Tesi sul tema "Cubic spline interpolation"

Numerical methods are essential in computational science, as analytic calculations for large datasets are impractical. Using numerical methods, one can approximate the problem to solve it with basic arithmetic operations. Interpolation is a commonly-used method, inter alia, constructing the value of new data points within an interval of known data points. Furthermore, polynomial interpolation with a sufficiently high degree can make the data set differentiable. One consequence of using high-degree polynomials is the oscillatory behaviour towards the endpoints, also known as Runge's Phenomenon. Spline interpolation overcomes this obstacle by connecting the data points in a piecewise fashion. However, its complex formulation requires nested iterations in higher dimensions, which is time-consuming. In addition, the calculations have to be repeated for computing each partial derivative at the data point, leading to further slowdown. The B-spline interpolation is an alternative representation of the cubic spline method, where a spline interpolation at a point could be expressed as the linear combination of piecewise basis functions. It was proposed that implementing this new formulation can accelerate many scientific computing operations involving interpolation. Nevertheless, there is a lack of detailed comparison to back up this hypothesis, especially when it comes to computing the partial derivatives. Among many scientific research fields, free energy calculations particularly stand out for their use of interpolation methods. Numerical interpolation was implemented in free energy methods for many purposes, from calculating intermediate energy states to deriving forces from free energy surfaces. The results of these calculations can provide insight into reaction mechanisms and their thermodynamic properties. The free energy methods include biased flat histogram methods, which are especially promising due to their ability to accurately construct free energy profiles at the rarely-visited regions of reaction spaces. Free Energies from Adaptive Reaction Coordinates (FEARCF) that was developed by Professor Kevin J. Naidoo has many advantages over the other flat histogram methods. iii Because of its treatment of the atoms in reactions, FEARCF makes it easier to apply interpolation methods. It implements cubic spline interpolation to derive biasing forces from the free energy surface, driving the reaction towards regions with higher energy. A major drawback of the method is the slowdown experienced in higher dimensions due to the complicated nature of the cubic spline routine. If the routine is replaced by a more straightforward B-spline interpolation, sampling and generating free energy surfaces can be accelerated. The dissertation aims to perform a comparative study between the cubic spline interpolation and B-spline interpolation methods. At first, data sets of analytic functions were used instead of numerical data to compare the accuracy and compute the percentage errors of both methods by taking the functions themselves as reference. These functions were used to evaluate the performances of the two methods at the endpoints, inflections points and regions with a steep gradient. Both interpolation methods generated identically approximated values with a percentage error below the threshold of 1%, although they both performed poorly at the endpoints and the points of inflection. Increasing the number of interpolation knots reduced the errors, however, it caused overfitting in the other regions. Although significant speed-up was not observed in the univariate interpolation, cubic spline suffered from a drastic slowdown in higher dimensions with up to 103 in 3D and 105 in 4D interpolations. The same results applied to the classical molecular dynamics simulations with FEARCF with a speed-up of up to 103 when B-spline interpolation was implemented. To conclude, the B-spline interpolation method can enhance the efficiency of the free energy calculations where cubic spline interpolation has been the currently-used method.

The surface of the Earth is subjected to vertical deformations caused by geophysical and geological processes which can be monitored by Global Positioning System (GPS) observations. The purpose of this work is to investigate GPS height time series to identify interannual signals affecting the Earth’s surface over the European and Mediterranean area, during the period 2001-2019. Thirty-six homogeneously distributed GPS stations were selected from the online dataset made available by the Nevada Geodetic Laboratory (NGL) on the basis of the length and quality of the data series. The Principal Component Analysis (PCA) is the technique applied to extract the main patterns of the space and time variability of the GPS Up coordinate. The time series were studied by means of a frequency analysis using a periodogram and the real-valued Morlet wavelet. The periodogram is used to identify the dominant frequencies and the spectral density of the investigated signals; the second one is applied to identify the signals in the time domain and the relevant periodicities. This study has identified, over European and Mediterranean area, the presence of interannual non-linear signals with a period of 2-to-4 years, possibly related to atmospheric and hydrological loading displacements and to climate phenomena, such as El Niño Southern Oscillation (ENSO). A clear signal with a period of about six years is present in the vertical component of the GPS time series, likely explainable by the gravitational coupling between the Earth’s mantle and the inner core. Moreover, signals with a period in the order of 8-9 years, might be explained by mantle-inner core gravity coupling and the cycle of the lunar perigee, and a signal of 18.6 years, likely associated to lunar nodal cycle, were identified through the wavelet spectrum. However, these last two signals need further confirmation because the present length of the GPS time series is still too short when compared to the periods involved.

