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Dissertations / Theses on the topic 'Interpolation'
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1
Rabut, Christophe. "B-splines polyharmoniques cardinales : interpolation, quasi-interpolation, filtrage." Toulouse 3, 1990. http://www.theses.fr/1990TOU30046.
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Les B-splines polynomiales sont couramment utilisées pour définir simplement une fonction spline qui passe "près" de points donnés. Dans le cas où les données sont régulièrement réparties, on apporte, par un traitement préalable des données (convolution avec certains vecteurs à support borné), plus de souplesse à cette opération : on peut alors obtenir une fonction qui passe très près des points -on parle alors de quasi-interpolation- ou au contraire qui filtre les bruits inhérents à ces données on parle alors de filtrage. On montre comment utiliser la méthode de validation croisée pour choisir de façon optimale la force d'un filtrage, qui peut être adaptative, et on propose une méthode de réduction de données, le taux de réduction étant lié à la bande passante du filtre. Ces notions sont ensuite généralisées en dimension quelconque par l'utilisation des B-splines polyharmoniques : après avoir defini les splines polyharmoniques pour des données qui peuvent être en nombre infini, on en donne une expression numériquement plus stable que celle généralement utilisée, et on montre un lien entre splines polyharmoniques d'ordre ou de dimension différents. On définit alors les B-splines polyharmoniques, et on présente leurs propriétés essentielles, très voisines de celles des B-splines polynomiales. On propose l'utilisation de ces B-splines d'une part pour quasi-interpoler ou filtrer des données régulièrement réparties, d'autre part pour déterminer rapidement, par une méthode de subdivision, la spline d'interpolation de ces données. On envisage enfin la généralisation de cette notion de B-spline à des noeuds quelconques et à toute famille de fonctions satisfaisant certaines équations différentielles.
2
Ramesh, Gayatri. "FRACTAL INTERPOLATION." Master's thesis, University of Central Florida, 2008. http://digital.library.ucf.edu/cdm/ref/collection/ETD/id/3128.
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This thesis is devoted to a study about Fractals and Fractal Polynomial Interpolation. Fractal Interpolation is a great topic with many interesting applications, some of which are used in everyday lives such as television, camera, and radio. The thesis is comprised of eight chapters. Chapter one contains a brief introduction and a historical account of fractals. Chapter two is about polynomial interpolation processes such as Newton s, Hermite, and Lagrange. Chapter three focuses on iterated function systems. In this chapter I report results contained in Barnsley s paper, Fractal Functions and Interpolation. I also mention results on iterated function system for fractal polynomial interpolation. Chapters four and five cover fractal polynomial interpolation and fractal interpolation of functions studied by Navascués. Chapter five and six are the generalization of Hermite and Lagrange functions using fractal interpolation. As a concluding chapter we look at the current applications of fractals in various walks of life such as physics and finance and its prospects for the future.<br>M.S.<br>Department of Mathematics<br>Sciences<br>Mathematical Science MS
3
Goggins, Dan. "Constraint-based interpolation /." Diss., CLICK HERE for online access, 2005. http://contentdm.lib.byu.edu/ETD/image/etd976.pdf.
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4
Yeung, R. Kacheong. "Stable rational interpolation." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1999. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape7/PQDD_0021/NQ46952.pdf.
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5
Christ, Jürgen [Verfasser], and Andreas [Akademischer Betreuer] Podelski. "Interpolation modulo theories." Freiburg : Universität, 2015. http://d-nb.info/1119805767/34.
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6
Fang, Quanlei. "Multivariable Interpolation Problems." Diss., Virginia Tech, 2008. http://hdl.handle.net/10919/28311.
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In this dissertation, we solve multivariable Nevanlinna-Pick type interpolation problems. Particularly, we consider the left tangential interpolation problems on the commutative or noncommutative unit ball. For the commutative setting, we discuss left-tangential operator-argument interpolation problems for Schur-class multipliers on the Drury-Arveson space and for the noncommutative setting, we discuss interpolation problems for Schur-class multipliers on Fock space. We apply the Krein-space geometry approach (also known as the Grassmannian Approach). To implement this approach J-versions of Beurling-Lax representers for shift-invariant subspaces are required. Here we obtain these J-Beurling-Lax theorems by the state-space method for both settings. We see that the Krein-space geometry method is particularly simple in solving the interpolation problems when the Beurling-Lax representer is bounded. The Potapov approach applies equally well whether the representer is bounded or not.<br>Ph. D.
7
Merrell, Jacob Porter. "Generalized Constrained Interpolation." Diss., CLICK HERE for online access, 2008. http://contentdm.lib.byu.edu/ETD/image/etd2360.pdf.
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8
Goggins, Daniel David. "Constraint-Based Interpolation." BYU ScholarsArchive, 2005. https://scholarsarchive.byu.edu/etd/610.
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Image reconstruction is the process of converting a sampled image into a continuous one prior to transformation and resampling. This reconstruction can be more accurate if two things are known: the process by which the sampled image was obtained and the general characteristics of the original image. We present a new reconstruction algorithm known as Constraint-Based Interpolation, which estimates the sampling functions found in cameras and analyzes properties of real world images in order to produce quality real-world image magnifications. To accomplish this, Constraint-Based Interpolation uses a sensor model that pushes the pixels in an interpolation to more closely match the data in the sampled image. Real-world image properties are ensured with a level-set smoothing model that smooths "jaggies" and a sharpening model that alleviates blurring. This thesis describes the three models, their methods and constraints. The effects of the various models and constraints are also shown, as well as a human observer test. A variation of a previous interpolation technique, Quad-based Interpolation, and a new metric, gradient weighted contour curvature, is presented and analyzed.
9
Ameur, Yacin. "Interpolation of Hilbert spaces." Doctoral thesis, Uppsala universitet, Matematiska institutionen, 2002. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-1753.
