogöra för det digitala objektets natur. Vidare produceras och konstitueras digitala objekt genom sina relationer, som kan delas upp i diskursiva (tekniska och logiska) och existentiella (förståelsen av att vara i världen) relationer. Digitala objekt måste förstås både som tekniska formella entiteter och som varelser i världen. Det första perspektivet är vanligt förekommande medan det andra – mer existentiella förhållningssättet – sällan tillämpas. Med hjälp av en metateoretisk och hermeneutisk metod diskuteras olika föreställningar om digitala records för att hitta likheter och möjligheter i relation till Huis teori. Digitala records som: en process av tillblivande, information som affordance, kontext, temporalitet och bevis är koncept som relateras och struktureras genom tillämpningen av Huis perspektiv.   Uppsatsen bidrar med en möjlig omdefinition av digitala records: Ett digitalt record konstitueras och konkretiseras som materiellt genom diskursiva och existentiella relationer vilka, tillsammans, fungerar som en ihållande representation. Vidare framhålls Huis teori för digitala objekt som en strukturerande och förenande teori. I relation till digitala records kan Huis teori om digitala objekt fungera som ett metatoretiskt mått och en metod för kritisk läsning.

No description available.

Les objets numériques 3D sont utilisés dans de nombreux domaines, les films d'animations, la visualisation scientifique, l'imagerie médicale, la vision par ordinateur.... Ces objets sont généralement représentés par des maillages à faces triangulaires avec un nombre énorme de triangles. La simplification de ces objets, avec préservation de la géométrie originale, a fait l'objet de nombreux travaux durant ces dernières années. Dans cette thèse, nous proposons un algorithme de simplification qui permet l'obtention d'objets simplifiés de grande précision. Nous utilisons des fusions de couples de sommets avec une relocalisation du sommet résultant qui minimise une métrique d'erreur. Nous utilisons deux types de mesures quadratiques de l'erreur : l'une uniquement entre l'objet simplifié et l'objet original (Accurate Measure of Quadratic Error (AMQE) ) et l'autre prend aussi en compte l'erreur entre l'objet original et l'objet simplifié ((Symmetric Measure of Quadratic Error (SMQE)) . Le coût calculatoire est plus important pour la seconde mesure mais elle permet une préservation des arêtes vives et des régions isolées de l'objet original par l'algorithme de simplification. Les deux mesures conduisent à des objets simplifiés plus fidèles aux originaux que les méthodes actuelles de la littérature. / Complex 3D digital objects are used in many domains such as animation films, scientific visualization, medical imaging and computer vision. These objects are usually represented by triangular meshes with many triangles. The simplification of those objects in order to keep them as close as possible to the original has received a lot of attention in the recent years. In this context, we propose a simplification algorithm which is focused on the accuracy of the simplifications. The mesh simplification uses edges collapses with vertex relocation by minimizing an error metric. Accuracy is obtained with the two error metrics we use: the Accurate Measure of Quadratic Error (AMQE) and the Symmetric Measure of Quadratic Error (SMQE). AMQE is computed as the weighted sum of squared distances between the simplified mesh and the original one. Accuracy of the measure of the geometric deviation introduced in the mesh by an edge collapse is given by the distances between surfaces. The distances are computed in between sample points of the simplified mesh and the faces of the original one. SMQE is similar to the AMQE method but computed in the both, direct and reverse directions, i.e. simplified to original and original to simplified meshes. The SMQE approach is computationnaly more expensive than the AMQE but the advantage of computing the AMQE in a reverse fashion results in the preservation of boundaries, sharp features and isolated regions of the mesh. For both measures we obtain better results than methods proposed in the literature.