Scientific machine learning (SciML) is a new branch of AI research at the edge of scientific computing (Sci) and machine learning (ML). It deals with efficient amalgamation of data-driven algorithms along with scientific computing to discover the dynamics of the time-evolving process. The output of such algorithms is represented in the form of a governing equation(s) (e.g., ordinary differential equation(s), ODE(s)), which one can solve then for any time point and, thus, obtain a rigorous prediction.  In this thesis, we present a methodology on how to incorporate the SciML approach in the context of clinical trials to predict IPF disease progression in the form of governing equation. Our proposed methodology also quantifies the uncertainties associated with the model by fitting 95\% high density interval (HDI) for the ODE parameters and 95\% posterior prediction interval for posterior predicted samples. We have also investigated the possibility of predicting later outcomes by using the observations collected at early phase of the study. We were successful in combining ML techniques, statistical methodologies and scientific computing tools such as bootstrap sampling, cubic spline interpolation, Bayesian inference and sparse identification of nonlinear dynamics (SINDy) to discover the dynamics behind the efficacy outcome as well as in quantifying the uncertainty of the parameters of the governing equation in the form of 95 \% HDI intervals. We compared the resulting model with the existed disease progression model described by the Weibull function. Based on the mean squared error (MSE) criterion between our ODE approximated values and population means of respective datasets, we achieved the least possible MSE of 0.133,0.089,0.213 and 0.057. After comparing these MSE values with the MSE values obtained after using Weibull function, for the third dataset and pooled dataset, our ODE model performed better in reducing error than the Weibull baseline model by 7.5\% and 8.1\%, respectively. Whereas for the first and second datasets, the Weibull model performed better in reducing errors by 1.5\% and 1.2\%, respectively. Comparing the overall performance in terms of MSE, our proposed model approximates the population means better in all the cases except for the first and second datasets, assuming the latter case's error margin is very small. Also, in terms of interpretation, our dynamical system model contains the mechanistic elements that can explain the decay/acceleration rate of the efficacy endpoint, which is missing in the Weibull model. However, our approach had a limitation in predicting final outcomes using a model derived from  24, 36, 48 weeks observations with good accuracy where as on the contrast, the Weibull model do not possess the predicting capability. However, the extrapolated trend based on 60 weeks of data was found to be close to population mean and the ODE model built on 72 weeks of data. Finally we highlight potential questions for the future work.

