Nonlinear Weighted Averaging Rule And Its Applications

Averaging rules are widely useful in many fields of application such as engineering,computer science,applied Mathematics and more.The arithmetic mean is one of the most practical averaging rule due to its simplicity in both calculation and implementation.However,simi-larly to many other averaging rules,this measure of central tendency lacks some flexibility in calculation and sometimes is not suitable especially if the data values to be averaged are not linearly related.This dissertation introduces a new flexible averaging rule,which is the nonlinear weighted averaging rule.This averaging rule is defined by using the monotonic functions and the weights.In this report,it is proved that this type of averaging technique satisfies all necessary and sufficient conditions to be called an averaging rule.As application,it is used to derive new recursive evaluation algorithm,formulating new schemes for subdivi-sion curves and surfaces and developing new techniques for approximating the data.For recursive evaluation algorithm,by using this averaging rule we derived the variants of de Casteljau evaluation algorithm for Bezier curves.The results show that these algorithms can generate not only polynomial functions but also transcendental functions that the classical algorithm is unable to generate.Moreover,the developed subdivision schemes for curves and surfaces mimic the existing schemes which apply the arithmetic mean in their topology.The subdivision schemes based on the nonlinear weighted averaging rule are advantageous since they provide a much free-dom to a designer for easily adjusting the limit shape.This happens by only modifying either the function or the weights,which define the averaging rule used to formulate the scheme.It is also proved that the smoothness of the curves generated by these schemes is derived from the smoothness of the functions,which induce the averaging rules used by the corresponding schemes.Furthermore,this averaging technique helped us to formulate an approach for approximating a function and express it into a simplified form by using the theory of the wavelet transform.In addition,with the nonlinear weighted average we developed a technique for resizing a digital image by averaging neighboring pixels'values.This resizing technique not only reduces the resolution of an image but also its storage size.We compared the images given by the proposed technique with the images given by the other existing resizing techniques and the results guarantee the efficiency of our technique.To assess the quality of images yielded by the proposed resizing technique,the structural similarity index measure(SSIM)is computed and the results show that the proposed technique is highly suitable for the grayscale images.