Constructing new control points for Bézier interpolating polynomials using new geometrical approach

Interpolation is a mathematical technique employed for estimating the value of missing data between data points. This technique assures that the resulting polynomial passes through all data points. One of the most useful interpolating polynomials is the parametric interpolating polynomial. Bézier interpolating curves and surfaces are parametric interpolating polynomials for two-dimensional (2D) and three-dimensional (3D) datasets, respectively, that produce smooth, flexible, and accurate functions. According to the previous studies, the most crucial component in deriving Bézier interpolating polynomials is the construction of control points. However, most of the existing strategies constructed control points that produce partial smooth functions. As a result, the approximate values of the missing data are not accurate. In this study, nine new strategies of geometrical approach for constructing new 2D and 3D Bézier control points are proposed. The obtained control points from each new strategies are substituted in the relevant Bézier curve and surface equations to derive Bézier piecewise and non-piecewise interpolating polynomials which leads to the development of nine new methods. The proposed methods are proven to preserve the stability and smoothness of the generated Bézier interpolating curves and surfaces. In addition, the numerical results show that most of the resulting polynomials are able to approximate the missing values more accurately compared to those derived by the existing methods. The Bézier interpolating surfaces derived by the proposed method with highest accuracy for 3D datasets are then applied to upscale grey and colour images by the factors of two and three. Not only does the proposed method produces higher quality upscaled images, the numerical results also show that it outperforms the existing methods in terms of accuracy. Therefore, this study has successfully proposed new strategies for constructing new 2D and 3D control points for deriving Bézier interpolating polynomials that are capable of approximating the missing values accurately. In terms of application, the derived Bézier interpolating surfaces have a great potential to be employed in image upscaling

An effective curriculum for teaching computer numerical control machining

The purpose of this project was to develop and document curricular content for Computer Numerical Control education program for Mt. San Jacinto Community College. The design of the curriculum focuses on showing students how skills learned in academic classes can be applied to the workplace

Improving Satellite Leaf Area Index Estimation Based On Various Integration Methods

Leaf Area Index (LAI) is an important land surface biophysical variable that is used to characterize vegetation amount and activity. Current satellite LAI products, however, do not satisfy the requirements of the modeling community due to their large uncertainties and frequent missing values. Each LAI product is currently generated from only one satellite sensor data. There is an urgent need for advanced methods to integrate multiple LAI products to improve the product's accuracy and integrality for various applications. To meet this need, this study proposes four methods, including the Optimal Interpolation (OI), Bayesian Maximum Entropy (BME), Multi-Resolution Tree (MRT) and Empirical Orthogonal Function (EOF), to integrate multiple LAI products. Three LAI products have been considered in this study: Moderate Resolution Imaging Spectroradiometer (MODIS), Multi-angle Imaging SpectroRadiometer (MISR) and Carbon cYcle and Change in Land Observational Products from an Ensemble of Satellites (CYCLOPES) LAI. As the basis of data integration, this dissertation first validates and intercompares MODIS and CYCLOPES LAI products and also evaluates their geometric accuracies. The CYCLOPES LAI product has smoother temporal profiles and fewer spatial variations, but tends to produce spurious large errors in winter. The Locally Adjusted Cubic-spline Capping algorithm is revised to smooth multiple years' average and variance. Although OI, BME and MRT based methods have been used in other fields, this is the first research to employ them in integrating multiple LAI products. This dissertation also presents a new integration method based on EOF to solve the problem of large data volume and inconsistent temporal resolution of different datasets. High resolution LAI reference maps generated with ground measurements are used to validate these algorithms. Validation results show that all of these four methods can fill data gaps and reduce the errors of the existing LAI products. The data gaps are filled with information from adjacent pixels and background. These algorithms remove the spurious large temporal and spatial variation of the original LAI products. The combination of multiple satellite products significantly reduces bias. OI and BME can reduce the RMSE from 1.0 (MODIS) to 0.7 and reduce the bias from +0.3 (MODIS) and -0.2 (CYCLOPES) to -0.1. MRT can produce similar results with OI but with significantly improved efficiency. EOF also generates the results with the RMSE of 0.7 but zero bias. Limited ground measurement data hardly prove which methods outperform the others. OI and BME theoretically produce statistically optimal results. BME relaxes OI's linear and Gaussian assumption and explicitly considers data error, but bears a much higher computational burden. MRT has improved efficiency but needs strict assumptions on the scale transfer function. EOF requires simpler model identification, while it is more "empirical" than "statistical". The original contributions of this study mainly include: 1) a new application of several different integration methods to incorporate multiple satellite LAI products to reduce uncertainties and improve integrality, 2) an enhancement of the Locally Adjusted Cubic-spline Capping by revising the end condition, 3) a novel comprehensive comparison of MODIS C5 LAI product with other satellite products, 4) the development of a new LAI normalization scheme by assuming the linear relationship between measurement error and LAI natural variance to account for the inconsistency between products, and finally, 5) the creation of a new data integration method based on EOF