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Companies working with bulk materials have appropriate locations for storage during the development of the production and storage of the final product, known as warehouses or storehouses. The values of stocks need to be periodically validated by comparing the control of receipts the and the physical situation (removal of the volume stored in the company). In this context, the calculation of physical inventory as the volume of bulk present in the warehouses is usually done manually with low credibility and prone to errors. The current audit procedures on the contents of warehouses involve inaccurate estimates, and often require emptying the warehouse. Considering the use of technologies which enable the electronic measurement of distances, angles, and automatic controls on actuators enabling mechanical movements on the supporting structures, we sought to develop a system capable of providing both computing solutions, and technology for the problem of calculation of irregular relief (products stocked in warehouses). The Automated Auditing Warehouse SisA3 intends to make this process automatic, fast and precise, without the need for emptying warehouses or having contact the products. To achieve this goal, we developed an integrated system composed of: (i) a scanner equipment, consoling the hybrid prototype of hardware and software called DigSisA3, in order to the measurement of points of relief non-uniform, formed by the products in stock, and (ii) a method for calculating the volume iCone, which combines techniques of scientific visualization, numerical interpolation points and iterative calculation of volume. The parallelization of the prototype iCone was also developed in order to satisfy the test of agility and performance of the method iCone in the audit process. The development for multiprocessor, multi-core, and distributed architectures was done over the DGM (Geometric Distributed Machine), which provides the formalities to ensure creation, management and application processing parallel and / or distributed scientific computing, with emphasis on the exploitation of data parallelism and synchronization steps. The prototype of software iCone was functionally validated, including analysis of error in the method. The analysis of performance in the prototype p-iCone showed satisfactory results. The development of this work strengthens the system SisA3, enabling automatic and reliable measurement of inventories, including broad market application
Empresas que trabalham com produtos a granel possuem locais para estocagem, durante o desenvolvimento do processo produtivo e no armazenamento do produto final, denominados armaz´ens ou silos. Os valores dos estoques devem ser validados periodicamente atrav´es da comparac¸ ao dos estoques fiscal (controle das notas fiscais) e f´ısico (levantamento do volume estocado na empresa). Neste contexto, o c´alculo do estoque f´ısico, ou seja, o volume de gran´eis presentes nos armaz´ens, ´e geralmente efetuado de forma manual e com baixa credibilidade, desta forma com propens ao a erros. Os atuais processos de auditoria no conte´udo de silos, al´em de envolverem estimativas inexatas, est ao frequentemente baseados no esvaziamento do silo. Considerando o uso de tecnologias que viabilizam a medic¸ ao eletr onica de dist ancias, angulos, e controles autom´aticos sobre atuadores que possibilitam movimentos mec anicos sobre estruturas de suporte, buscou-se o desenvolvimento de um sistema capaz de prover tanto soluc¸ oes computacionais, quanto tecnol´ogicas para o problema de c´alculo do volume de relevos irregulares, no caso dos produtos estocados nos armaz´ens. O Sistema Automatizado de Auditoria em Armaz´ens (SisA3) pretende tornar este processo autom´atico, r´apido e preciso, sem a necessidade de esvaziamento ou contato com os produtos. Para alcanc¸ar este objetivo, tem-se um sistema integrado composto de: (i) um equipamento digitalizador, consolidando o prot´otipo h´ıbrido de hardware e software denominado Dig-SisA3 , para a medic¸ ao de pontos do relevo n ao-uniforme, formado pelos produtos estocados; e (ii) m´etodo para o c´alculo do volume (iCone), que combina t´ecnicas de visualizac¸ ao cient´ıfica, interpolac¸ ao num´erica de pontos e c´alculo iterativo de volume. Al´em disto, introduz-se a paralelizac¸ ao do prot´otipo iCone, para diminuir o tempo da obtenc¸ ao dos resultados do m´etodo iCone no processo de auditoria. A an´alise sobre as perspectivas em arquiteturas multiprocessadas, multi-core e paralela distribu´ıda, utiliza o ambiente D-GM (Distributed Geometric Machine), a qual prov e os formalismos para garantir criac¸ ao, gerenciamento e processamento de aplicac¸ oes paralelas e/ou distribu´ıdas da computac¸ ao cient´ıfica, com enfase na explorac¸ ao do paralelismo de dados e nas etapas de sincronizac¸ oes. O prot´otipo de software iCone apresenta-se funcionalmente validado, incluindo an´alise de erro na execuc¸ ao do m´etodo. As an´alises de desempenho no prot´otipo p-iCone apresentaram resultados satisfat´orios. O desenvolvimento deste trabalho consolida o sistema SisA3, viabilizando aferic¸ ao autom´atica e confi´avel de estoques, incluindo ampla aplicac¸ ao no mercado

In this master’s thesis we focus on the basic properties of computer curves and their practical applicability. We explain how the curve can be understood in general, what are polynomial curves and their composing possibilities. Then we focus on the description of Bezier curves, especially the Bezier cubic. We discuss in more detail some of fundamental algorithms that are used for modelling these curves on computers and then we will show their practical interpretation. Then we explain non uniform rational B-spline curves and De Boor algorithm. In the end we discuss topic rasterization of segment, thick line, circle and ellipse. The aim of master’s thesis is the creation of the set of interactive applets, simulating some of the methods and algorithm we discussed in theoretical part. This applets will help facilitate understanding and will make the teaching more effective.