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(i) We prove that intermediate Banach spaces A, B with respect to arbitrary Hilbert couples H, K are exact interpolation iff they are exact K-monotonic, i.e. the condition f0∊A and the inequality K(t,g0;K)≤K(t,f0;H), t>0 imply g0∊B and ||g0||B≤||f0||A (K is Peetre's K-functional). It is well-known that this property is implied by the following: for each ρ>1 there exists an operator T : H→K such that Tf0=g0, and K(t,Tf;K)≤ρK(t,f;H), f∊H0+H1, t>0.Verifying the latter property, it suffices to consider the "diagonal" case where H=K is finite-dimensional. In this case, we construct the relevant operators by a method which allows us to explicitly calculate them. In the strongest form of the theorem, it is shown that the statement remains valid when substituting ρ=1. (ii) A new proof is given to a theorem of W. F. Donoghue which characterizes certain classes of functions whose domain of definition are finite sets, and which are subject to certain matrix inequalities. The result generalizes the classical Löwner theorem on monotone matrix functions, and also yields some information with respect to the finer study of monotone functions of finite order. (iii) It is shown that with respect to a positive concave function ψ there exists a function h, positive and regular on ℝ+ and admitting of analytic continuation to the upper half-plane and having positive imaginary part there, such that h≤ψ≤ 2h. This fact is closely related to a theorem of Foiaş, Ong and Rosenthal, which states that regardless of the choice of a concave function ψ, and a weight λ, the weighted l2-space l2(ψ(λ)) is c-interpolation with respect to the couple (l2,l2(λ)), where we have c≤√2 for the best c. It turns out that c=√2 is best possible in this theorem; a fact which is implicit in the work of G. Sparr. (iv) We give a new proof and new interpretation (based on the work (ii) above) of Donoghue's interpolation theorem; for an intermediate Hilbert space H* to be exact interpolation with respect to a regular Hilbert couple H it is necessary and sufficient that the norm in H* be representable in the form ||f||*= (∫[0,∞] (1+t-1)K2(t,f;H)2dρ(t))1/2 with some positive Radon measure ρ on the compactified half-line [0,∞]. (v) The theorem of W. F. Donoghue (item (ii) above) is extended to interpolation of tensor products. Our result is related to A. Korányi's work on monotone matrix functions of several variables.
10
Ameur, Yacin. "Interpolation of Hilbert spaces /." Uppsala : Matematiska institutionen, Univ. [distributör], 2001. http://publications.uu.se/theses/91-506-1531-9/.
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Kristoffersen, Stian. "The Empirical Interpolation Method." Thesis, Norges teknisk-naturvitenskapelige universitet, Institutt for matematiske fag, 2013. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-23378.
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In this thesis we look at the Empirical Interpolation Method (EIM) and how it can be used in different applications. We propose a new formulation of EIM to make it easier to perform analytical operations like differentiation and integration of the basis functions as well as to apply EIM to a variety of problems. The new formulation is used to develop quadrature rules for the circle and semicircle, as well as for arbitrary simple polygons. The new formulation is also used to solve partial differential equations using a collocation approach on various domains including the circle, semicircle and triangle. The framework is briefly applied to compression of 3D animation in addition to recognition of images and sound.Several of the methods show great potential, with exponential convergence for quadrature and collocation for regular problems. However, there are also serious issues that must be addressed if the methods are to be developed further. These issues are related to making the methods more robust and stable.
12
FERREIRA, CARLOS ROBERTO DA COSTA. "MODIFIED INTERPOLATION OF LSFNULLS." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2006. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=9195@1.
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COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR<br>Os novos serviços de telecomunicações têm impulsionado o
desenvolvimento de melhorias nos algoritmos de codificação
de voz, devido à
necessidade de se melhorar a qualidade da voz codificada,
utilizando a menor taxa
de transmissão possível. Esta dissertação analisa e
propõem melhorias em um
método para o ajuste de parâmetros LSFs de modo a torná-
los mais precisos,
minimizando as perdas no processo de interpolação de LSFs
codificadas. Com
isso, a percepção de qualidade da voz sintetizada na saída
do decodificador é
aumentada, sem que seja necessário aumento da taxa de
transmissão. É
apresentada de modo detalhado toda a dedução matemática do
método citado.
Para a avaliação de desempenho das melhorias propostas, o
processo de ajuste é
implementado em um codificador a taxas médias inferiores a
2 kb/s. Os resultados
confirmam que é possível obter redução significativa nas
medidas de distorção
com a utilização do ajuste de LSFs.<br>The new telecommunications services have been pushing
forward the
development of improvements in speech coding, because of
the need to improve
coded speech quality, using the smallest transmission rate
possible. This
thesis analyzes and proposes improvements in a method to
adjust LSF parameters
so they get more accurate, minimizing the losses in the
coded LSFs interpolation
process. With this, the synthesized speech perceptual
quality
in the decoder exit is increased, without having to
increase the transmission rate.
The mathematical deduction of the method is presented in a
detailed way. To
evaluate the performance of the proposed improvements, the
adjust process is
implemented in a speech coder with mean rates less than 2
kb/s. The results
confirmed that is possible to obtain significant reduction
in distortion measures using the adjustment of LSFs.
13
Jin, Shangzhu. "Backward fuzzy rule interpolation." Thesis, Aberystwyth University, 2015. http://hdl.handle.net/2160/e4e6acbd-914f-4c8a-b40c-91b13a017c69.
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Naik, Nitin. "Dynamic fuzzy rule interpolation." Thesis, Aberystwyth University, 2015. http://hdl.handle.net/2160/1caf8126-23c0-4c9c-9d8d-16c65f2f9878.
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Designers of effective and efficient fuzzy systems have long recognised the value of inferential hybridity in the implementation of sparse fuzzy rule based systems. Which is to say: such systems should have recourse to fuzzy rule interpolation (FRI) only when no rule matches a given observation; otherwise, when an observation partially or exactly matches at least one of the rules of the sparse rule base, a compositional rule of inference (CRI) should be used in order to avoid the computational overheads of interpolation. Sparse fuzzy rule bases are constructed by experts or derived from data and may support FRI reasoning in long run. However, two potential problems arise: (1) a system's requirements may change over time leading to rule redundancy; and (2) the system may cease in the long run to provide precise and pertinent results. The need to maintain the concurrency and accuracy of a sparse fuzzy rule base, in order that it generates the most precise and relevant results possible, motivates consideration of a dynamic (real-time) fuzzy rule base. This thesis therefore presents a framework of dynamic fuzzy rule interpolation (D-FRI), integrated with general fuzzy inference (CRI), which uses the FRI result set itself for the selection, combination and promotion of informative, frequentlyused intermediate rules into the existing rule base. Here two versions of the D-FRI approach are presented:k-means-based and GA-aided. Integration uses the concept of -cut overlapping between fuzzy sets to decide an exact or partial matching between rules and observation so that CRI can be utilised for reasoning. Otherwise, the best closest rules are selected for FRI by exploiting the centre of gravity (COG), Hausdorff distance (HD) and earth mover's distance (EMD) metrics. Testing seeks to show that dynamically-promoted rules generate results of greater accuracy and robustness than would be achievable through conventional FRI tout court, and to support the claim that the D-FRI approach results in a more effective interpolative reasoning system. To this end, an implementation of D-FRI is applied to the problem domain of intrusion detection systems (IDS), by integrating it with Snort in order to improve port-scanning detection and increase the level of accuracy of alert predictions.