A Chronology of Interpolation: From Ancient Astronomy to Modern Signal and Image Processing

This paper presents a chronological overview of the developments in interpolation theory, from the earliest times to the present date. It brings out the connections between the results obtained in different ages, thereby putting the techniques currently used in signal and image processing into historical perspective. A summary of the insights and recommendations that follow from relatively recent theoretical as well as experimental studies concludes the presentation

Wavelet-Based Enhancement Technique for Visibility Improvement of Digital Images

Image enhancement techniques for visibility improvement of color digital images based on wavelet transform domain are investigated in this dissertation research. In this research, a novel, fast and robust wavelet-based dynamic range compression and local contrast enhancement (WDRC) algorithm to improve the visibility of digital images captured under non-uniform lighting conditions has been developed. A wavelet transform is mainly used for dimensionality reduction such that a dynamic range compression with local contrast enhancement algorithm is applied only to the approximation coefficients which are obtained by low-pass filtering and down-sampling the original intensity image. The normalized approximation coefficients are transformed using a hyperbolic sine curve and the contrast enhancement is realized by tuning the magnitude of the each coefficient with respect to surrounding coefficients. The transformed coefficients are then de-normalized to their original range. The detail coefficients are also modified to prevent edge deformation. The inverse wavelet transform is carried out resulting in a lower dynamic range and contrast enhanced intensity image. A color restoration process based on the relationship between spectral bands and the luminance of the original image is applied to convert the enhanced intensity image back to a color image. Although the colors of the enhanced images produced by the proposed algorithm are consistent with the colors of the original image, the proposed algorithm fails to produce color constant results for some pathological scenes that have very strong spectral characteristics in a single band. The linear color restoration process is the main reason for this drawback. Hence, a different approach is required for tackling the color constancy problem. The illuminant is modeled having an effect on the image histogram as a linear shift and adjust the image histogram to discount the illuminant. The WDRC algorithm is then applied with a slight modification, i.e. instead of using a linear color restoration, a non-linear color restoration process employing the spectral context relationships of the original image is applied. The proposed technique solves the color constancy issue and the overall enhancement algorithm provides attractive results improving visibility even for scenes with near-zero visibility conditions. In this research, a new wavelet-based image interpolation technique that can be used for improving the visibility of tiny features in an image is presented. In wavelet domain interpolation techniques, the input image is usually treated as the low-pass filtered subbands of an unknown wavelet-transformed high-resolution (HR) image, and then the unknown high-resolution image is produced by estimating the wavelet coefficients of the high-pass filtered subbands. The same approach is used to obtain an initial estimate of the high-resolution image by zero filling the high-pass filtered subbands. Detail coefficients are estimated via feeding this initial estimate to an undecimated wavelet transform (UWT). Taking an inverse transform after replacing the approximation coefficients of the UWT with initially estimated HR image, results in the final interpolated image. Experimental results of the proposed algorithms proved their superiority over the state-of-the-art enhancement and interpolation techniques