博士
國立中山大學
資訊工程學系研究所
89
In this dissertation, a new cubic-spline interpolation (CSI) for both one-dimensional and two-dimensional signals is developed to sub-sample signal, image and video compression data. This new interpolation scheme that is based on the least-squares method with a cubic-spline function can be implemented by the fast Fourier transform (FFT). The result is a simpler and faster interpolation scheme than can be obtained by other conventional means. It is shown by computer simulation that such a new CSI yields a very accurate algorithm for smoothing. Linear interpolation, linear-spline interpolation, cubic-convolution interpolation and cubic B-spline interpolation tend to be inferior in performance. In addition it is shown in this dissertation that the CSI scheme can be performed by a fast and efficient computation. The proposed method uses a simpler technique in the decimation process. It requires substantially fewer additions and multiplications than the original CSI algorithm. Moreover, a new type of overlap-save scheme is utilized to solve the boundary-condition problems that occur between two neighboring subimages in the actual image. It is also shown in this dissertation that a very efficient 9-point Winograd discrete Fourier transform (Winograd DFT) can be used to replace the FFT needed to implement the CSI scheme. Furthermore, the proposed fast new CSI scheme is used along with the Joint Photographic Experts Group (JPEG) standard to design a modified JPEG encoder- decoder for image data compression. As a consequence, for higher compression ratios the proposed modified JPEG encoder-decoder obtains a better quality of reconstructed image and also requires less computational time than both the conventional JPEG method and the America on Line (AOL) algorithm. Finally, the new fast CSI scheme is applied to the JPEG 2000, MPEG-1 and MPEG-2 algorithms, respectively. A computer simulation shows that in the encoding and decoding, the proposed modified JPEG 2000 encoder-decoder speeds up the JPEG 2000 standard, respectively, and still obtains a good quality of reconstructed image that is similar to JPEG 2000 standard for high compression ratios. Additionally, the reconstructed video using the modified MPEG encoder-decoder indicates a better quality than the conventional MPEG-1 and MPEG-2 algorithms for high compression ratios or low-bit rates.

碩士
國立屏東商業技術學院
資訊管理系
95
With the digital image processing technology growing up, the applications in this domain such as high definition television (HDTV) and video-phone, have already gone deep in our life. When the digital image enlargement, it causes a blur problem. To improve the quality of the blurred image, many digital image processing approaches have been developed. Image enhancement is an indispensable post-processing method. Among many image enhancement approaches, Nonlinear Image Enhancement method is not only a simple structure but also has better effect, this method uses a Low-Pass Filter with an image enhancement scheme to improve the blurred image. Cubic-Spline Interpolation (CSI) has been proposed based on the Least-Squares method with a Cubic-Spline function for the smoothing of image data. In this paper, it is shown that this CSI can be used to improve the nonlinear image enhancement method. In addition, the two important parameters c (Clipping) and s (Scaling) are discussed in the Nonlinear Image Enhancement scheme to improve the blurred image problem. Finally, experimental results show that the proposed method yields a better quality of the reconstructed image than other nonlinear enhancement methods.

碩士
玄奘大學
資訊管理學系碩士班
98
Audio information hiding methods are widely discussed during this few years. WAVE or WAV is one of the most popular sound storage standard formats. In this thesis we use the spline interpolation technique to embed the secret data in the wave form of the original audio data. We set the control points, such as the odd number points to rebuild the spline curve to fit the original curve. We use the different between original and spline curve to embed the secret data to be a stego-audio. The extracting steps are simply to use the unchanged control points to compute the spline curve then find the different between spline curve and the stego-audio, to obtain the embedded secret data. The experiments show the sound is inaudible to be distinguished between stego-audio and original audio.

碩士
國立中正大學
應用數學研究所
97
In there, we persent B-spline curves and C2 cubic B-spline interpolating curves. Next, we do curve design by C2 cubic B-spline interpolating curves to cover irregular shapes.

碩士
玄奘大學
資訊管理學系碩士班
99
In this thesis we use the spline interpolation technique to embed the secret data in the wave form of the original audio data. We set the control points,adjust the control points of the partitions, to rebuild the spline curve to fit the original curve. We use the different between original and spline curve to embed the secret data to be a stego-audio. The extracting steps are simply to use the unchanged control points to compute the spline curve then find the different between spline curve and the stego-audio, to obtain the embedded secret data. The experiments show the control points to adjust the spacing can be embedded in a larger volume.

碩士
國立高雄應用科技大學
金融資訊研究所
97
The bond instrument is an important public financing tool for governments, financial institutions, industrial and commercial enterprises. Bonds are issued to investors, together with a face value of contract legislation, at a certain interest rate and with periodic payments of interest according to the agreed upon conditions of the outstanding principal on the maturity of the debt certificate. Due to the uncertainty nature of interest rates, accurate pricing of the bond is difficult because of the frequent fluctuation of long-term interest rates. This paper proposes a dynamic programming approach for pricing options embedded in bonds, with the focus placed on the calculation of the call and put options with executable period. This study uses the dynamic programming model with cubic spline interpolation for pricing options embedded in bonds. An efficient procedure is developed for the case where the interest rate process follows the Vasicek and Cox-Inegersoll-Ross(CIR) models. A dynamic programming methodology uses the exact joint distribution of the interest rate to integrate interest rates of future dates, which are conditional depending on the current value of interest rates. This study provides numerical illustrations for the Vasicek and CIR models, comparing pecewise interpolation method with cubic spline interpolation method. The dynamic programming procedure compares quite favorably in terms of both efficiency and accuracy as a result.