15
Du, Toit Wilna. "Radial basis function interpolation." Thesis, Stellenbosch : Stellenbosch University, 2008. http://hdl.handle.net/10019.1/2002.
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Thesis (MSc (Applied Mathematics))--Stellenbosch University, 2008.<br>A popular method for interpolating multidimensional scattered data is using
radial basis functions. In this thesis we present the basic theory of radial basis
function interpolation and also regard the solvability and stability of the
method. Solving the interpolant directly has a high computational cost for
large datasets, hence using numerical methods to approximate the interpolant
is necessary. We consider some recent numerical algorithms. Software to implement
radial basis function interpolation and to display the 3D interpolants
obtained, is developed. We present results obtained from using our implementation
for radial basis functions on GIS and 3D face data as well as an image
warping application.
16
Goosen, Karin M. (Karin Michelle). "Subdivision, interpolation and splines." Thesis, Stellenbosch : Stellenbosch University, 2000. http://hdl.handle.net/10019.1/51924.
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Thesis (MSc)--University of Stellenbosch, 2000.<br>ENGLISH ABSTRACT: In this thesis we study the underlying mathematical principles of stationary subdivision,
which can be regarded as an iterative recursion scheme for the generation of smooth curves
and surfaces in computer graphics. An important tool for our work is Fourier analysis, from
which we state some standard results, and give the proof of one non-standard result. Next,
since cardinal spline functions have strong links with subdivision, we devote a chapter to this
subject, proving also that the cardinal B-splines are refinable, and that the corresponding
Euler-Frobenius polynomial has a certain zero structure which has important implications
in our eventual applications. The concepts of a stationary subdivision scheme and its convergence
are then introduced, with as motivating example the de Rahm-Chaikin algorithm.
Standard results on convergence and regularity for the case of positive masks are quoted and
graphically illustrated.
Next, we introduce the concept of interpolatory stationary subdivision, in which case
the limit curve contains all the original control points. We prove a certain set of sufficient
conditions on the mask for convergence, at the same time also proving the existence and
other salient properties of the associated refinable function. Next, we show how the analysis
of a certain Bezout identity leads to the characterisation of a class of symmetric masks which
satisfy the abovementioned sufficient conditions. Finally, we show that specific special cases
of the Bezout identity yield convergent interpolatory symmetric subdivision schemes which
are identical to choosing the corresponding mask coefficients equal to certain point evaluations
of, respectively, a fundamental Lagrange interpolation polynomial and a fundamental
cardinal spline interpolant. The latter procedure, which is known as the Deslauriers-Dubuc
subdivision scheme in the case of a polynomial interpolant, has received attention in recent
work, and our approach provides a convergence result for such schemes in a more general
framework.
Throughout the thesis, numerical illustrations of our results are provided by means of
graphs.<br>AFRIKAANSE OPSOMMING: In hierdie tesis ondersoek ons die onderliggende wiskundige beginsels van stasionêre onderverdeling,
wat beskou kan word as 'n iteratiewe rekursiewe skema vir die generering van
gladde krommes en oppervlakke in rekenaargrafika. 'n Belangrike stuk gereedskap vir ons
werk is Fourieranalise, waaruit ons sekere standaardresuJtate formuleer, en die bewys gee van
een nie-standaard resultaat. Daarna, aangesien kardinale latfunksies sterk bande het met
onderverdeling, wy ons 'n hoofstuk aan hierdie onderwerp, waarin ons ook bewys dat die
kardinale B-Iatfunksies verfynbaar is, en dat die ooreenkomstige Euler-Frobenius polinoom
'n sekere nulpuntstruktuur het wat belangrike implikasies het in ons uiteindelike toepassings.
Die konsepte van 'n stasionêre onderverdelingskema en die konvergensie daarvan word
dan bekendgestel, met as motiverende voorbeeld die de Rahm-Chaikin algoritme. Standaardresultate
oor konvergensie en regulariteit vir die geval van positiewe maskers word
aangehaal en grafies geïllustreer.
Vervolgens stelons die konsep van interpolerende stasionêre onderverdeling bekend, in
welke geval die limietkromme al die oorspronklike kontrolepunte bevat. Ons bewys 'n sekere
versameling van voldoende voorwaardes op die masker vir konvergensie, en bewys terselfdertyd
die bestaan en ander toepaslike eienskappe van die ge-assosieerde verfynbare funksie.
Daarna wys ons hoedat die analise van 'n sekere Bezout identiteit lei tot die karakterisering
van 'n klas simmetriese maskers wat die bovermelde voldoende voorwaardes bevredig. Laastens
wys ons dat spesifieke spesiale gevalle van die Bezout identiteit konvergente interpolerende
simmetriese onderverdelingskemas lewer wat identies is daaraan om die ooreenkomstige
maskerkoëffisientegelyk aan sekere puntevaluasies van, onderskeidelik, 'n fundamentele
Lagrange interpolasiepolinoom en 'n kardinale latfunksie-interpolant te kies. Laasgenoemde
prosedure, wat bekend staan as die Deslauriers-Dubuc onderverdelingskema in die geval van
'n polinoominterpolant, het aandag ontvang in onlangse werk, en ons benadering verskaf 'n
konvergensieresultaat vir sulke skemas in 'n meer algemene raamwerk.
Deurgaans in die tesis word numeriese illustrasies van ons resultate met behulp van
grafieke verskaf.
