The main problems of numerical analysis involve performing analytical operations, such as integration, differentiation, finding zeroes, interpolation, and so forth, of a function when all the data available are some samples of the function. Therefore, the purpose of this paper is to investigate the following problem: given a set of data points (x[sub i], y[sub i]) which are samples of some function, determine an approximating function. Further, extend the problem to that of determining an approximating function for a surface given some samples (x[sub i], y[sub j], z[sub ij]) of the surface.

Due to the fast development in 3D technology during recent decades, many approaches in 3D representation technologies have been proposed worldwide. In order to get an accurate information to render a 3D representation, more data need to be recorded compared to normal video sequence. In this case, how to find an efficient way to transmit the 3D representation data becomes an important part in the whole 3D representation technology. Recent years, many coding schemes based on the principle of encoding the depth have been proposed. Compared to the traditional multiview coding schemes, those new proposed schemes can achieve higher compression efficiency. Due to the development of depth capturing technology, the accuracy and quality of the reconstructed depth image also get improved. In this thesis we propose an efficient depth data compression scheme for 3D images. Our proposed depth data compression scheme is platelet based coding using Lagrangian optimization, quadtree decomposition and quadratic curve fitting. We study and improve the original platelet based coding scheme and achieve a compression improvement of 1-2 dB compared to the original platelet based scheme. The experimental results illustrate the improvement provided by our scheme. The quality of the reconstructed results of our proposed curve fitting based platelet coding scheme are better than that of the original scheme.

Today's modern agriculture is totally dependent on phosphorus to sustain their large yields. Several studies have recently expressed a concern for a future phosphorus deficiency. These studies are based on data for estimated reserves which have been increased with more than a fourfold since 2010. Some argue that these concerns are unfounded, despite the fact that only Morocco account for the bulk of these new reserves. This report provides new forecast for the world phosphorus production based on the new available reserve data. These forecasts are using bell shaped curve models to examine how individual countries' future production of phosphate rock affects a global production peak. Estimates of the size of several reserves are highly uncertain and it is therefore difficult to make an accurate forecast of future phosphorus extraction. Despite this uncertainty, a global production peak is likely to occur within this century. The global production will depend largely on China and Morocco's production as they hold a large share of the reserves and the current production. China's production will probably peak in 10-20 years at current production trend. It is uncertain if Morocco can increase production enough to replace China's production in the future. It is not likely that Morocco will be able to produce as much as would be required to meet the highest scenarios. This is mainly due to a number of bottlenecks in production such as water scarcity, increasing proportion of impurities and a decreasing concentration of phosphorus in the phosphate rock.

In this study, three different MR pulse sequences, IR-FSE, MP2RAGE, and firstly proposed MP3RAGE, were applied to obtain high-resolution 3D T1 mapping of whole brain at 1.5 Tesla. Among these three sequences, MP2RAGE uses fast gradient echo as readout module. Signals of two different inversion times are acquired at once and can be used to calculate T1 relaxation time according to Bloch equation. However, the magnetization was also influenced by the excitation efficiency of inversion adiabatic pulse, which was usually estimated by numerical simulation and taken as a constant over the field of view in the literature. However, this might not be true in practice. Therefore, a newly modified pulse sequence, MP3RAGE, was proposed to acquire data of three distinct inversion times without increasing scanning time. As a result, the spatial distribution of T1 and inversion efficiency can be assessed by solving nonlinear least square problem. In addition, the IR-FSE sequence with six inversion times was also applied in every experiment to provide T1 value for reference. Results showed that the T1 estimation obtained by MP2RAGE is close to, but slightly lower than that by IR-FSE, which is in agreement with those reported in literatures. In addition, the 3D high-resolution maps of T1 and efficiency were successfully estimated with the use of MP3RAGE. Spatial smoothing on inversion efficiency helps reducing the sensitivity to noise in the nonlinear approach, leading to T1 values closer to those by IR-FSE.

The kinematics characteristics of the human knee joint can be represented by its centrodes. This paper provides a method, called the photograph method, to construct the centrodes by using commercial digital camera to take the sequential pictures. In order to eliminate the undesired movements, a testing chair and a brace are specially designed. Two types of curve fitting methods are introduced to smooth the measured data. The differential method is applied to construct the centrodes of knee joints from the measured data. This paper provides a safe way to measure the centrodes of human knee joints.

New elastographic techniques such as poroelastography and viscoelasticity imaging aim at imaging the temporal mechanical behavior of tissues. These techniques usually involve the use of curve fitting methods as applied to noisy data to estimate new elastographic parameters. As of today, however, image quality performance of these new elastographic imaging techniques is still largely unknown due to a paucity of data and the lack of systematic studies that analyze performance limitations of estimators suitable for these novel applications. Furthermore, current elastographic implementations of poroelasticity and viscoelasticity imaging methods are in general too slow and not optimized for clinical applications. In this paper, we propose a new elastographic time constant (TC) estimator, which is based on the use of the Least Square Error (LSE) curve-fitting method and the Levenberg-Marquardt (LM) optimization rule as applied to noisy elastographic data obtained from a tissue under creep compression. The estimator's performance is analyzed using simulations and quantified in terms of accuracy, precision, sensitivity, signal-to-noise ratio (SNR) and speed. Experiments are performed as a proof of principle of the technical applicability of the new estimator on real experimental data. The results of this study demonstrate that the new elastographic estimator described in this thesis can produce highly accurate, sensitive and precise time constant estimates in real-time and at high SNR. In the future, the use of this estimator could allow real-time imaging of the temporal behavior of complex tissues and provide advances in lymphedema and cancer imaging.

Thesis (M.S.)--Ohio University, August, 2000. / Title from PDF t.p.

Thesis (Ph. D.)--Ohio State University, 2007. / Title from first page of PDF file. Includes bibliographical references (p. 129-135).

Due to the fast development in 3D technology during recent decades, many approaches in 3D representation technologies have been proposed worldwide. In order to get an accurate information to render a 3D representation, more data need to be recorded compared to normal video sequence. In this case, how to find an efficient way to transmit the 3D representation data becomes an important part in the whole 3D representation technology. Recent years, many coding schemes based on the principle of encoding the depth have been proposed. Compared to the traditional multiview coding schemes, those new proposed schemes can achieve higher compression efficiency. Due to the development of depth capturing technology, the accuracy and quality of the reconstructed depth image also get improved. In this thesis we propose an efficient depth data compression scheme for 3D images. Our proposed depth data compression scheme is platelet based coding using Lagrangian optimization, quadtree decomposition and quadratic curve fitting. We study and improve the original platelet based coding scheme and achieve a compression improvement of 1-2 dB compared to the original platelet based scheme. The experimental results illustrate the improvement provided by our scheme. The quality of the reconstructed results of our proposed curve fitting based platelet coding scheme are better than that of the original scheme.

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