碩士
國立清華大學
資訊工程學系
101
Image up-scaling is an important technique to increase the resolution of an image. While earlier interpolation based approaches such as the bilinear and the bicubic method cause blurring and ringing artifacts in edge regions of the up-scaled image due to the loss of high frequency details. Recent approaches such as the local-self example super resolution can achieve very promising up-scaling results while their computation cost are high because they recover high frequency components of the whole image. In this paper, we proposed an image up-scaling method via an up-scaled edge map. By predicting edge regions of the up-scaled image, we recover high frequency components of edge regions of the up-scaled image to improve the sharpness and reduce ringing artifacts. We propose an edge curve scaling method with cubic spline interpolation to up-scale an edge map. If an edge curve is directly applied to the cubic spline interpolation function for edge curve up-scaling , the edge curve scaling results have zigzag artifacts. We also propose a simple smoothing function to avoid the zigzag problems and maintain the contour shape of images. Our methods can reduce execution time by 90% because we only perform high frequency components recovery on edge regions while other methods adopt to recover the high frequency components of every points in the up-scaled image. Experimental results show that we can achieve similar performances with the local self example super resolution method.

碩士
國立中山大學
資訊工程學系研究所
89
This thesis applies the one-dimensional (1-D) and two-dimensional (2-D) cubic-spline interpolation (CSI) schemes to MPEG standard for very low-bit rate video coding. In addition, the CSI scheme is used to implement the scalable video compression scheme in this thesis. The CSI scheme is based on the least-squares method with a cubic convolution function. It has been shown that the CSI scheme yields a very accurate algorithm for smoothing and obtains a better quality of reconstructed image than linear interpolation, linear-spline interpolation, cubic convolution interpolation, and cubic B-spline interpolation. In order to obtain a very low-bit rate video, the CSI scheme is used along with the MPEG-1 standard for video coding. Computer simulations show that this modified MPEG not only avoids the blocking effect caused by MPEG at high compression ratio but also gets a very low-bit rate video coding scheme that still maintains a reasonable video quality. Finally, the CSI scheme is also used to achieve the scalable video compression. This new scalable video compression scheme allows the data rate to be dynamically changed by the CSI scheme, which is very useful when operates under communication networks with different transmission capacities.

碩士
樹德科技大學
資訊工程學系
95
Multimedia communication has become the main application of modern network system. Among these digital data, the image and video have the greatest data size. Therefore, the image and video compression algorithms are the key points of the multimedia communication system. A well-designed interpolation can reduce image / video data size before compression. It has been shown that the Cubic Spline Interpolation (CSI) is one of the best interpolation algorithms in the world. One of its applications is to cooperate with MPEG-4 video codec and result in a modified CSI_MPEG-4 video codec. It follows from literatures that the reconstructed video quality of the modified CSI_MPEG-4 video codec is better than that of standard MPEG-4 video codec under the same bit-rates. This thesis studies the half/full pixel motion estimation algorithm in the modified CSI_MPEG-4 video codec. The performance of the half/full pixel motion estimation algorithm in the modified CSI_MPEG-4 video codec is evaluated by using various video clips with different resolutions. Experimental results show that the use of half pixel motion estimation algorithm in the modified CSI_MPEG-4 video codec cannot improve video quality substantially as it works in the standard MPEG-4 video codec. Furthermore, the benefit of using half pixel motion estimation will degrade rapidly when the encoded video bit-rate is increased. In other words, it is unnecessary to use half pixel motion estimation in the high bit-rate situations. These results can be applied to set the encoding parameters of the modified CSI_MPEG-4 video codec.