17
Calvi, Jean-Paul. "Interpolation et fonctionnelles analytiques." Toulouse 3, 1993. http://www.theses.fr/1993TOU30014.
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Nous etudions quelques problemes d'interpolation (essentiellement polynomiale). Dans le chapitre premier, nous etudions la reconstruction d'une fonction de plusieurs variables complexes au moyen de polynomes d'interpolation obtenus par la donnee de certaines fonctionnelles des derivees. Plusieurs methodes classiques d'interpolation se laissent mettre sous la forme etudiee. Le second chapitre traite d'un probleme de convergence du polynome d'interpolation de kergin. Etant donnes trois ensembles compacts e,f,g, nous disons que la propriete (e,f,g) est verifiee lorsque pour chaque tableau d'interpolation dans f, chaque fonction f analytique dans un voisinage de g, le polynome de kergin de f converge uniformement vers f sur e. Etant donnes deux de ces trois ensembles, il s'agit de construire le troisieme de telle sorte que (e,f,g) soit verifiee. Dans le dernier chapitre nous etudions le probleme de l'interpolation sous une forme plus abstraite. Nous donnons une condition necessaire et suffisante pour qu'une suite de formes lineaires continues sur un espace de frechet soit d'interpolation. En particulier nous proposons un critere pratique qui est applique a plusieurs problemes naturels de la theorie classique des fonctions
18
Eastwood, Alan. "Interpolation à plusieurs variables." Nice, 1988. http://www.theses.fr/1988NICE4170.
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On caractérise les sous-schémas génériques localement rectilignes de dimension zéro rangés de l'espace projectif de dimension finie quelconque sur un corps algébriquement clos. On en tire quelques corollaires
19
Iebesh, Abdulhamid. "Interpolation of Yield curves." Thesis, Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-48968.
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In this thesis we survey several interpolation methods that are used to construct the yield curves. We also review the bootstrapping and show that the bootstrap is closely connected to the interpolation in the case of bootstrapping yield curve. The most effort is dedicated, in this thesis, on the monotone convex method and on investigation of the difficulties to get accurate yield curves.
20
Eastwood, Alan. "Interpolation à plusieurs variables." Grenoble 2 : ANRT, 1988. http://catalogue.bnf.fr/ark:/12148/cb37613442b.
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Miranda, Gerald N. "Interpolation weights of algebraic multigrid." Monterey, Calif. : Springfield, Va. : Naval Postgraduate School ; Available from National Technical Information Service, 1997. http://handle.dtic.mil/100.2/ADA334079.
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22
Gandhi, Sonia. "ENO interpolation for image compression." Diss., Connect to online resource, 2005. http://wwwlib.umi.com/cr/colorado/fullcit?p1425778.
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23
Subhan, Fazli. "Multilevel sparse kernel-based interpolation." Thesis, University of Leicester, 2011. http://hdl.handle.net/2381/9894.
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Radial basis functions (RBFs) have been successfully applied for the last four decades for fitting scattered data in Rd, due to their simple implementation for any d. However, RBF interpolation faces the challenge of keeping a balance between convergence performance and numerical stability. Moreover, to ensure good convergence rates in high dimensions, one has to deal with the difficulty of exponential growth of the degrees of freedom with respect to the dimension d of the interpolation problem. This makes the application of RBFs limited to few thousands of data sites and/or low dimensions in practice. In this work, we propose a hierarchical multilevel scheme, termed sparse kernel-based interpolation (SKI) algorithm, for the solution of interpolation problem in high dimensions. The new scheme uses direction-wise multilevel decomposition of structured or mildly unstructured interpolation data sites in conjunction with the application of kernel-based interpolants with different scaling in each direction. The new SKI algorithm can be viewed as an extension of the idea of sparse grids/hyperbolic cross to kernel-based functions. To achieve accelerated convergence, we propose a multilevel version of the SKI algorithm. The SKI and multilevel SKI (MLSKI) algorithms admit good reproduction properties: they are numerically stable and efficient for the reconstruction of large data in Rd, for d = 2, 3, 4, with several thousand data. SKI is generally superior over classical RBF methods in terms of complexity, run time, and convergence at least for large data sets. The MLSKI algorithm accelerates the convergence of SKI and has also generally faster convergence than the classical multilevel RBF scheme.
24
Arnesen, Martin. "The Generalized Empirical Interpolation Method." Thesis, Norges teknisk-naturvitenskapelige universitet, Institutt for matematiske fag, 2014. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-24676.
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The empirical interpolation method is an interpolation scheme with problem dependent basis functions and interpolation nodes, originally developed for parameter dependent functions. It was developed in connection with the reduced basis framework for fast evaluation of output from parameterized partial differential equations, but the procedure may be applicable to a variety of problems, such as image and pattern recognition, numerical integration and data compression. We present the theoretical background and implementation of the method, and give examples to verify exponential convergence for analytic problems. An extension of the method was proposed recently, denoted as the generalized empirical interpolation method (GEIM). The GEIM considers a parametric manifold of functions, with a set of linear functionals. Further, we explore how the interpolation points can be used as measurement points in the estimation of parameters from noisy data. We present the statistical framework, and we show how we can identify a set of parameter values that are consistent with our measurements.
25
Asaturyan, Souren. "Shape preserving surface interpolation schemes." Thesis, University of Dundee, 1989. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.278368.
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Chen, Chengyuan. "Higher order fuzzy rule interpolation." Thesis, Aberystwyth University, 2015. http://hdl.handle.net/2160/28b8a37d-9833-4afb-b0c8-bef53749a9d2.
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Fuzzy inference is an effective means for representing and handling vagueness and imprecision. As a particular type of fuzzy inference, fuzzy rule interpolation enhances the performance of the inference when a given observation has no overlap with the antecedent values of any of the existing rules. In such cases, conventional fuzzy inference methods cannot derive a conclusion, but fuzzy rule interpolation methods can still obtain a certain conclusion. Unfortunately, very little of the existing work on fuzzy rule interpolation can conjunctively handle more than one form of uncertainty in the rules or observations. In particular, the difficulty in defining the required precise-valued membership functions for the fuzzy sets that are used by conventional fuzzy rule interpolation techniques significantly restricts their application. In this thesis, a novel framework termed 'higher order fuzzy rule interpolation' is proposed in an attempt to address such difficulties. The proposed framework allows the representation, handling and utilisation of different types of uncertainty in knowledge. This allows transformation-based fuzzy rule interpolation techniques to harness and utilise the additional uncertainty in order to implement a fuzzy interpolative reasoning system. Final conclusions can then be derived by performing higher order interpolation over this representation. The techniques for the representation and handling of uncertainty are organised in this framework such that in circumstances when different types of uncertainty are encountered the inference process can deal with them in an appropriate way. A roughfuzzy set based rule interpolation approach is proposed in this work, by exploiting the concept of rough-fuzzy sets and generalising scale and move transformation-based fuzzy interpolation. A type-2 fuzzy set based interpolation approach is also presented as an alternative implementation of the framework. The effectiveness of this work in improving the robustness of fuzzy rule interpolation is demonstrated through the practical application to the prediction of disease rates in remote villages. Moreover, this framework is also further evaluated with application to other realistic decision making problems. The resultant accuracy reveals the efficacy of this research.
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Yang, Longzhi. "Fuzzy interpolation and its adaptation." Thesis, Aberystwyth University, 2011. http://hdl.handle.net/2160/d24357c7-91a1-400d-83ce-23d1b1aa01cf.
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Fuzzy inference has been widely used to represent and manage the imprecision and incompleteness in commonsense reasoning with high performance and comprehensibility. Fuzzy interpolation, a particular type of fuzzy inference, strengthens the power of fuzzy inference in two respects. Firstly, it reduces system complexity by omitting those rules that can be approximated by their neighbouring ones. Secondly, it enhances the robustness of fuzzy systems by guaranteeing a certain conclusion always being generated. However, it is possible that multiple object values for a common variable are inferred from complex real-world applications by fuzzy inference. Particularly for fuzzy interpolation, due to the high incompleteness of knowledge bases and the imprecision of observations and rules used for fuzzy interpolation, these values are likely to be inconsistent. Such inconsistencies may result from incorrect observations, incorrect rules in the given rule base or defective interpolation procedures. This PhD work presents, as its main part, a novel approach for identification and correction of multiple simultaneous faults, including incorrect observations, incorrect rules or/and defective interpolation procedures, during the interpolation process in an effort to remove all the inconsistencies. In particular, the assumption-based truth maintenance system (ATMS) is employed to record the dependencies of reasoning conclusions and system inconsistencies, while the underlying technique that the classical general diagnostic engine (GDE) employs for fault localisation is adapted to isolate possible sets of faults. From this, a modification mechanism is introduced to correct a set of identified faults in interpolation, thereby removing inconsistencies. This approach is applied to a real-world problem, which predicates the diarrhoeal disease rates in remote villages, to demonstrate the potential of this work in improving the effectiveness of fuzzy interpolation. The scale and move transformation-based fuzzy interpolation is utilised as the foundation of this work. This approach has been extended to deal with interpolation and extrapolation with multiple multi-antecedent rules. However, the generalised approach may not be able to degenerate back to the basic crisp interpolation and extrapolation based on two rules; and the approximate function of the extended approach may not be continuous. In order to address these limitations, a new approach to generalising the basic fuzzy interpolation techniques is also proposed in this work. Experimental results show that the proposed extension not only successfully removes all the existing shortcomings, but also leads to more reasonable conclusions. In addition, as a supplementary to ordinary fuzzy reasoning, results derived by fuzzy interpolation are expected to be compatible with those obtained by ordinary fuzzy reasoning whenever both approaches are applicable, but this is often not the case. Another further development of fuzzy interpolation made in this work is a different fuzzy interpolation approach based on a direct use of the Extension Principle, which bears a close relationship with the compositional rule of inference, which in turn plays a crucial rule in ordinary fuzzy reasoning. The proposed fuzzy interpolation approach has been demonstrated to be compatible with ordinary fuzzy inference for situations where ordinary fuzzy reasoning can be performed while still entailing interpolative inference for situations where an observation matches no fuzzy rules.
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Setterqvist, Eric. "Taut Strings and Real Interpolation." Doctoral thesis, Linköpings universitet, Matematik och tillämpad matematik, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-132421.
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The taut string problem concerns finding the function with the shortest graph length, i.e. the taut string, in a certain set of continuous piecewise linear functions. It has appeared in a broad range of applications including statistics, image processing and economics. As it turns out, the taut string has besides minimal graph length also minimal energy and minimal total variation among the functions in Ω. The theory of real interpolation is based on Peetre’s K-functional. In terms of the K-functional, we introduce invariant K-minimal sets and show a close connection between taut strings and invariant K-minimal sets. This insight leads to new problems of interpolation theory, gives possibility to generalize the notion of taut strings and provides new applications. The thesis consists of four papers. In paper I, connections between invariant K-minimal sets and various forms of taut strings are investigated. It is shown that the set Ω′ of the derivatives of the functions in can be interpreted as an invariant K-minimal set for the Banach couple (ℓ1, ℓ∞) on Rn. In particular, the derivative of the taut string has minimal K-functional in Ω′. A characterization of all bounded, closed and convex sets in Rn that are invariant K-minimal for (ℓ1, ℓ∞) is established. Paper II presents examples of invariant K-minimal sets in Rn for (ℓ1, ℓ∞). A convergent algorithm for computing the element with minimal K-functional in such sets is given. In the infinite-dimensional setting, a sufficient condition for a set to be invariant K-minimal with respect to the Banach couple L1 ([0,1]m) ,L∞ ([0,1]m) is established. With this condition at hand, different examples of invariant K-minimal sets for this couple are constructed. Paper III considers an application of taut strings to buffered real-time communication systems. The optimal buffer management strategy, with respect to minimization of a class of convex distortion functions, is characterized in terms of a taut string. Further, an algorithm for computing the optimal buffer management strategy is provided. In paper IV, infinite-dimensional taut strings are investigated in connection with the Wiener process. It is shown that the average energy per unit of time of the taut string in the long run converges, if it is constrained to stay within the distance r > 0 from the trajectory of a Wiener process, to a constant C2/r2 where C ∈ (0,∞). While the exact value of C is unknown, the numerical estimate C ≈ 0.63 is obtained through simulations on a super computer. These simulations are based on a certain algorithm for constructing finite-dimensional taut strings.