碩士
國立嘉義大學
電機工程學系研究所
107
This thesis proposes to use a metal-oxide-semiconductor field-effect transistor (MOSFET) to construct a negative differential resistance (NDR) circuit to replace the traditional resonant tunneling diode (RTD) component. The negative differential resistance circuit and the divider circuit are implemented by the TSMC 0.18μm CMOS process provided by the National Wafer Center (CIC). In this thesis, the cubic spline interpolation is used to operate the bifurcation diagram of the MATLAB software, and the HSPICE software is used to verify the periodic state, quasi-periodic state and chaotic phenomenon. In this paper, the measurement results of the frequency divider are included, and the parameters are changed and the results are compared and discussed.

碩士
國立成功大學
電機工程學系
104
This research proposed a pulse wave processing framework of baseline removal based on wavelet and cubic-spline interpolation. With the consideration of subject movement, sensor hardness and other situations, the framework processes pulse signal in each channel adaptively for good construction of the 3DPM. The result shows that the proposed framework has good performance for pulse wave signals with different qualities. However, the high-frequency suppression of this work can be improved on future research. This study has modified the L-Cube recognition proposed previously and improved the methods of fitting and evaluation. Hyperplanes in both x- and y-direction are added into the L-Cube function for a better flexibility when describing an asymmetric 3DPM. Also, the shape parameter a and b are replaced with the axis ratio r and the decay parameter σ. As the fitting process completes, the axis ratio is found to be the new string-like index which is more sensitive than the eccentricity ε. However, the new L-Cube is still not able to describe non-uniform 3DPMs well and further research can be done on this. On the topic of fitting method and evaluation, we abandon the volume error (VE) and R2 but keep the root mean squared error (RMSE). Besides, the weighted RMSE, rotation angle error and the center point deviation are introduced for the fitting and its evaluation. From the experiment, we have found the weights of above terms are suggested to be 1 : 1 : 0.01 : 0. Using the weights concluded above, a cold pressor data (CPT) was analyzed with the new L-Cube function. The resultant string-like index increased as the cold stimulation applied and went back down as the of ice bag removed. This verified the work related to making blood pressure higher temporally by CPT so that the pulse condition tends to get more string-like as well. Besides, the variation of r indicates the individual differences may lead the pulse condition fail to get more string-like through the CPT.

Two extensions to the proper orthogonal decomposition (POD) technique are considered for steady transonic aerodynamic applications. The first is to couple the POD approach with a cubic spline interpolation procedure in order to develop fast, low-order models that accurately capture the variation in parameters, such as the angle of attack or inflow Mach number. The second extension is a POD technique for the reconstruction of incomplete or inaccurate aerodynamic data. First, missing flow field data is constructed with an existing POD basis constructed from complete aerodynamic data. Second, a technique is used to develop a complete snapshots from an incomplete set of aerodynamic snapshots.
Singapore-MIT Alliance (SMA)

The emperical pseudopotential method in the large basis approach was used to calculate the electronic bandstructures of bulk semiconductor materials and layered semiconductor heterostructures. The crucial continuous atomic form factor potentials needed to carry out such calculations were determined by using Levenberg-Marquardt optimization in order to obtain optimal cubic spline interpolations of the potentials. The optimized potentials were not constrained by any particular functional form (such as a linear combination of Gaussians) and had better convergence properties for the optimization. It was demonstrated that the results obtained in this work could potentially lead to better agreement between calculated and empirically determined band gaps via optimization
Physics
M. Sc. (Physics)