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Nordenfors, Oskar. "The Riesz-Thorin Interpolation Theorem." Thesis, Umeå universitet, Institutionen för matematik och matematisk statistik, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-149503.
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In this essay we present some elementary measure theory and some theory of Lp-spaces with the goal of proving the Riesz-Thorin interpolation theorem.<br>I denna uppsats presenteras grundläggande måtteori och något kring teorin om Lp-rum med målet att bevisa Riesz-Thorins interpolationssats.
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Dehlbom, Gustaf. "Interpolation of the yield curve." Thesis, Uppsala universitet, Tillämpad matematik och statistik, 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-423128.
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Severo, Franco. "Interpolation schemes in percolation theory." Thesis, université Paris-Saclay, 2020. http://www.theses.fr/2020UPASM004.
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Cette thèse fournit de nouveaux résultats concernant la transition de phase des modèles de percolation, en particulier la percolation de Bernoulli et les lignes de niveau du champ libre gaussien. La technique commune utilisée dans ces résultats consiste à comparer deux modèles de percolation différents en construisant une famille de modèles interpolant entre les deux. L’objectif principal de cette thèse est d’illustrer comment cette technique peut être appliquée dans un large contexte<br>This thesis provides new results concerning the phase transition of percolation models, specially Bernoulli percolation and level-sets of the Gaussian free field. The common technique used in theses results consists in comparing two different percolation models by continuously interpolating between them. The main purpose of this thesis is to illustrate how this technique can be applied to a wider variety of contexts than those previously studied
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Leung, Nim Keung. "Convexity-Preserving Scattered Data Interpolation." Thesis, University of North Texas, 1995. https://digital.library.unt.edu/ark:/67531/metadc277609/.
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Surface fitting methods play an important role in many scientific fields as well as in computer aided geometric design. The problem treated here is that of constructing a smooth surface that interpolates data values associated with scattered nodes in the plane. The data is said to be convex if there exists a convex interpolant. The problem of convexity-preserving interpolation is to determine if the data is convex, and construct a convex interpolant if it exists.
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Taylor, Rodney. "Lagrange interpolation on Leja points." [Tampa, Fla] : University of South Florida, 2008. http://purl.fcla.edu/usf/dc/et/SFE0002363.
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Dinh, Andreas. "Lokale Lagrange-Interpolation mit Splineoberflächen." [S.l. : s.n.], 2006. http://nbn-resolving.de/urn:nbn:de:bsz:180-madoc-13537.
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Martin, Peter. "Spatial interpolation in other dimensions /." Connect to this title online, 2004. http://hdl.handle.net/1957/4063.
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Good, Jennifer Rose. "Weighted interpolation over W*-algebras." Diss., University of Iowa, 2015. https://ir.uiowa.edu/etd/1843.
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An operator-theoretic formulation of the interpolation problem posed by Nevanlinna and Pick in the early twentieth century asks for conditions under which there exists a multiplier of a reproducing kernel Hilbert space that interpolates a specified set of data. Paul S. Muhly and Baruch Solel have shown that their theory for operator algebras built from W*-correspondences provides an appropriate context for generalizing this classic question. Their reproducing kernel W*-correspondences are spaces of functions that generalize the reproducing kernel Hilbert spaces. Their Nevanlinna-Pick interpolation theorem, which is proved using commutant lifting, implies that the algebra of multipliers of the reproducing kernel W*-correspondence associated with a certain W*-version of the classic Szegö kernel may be identified with their primary operator algebra of interest, the Hardy algebra.
To provide a context for generalizing another familiar topic in operator theory, the study of the weighted Hardy spaces, Muhly and Solel have recently expanded their theory to include operator-valued weights. This creates a new family of reproducing kernel W*-correspondences that includes certain, though not all, classic weighted Hardy spaces. It is the purpose of this thesis to generalize several of Muhly and Solel's results to the weighted setting and investigate the function-theoretic properties of the resulting spaces.
We give two principal results. The first is a weighted version of Muhly and Solel's commutant lifting theorem, which we obtain by making use of Parrott's lemma. The second main result, which in fact follows from the first, is a weighted Nevanlinna-Pick interpolation theorem. Other results, several of which follow from the two primary results, include the construction of an orthonormal basis for the nonzero tensor product of two W*-corrrespondences, a double commutant theorem, the identification of several function-theoretic properties of the elements in the reproducing kernel W*-correspondence associated with a weighted W*-Szegö kernel as well as the elements in its algebra of mutlipliers, and the presentation of a relationship between this algebra of multipliers and a weighted Hardy algebra. In addition, we consider a candidate for a W*-version of the complete Pick property and investigate the aforementioned weighted W*-Szegö kernel in its light.
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Dudziak, William James. "PRESENTATION AND ANALYSIS OF A MULTI-DIMENSIONAL INTERPOLATION FUNCTION FOR NON-UNIFORM DATA: MICROSPHERE PROJECTION." University of Akron / OhioLINK, 2007. http://rave.ohiolink.edu/etdc/view?acc_num=akron1183403994.
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Ermis, Evren [Verfasser], and Andreas [Akademischer Betreuer] Podelski. "Interpolation in software model checking and fault localization = Interpolation in Software Model Checking und Defektlokalisierung." Freiburg : Universität, 2014. http://d-nb.info/1123479259/34.
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Khosravan, Najafabadi Shohreh. "Optimal vector interpolation of asynoptic spatial survey of vector quantities for interpolating ADCP water velocity measurements." Thesis, University of Ottawa (Canada), 2006. http://hdl.handle.net/10393/27381.
Full textAbstract:
In fields of study such as geophysics and hydraulics, many random variables are vector quantities, not scalars. Vector quantities require statistical techniques that are independent of choice of coordinate system. In this research a new optimal vector interpolation method, suitable for interpolation of asynoptically measured spatial vector fields, was developed and tested. The new method was compared to scalar interpolation by kriging. The test data were spatial Acoustic Doppler Current Profiler (ADCP) surveys of depth average fluvial water velocity in reaches upstream and downstream of a bridge. The interpolation procedures were evaluated by interpolating the fields with various amounts of data removal, and comparing to the actual measured field using a vector correlation coefficient previously developed by Crosby et al. (1993). The new optimal vector interpolation method was superior to kriging when all data were utilized (upstream reach) and for data removal rates of up to 30% (downstream reach).
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Apel, Thomas, and Cornelia Pester. "Clément-type interpolation on spherical domains - interpolation error estimates and application to a posteriori error estimation." Universitätsbibliothek Chemnitz, 2006. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200601335.
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In this paper, a mixed boundary value problem for
the Laplace-Beltrami operator is considered for
spherical domains in $R^3$, i.e. for domains on
the unit sphere. These domains are parametrized
by spherical coordinates (\varphi, \theta),
such that functions on the unit sphere are
considered as functions in these coordinates.
Careful investigation leads to the introduction
of a proper finite element space corresponding to
an isotropic triangulation of the underlying
domain on the unit sphere. Error estimates are
proven for a Clément-type interpolation operator,
where appropriate, weighted norms are used.
The estimates are applied to the deduction of
a reliable and efficient residual error estimator
for the Laplace-Beltrami operator.
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Hecklin, Gero. "Interpolation mit bivariaten und trivariaten Splineräumen." [S.l.] : [s.n.], 2005. http://deposit.ddb.de/cgi-bin/dokserv?idn=974189065.
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Facciolo, Furlan Gabriele. "Irregularly sampled image resortation and interpolation." Doctoral thesis, Universitat Pompeu Fabra, 2011. http://hdl.handle.net/10803/22714.
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The generation of urban digital elevation models from satellite images using stereo
reconstruction techniques poses several challenges due to its precision requirements.
In this thesis we study three problems related to the reconstruction of urban models
using stereo images in a low baseline disposition. They were motivated by the MISS project,
launched by the CNES (Centre National d'Etudes Spatiales), in order to develop a low
baseline acquisition model.
The first problem is the restoration of irregularly sampled images and image fusion
using a band limited interpolation model. A novel restoration algorithm is proposed,
which incorporates the image formation model as a set of local constraints, and uses
of a family of regularizers that allow to control the spectral behavior of the solution.
Secondly, the problem of interpolating sparsely sampled images is addressed using a
self-similarity prior. The related problem of image inpainting is also considered,
and a novel framework for exemplar-based image inpainting is proposed. This framework is
then extended to consider the interpolation of sparsely sampled images. The third problem
is the regularization and interpolation of digital elevation models imposing geometric
restrictions. The geometric restrictions come from a reference image. For this problem
three different regularization models are studied: an anisotropic minimal surface
regularizer, the anisotropic total variation and a new piecewise affine interpolation
algorithm.<br>La generación de modelos urbanos de elevación a partir de imágenes de satélite mediante
técnicas de reconstrucción estereoscópica presenta varios retos debido a sus requisitos
de precisión. En esta tesis se estudian tres problemas vinculados a la generación de
estos modelos partiendo de pares estereoscópicos adquiridos por satélites en una configuración
con baseline pequeño. Estos problemas fueron motivados por el proyecto MISS,
lanzado por el CNES (Centre National d'Etudes Spatiales) con el objetivo de desarrollar las
técnicas de reconstrucción para imágenes adquiridas con baseline pequeños. El primer
problema es la restauración de imágenes muestreadas irregularmente y la fusión de imágenes
usando un modelo de interpolación de banda limitada. Se propone un nuevo método de
restauración, el cual usa una familia de regularizadores que permite controlar el
decaimiento espectral de la solución e incorpora el modelo de formación de imagen como un
conjunto de restricciones locales. El segundo problema es la interpolación de imágenes
muestreadas en forma dispersa usando un prior de auto similitud, se considera también el
problema relacionado de inpainting de imágenes. Se propone un nuevo framework para
inpainting basado en ejemplares, el cual luego es extendido a la interpolación de imágenes
muestreadas en forma dispersa. El tercer problema es la regularización e interpolación de
modelos digitales de elevación imponiendo restricciones geométricas las cuales se extraen de
una imagen de referencia. Para este problema se estudian tres modelos de regularización: un
regularizador anisótropo de superficie mínima, la variación total anisótropa y un nuevo
algoritmo de interpolación afín a trozos.
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Riehle, Thomas J. "Adaptive bilateral extensor for image interpolation." Diss., Columbia, Mo. : University of Missouri-Columbia, 2006. http://hdl.handle.net/10355/4555.
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Thesis (M.S.)--University of Missouri-Columbia, 2006.<br>The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file viewed on (February 23, 2007) Includes bibliographical references.
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Ounaïes, Myriam. "Interpolation dans les algèbres de Hörmander." Habilitation à diriger des recherches, Université Louis Pasteur - Strasbourg I, 2008. http://tel.archives-ouvertes.fr/tel-00338027.
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Nous traitons des problèmes d'interpolation dans les espaces ${\mathcal A}_p(\C)$ des fonctions entières telles que $\sup_{z\in \C}\vert f(z)\vert e^{-Bp(z)}<\infty$, où $p$ est une fonction poids et $B$ est une constante positive qui peut varier. Ces espaces sont des algèbres, qu'on appelle algèbres de Hörmander. Le problème peut être formulé de la manière suivante : étant donnée une suite discrète de nombres complexes $\{\alpha_j\}$ et une suite de valeurs complexes $\{w_j\}$ vérifiant $\sup_j\vert w_j\vert e^{-B'p(\alpha_j)}<\infty$ avec une certaine constante $B'>0$, à quelles conditions existe-t-il une fonction $f\in {\mathcal A}_p(\C)$ telle que, pour tout $j$,$f(\alpha_j)=w_j $?Ce problème a été motivé par ses applications à l'analyse harmonique et particulièrement aux équations de convolution. Nous explorons cet aspect en appliquant certains de nos résultats sur l'interpolation aux fonctions moyenne-périodiques. Nous nous intéressons également à la question de l'interpolation en plusieurs variables complexes.
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Hartmann, Andreas. "Interpolation libre et opérateurs de Toeplitz." Habilitation à diriger des recherches, Université Sciences et Technologies - Bordeaux I, 2005. http://tel.archives-ouvertes.fr/tel-00281652.