Tese no âmbito do Programa Interuniversitário de Doutoramento em Matemática, apresentada à Faculdade de Ciências e Tecnologia da Universidade de Coimbra
Interpolating data in non-Euclidean spaces plays an important role in different areas of knowledge. The main goal of this thesis is to present, in detail, two different approaches for solving interpolation problems on the Generalized Essential manifold Gk,n ×SO(n), consisting of the product of the Grassmann manifold of all k-dimensional subspaces of R^n and the Lie group of rotations in R^n. The first approach to be considered is a generalization to manifolds of the De Casteljau algorithm and the second is based on rolling motions. In order to achieve our objective, we first gather information of all the essential topics of Riemannian geometry and Lie theory necessary for a complete understanding of the geometry of the fundamental manifolds involved in this work, with particular emphasis on the Grassmann manifold and on the Normalized Essential manifold. To perform the De Casteljau algorithm in the manifold Gk,n×SO(n) we adapt a procedure already developed for connected and compact Lie groups and for spheres, and accomplish the implementation of that algorithm, first for the generation of geometric cubic polynomials in the Grassmann manifold Gk,n, and then extending it to generate cubic splines in the same manifold. New expressions for the velocity vector field along geometric cubic polynomials and for its covariant derivative are derived in order to obtain admissible curves that also fulfil appropriate boundary conditions. To solve the interpolation problem using the second approach, we propose an algorithm inspired in techniques that combine rolling/unrolling with unwrapping/wrapping, but accomplishing the objective using rolling motions only. Interpolating curves given in explicit form are obtained for the manifold Gk,n ×SO(n), which also prepares the ground for applications using the Normalized Essential manifold. The definition of rolling map is a crucial tool in this approach. We present a geometric interpretation of all the conditions present in that definition, including a refinement of the non-twist conditions which allows to prove interesting properties of rolling and, consequently, simplifies the study of rolling motions. In particular, the non-twist conditions are rewritten in terms of parallel vector fields, allowing for a clear connection between rolling and parallel transport. When specializing to the rolling manifold Gk,n ×SO(n) the definition of rolling map is adjusted in order to avoid destroying the matrix structure of that manifold. We also address controllability issues for the rolling motion of the Grassmann manifold Gk,n. In parallel with a theoretical proof, we present a constructive proof of the controllability of the kinematic equations that describe the pure rolling motions of the Grassmann manifold Gk,n over the affine tangent space at a point. We make connections with other known approaches to generate interpolating curves in manifolds and point out some directions for future work.
A interpolação de dados em espaços não Euclidianos desempenha um papel importante em diferentes áreas do conhecimento. O objetivo principal desta tese é apresentar, em detalhe, duas abordagens diferentes para resolver problemas de interpolação na variedade Essencial Generalizada Gk,n×SO(n), que consiste no produto cartesiano da variedade de Grassmann formada por todos os subespaços k-dimensionais de R^n e o grupo de Lie das rotações em R^n. A primeira abordagem a ser considerada é uma generalização para variedades do algoritmo de De Casteljau e a segunda é baseada em certos movimentos de rolamento. A fim de alcançar o nosso objetivo, primeiro reunimos informações de todos os tópicos essenciais de geometria Riemanniana e de teoria de Lie necessários para uma completa compreensão da geometria das variedades fundamentais envolvidas neste trabalho, com particular ênfase na variedade de Grassmann e na variedade Essencial Normalizada. Para implementar o algoritmo de De Casteljau na variedade Gk,n ×SO(n), adaptamos um procedimento já conhecido para grupos de Lie conexos e compactos e para esferas, e realizamos a implementação desse algoritmo, primeiro para a geração de polinómios geométricos cúbicos na variedade de Grassmann Gk,n, e depois estendemo-lo para gerar splines cúbicos na mesma variedade. São deduzidas novas expressões para o campo de vetores velocidade ao longo dessas curvas e para a sua derivada covariante, a fim de obter curvas admissíveis que também satisfaçam condições de fronteiras apropriadas. Para resolver o problema de interpolação utilizando a segunda abordagem, propomos um algoritmo inspirado em técnicas que combinam rolling/unrolling com unwrapping/wrapping, mas cumprindo o objetivo utilizando apenas movimentos de rolamento. As curvas de interpolação para a variedade Gk,n×SO(n) são obtidas de forma explícita, o que também prepara o terreno para aplicações utilizando a variedade Essencial Normalizada. A definição de aplicação rolamento é uma ferramenta crucial nesta abordagem. Apresentamos uma interpretação geométrica de todas as condições presentes nessa definição, incluindo um refinamento das condições de non-twist o que permite provar propriedades interessantes de rolamento e, consequentemente, simplifica o estudo dos movimentos de rolamento. Em particular, as condições de non-twist são reescritas em termos de campos vectoriais paralelos, permitindo uma ligação clara entre o rolamento e o transporte paralelo. Quando é especificada para a variedade de rolamento Gk,n×SO(n), a definição de aplicação rolamento é ajustada de forma a evitar destruir a estrutura matricial dessa variedade. Também abordamos questões de controlabilidade para o movimento de rolamento da variedade de Grassmann Gk,n. Em paralelo com uma prova teórica, apresentamos uma prova construtiva da controlabilidade das equações da cinemática que descrevem os movimentos de rolamento puro da variedade de Grassmann Gk,n sobre o espaço afim associado ao espaço tangente num ponto. Estabelecemos algumas relações com outras abordagens conhecidas para gerar curvas interpoladoras em variedades e apresentamos algumas direções para o trabalho futuro.