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Les travaux présentés dans cette habilitation sont articulés autour d'un thème fédérateur : interpolation. Le cas le plus classique consiste à déterminer la trace d'un ensemble de fonctions sur un sous ensemble du domaine de définition commun de notre ensemble de fonctions intial. En particulier les aspects suivants seront étudiés.<br /><br />1) Interpolation simple : interpolation des valeurs en des points ;<br /><br />2) Interpolation généralisée : p.ex. interpolation des dérivées, interpolation sur des points proches, interpolation tangentielle, etc. ;<br /><br />3) Interpolation classique : l'interpolation est définie à partir d'un espace des traces déterminé a priori ;<br /><br />4) Interpolation libre : l'interpolation est définie à partir d'une propriété de la trace (à savoir d'être un idéal d'ordre) ;<br /><br />5) Interpolation libre et fonctions extrémales : caractérisation de l'interpolation en termes de fonctions extrémales ;<br /><br />6) Interpolation libre et opérateurs de Toeplitz.<br /><br />Le dernier point nous éloignera un peu des problèmes d'interpolation. Même s'il existe un lien étroit entre les problèmes d'interpolation libre (en particulier dans les espaces de type Paley-Wiener ou plus généralement les espaces modèles, voir Section 4.1), nous allons nous intéresser de plus près à certaines propriétés des opérateurs de Toeplitz qui se révèlent importantes dans le contexte de l'interpolation. Cependant, notre étude sera menée détachée du contexte de l'interpolation. Ce sera l'occasion de rencontrer à nouveau des fonctions extrémales. Nous allons en effet étudier les fonctions extrémales des noyaux d'opérateurs de Toeplitz (supposés non triviaux). Celles-ci s'avèrent posséder beaucoup de propriétés intéressantes.<br /><br />Une remarque concernant les techniques utilisées. Les problèmes d'interpolation étant abordés dans des situations très variées (espaces de Hilbert et de Banach comme par exemple Bergman et Hardy, algèbres de Fréchet, et même des espaces vectoriels qui ne sont pas topologiques ; interpolation classique, libre et généralisée) nécessitent des méthodes très difféerentes. Par ailleurs, les problèmes connexes sont motivés par des problèmes d'interpolation mais ils sont considérés dans un contexte déconnecté de l'interpolation. Nous verrons ainsi de l'analyse complexe classique (espaces de Hardy, factorisation de Riesz-Nevanlinna, mesures de Carleson, majorantes harmoniques) et harmonique (toujours présente dans le contexte de l'interpolation et du sampling), de la géométrie des espaces de Banach (bases, bases inconditionnelles, espaces d'interpolation, indices de Boyd), de l'analyse fonctionnelle (principes variationnels, certains aspects topologiques) et convexe (Lemme de Minkowski-Farkas) en passant par la théorie des opérateurs (Théorème du relèvement du commutant, sous-espaces invariants), ainsi que de l'analyse complexe d'une et plusieurs variables (méthodes du d-bar) jusqu'aux espaces de de Branges-Rovnyak.
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Shi, Feng. "Panorama interpolation for image-based navigation." Thesis, University of Ottawa (Canada), 2008. http://hdl.handle.net/10393/27607.
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This thesis presents methods for novel image synthesis from cubic panoramas taken with multi-sensor cameras. The pre-captured cubic panoramas are used to interpolate arbitrary views to allow a virtual walkthrough of the remote real environment. In our approach, the "transfer" and "triangulation" methods are adopted to analyse the geometry of cubic panoramas and recover an accurate essential matrix between cubes. To generate a novel view between two aligned cubes, a warping model is applied to warp cubes to approximate navigation. This technique, called cube warping, works by simplifying the model of pixel displacements between cubes. A new raytracing-like image-based interpolation method is also proposed for free-viewpoint cube synthesis. Instead of attempting to recover dense reconstruction precisely, our method tries to reconstruct colours with colour invariance constraints. Due to the fact that photo consistency has more to do with colour than shape, our algorithm can generate a complete novel scene view with maximized photo consistency.
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Filipsson, Lars. "On polynomial interpolation and complex convexity /." Stockholm : Tekniska högsk, 1999. http://www.lib.kth.se/abs99/fili0604.pdf.
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Martin, Russell McAnally Ken. "Interpolation of head-related transfer functions." Fishermans Bend,Victoria : Defence Science and Technology Organisation, 2007. http://hdl.handle.net/1947/8028.
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Siu, Anthony. "Real time trajectory generation and interpolation." Thesis, University of British Columbia, 2011. http://hdl.handle.net/2429/35898.
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This thesis presents a continuous tool motion trajectory generation algorithm for high speed free form surface machining. A NURBS toolpath generation algorithm is presented to fit the discrete motion commands generated from free-form CAD-models. By using a NURBS representation of the machine part, the toolpath is interpolated continuously to direct the synchronized motion of the 5-axis CNC machine. The higher continuity of the motion trajectory allowed for tighter machining tolerances and reduced feedrate fluctuations and the undesired acceleration harmonics in the overall feed motion and in each of the motor motions. An optimal and feasible feedrate profile have been used to continuously maneuver the cutting tool with the interpolated reference tool position and tool orientation commands such that the kinematic constraints of the drives are not violated.
Commonly used least squares curve fitting of discrete data points forces the curve to weave through the data points and results in a fluctuating toolpath. By making use of the defined basis function distributions of the NURBS control points, a higher smoothness fit has been achieved through a minimization on the chord error and the third derivative of the curve. The feasibility of this toolpath generation algorithm has been extended using the double spline representation to represent both the tool position and the tool orientation with minimal fitting error.
The real time interpolation of the fitted NURBS toolpath has also been implemented using the multi-segment Feed Correction Polynomial. This method provides an adaptive mapping between the nonlinear relationship of the NURBS curve parameter and the curve displacement to allow for a consistent feedrate in the cutting motion. Additionally, the kinematic compatibility conditions are considered based on the inverse kinematics of the 5-axis CNC machine. The proposed algorithm ensures that an overall efficient feed constraint is placed such that none of the individual drives are overdriven. The results from experiments and simulations are presented to demonstrate the effectiveness of the developed trajectory generation algorithms.
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Price, Jeffery Ray. "A framework for adaptive image interpolation." Diss., Georgia Institute of Technology, 1999. http://hdl.handle.net/1853/13718.
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