Approximation of certain bivariate functions by almost Euler means of double Fourier series

Abstract In this paper, we study the degree of approximation of 2π-periodic functions of two variables, defined on T2=[−π,π]×[−π,π] $T^{2}=[-\pi,\pi]\times[-\pi,\pi]$ and belonging to certain Lipschitz classes, by means of almost Euler summability of their Fourier series. The degree of approximation obtained in this way depends on the modulus of continuity associated with the functions. We also derive some corollaries from our theorems

DEGENERATIVE SPINE DETECTION

Spinal Misalignment is a chronic disease that is widespread across the world. It causes different diseases such as Stenosis, Scoliosis, Osteoporotic Fractures, Thoracolumbar fractures, Disc degeneration, etc. The diagnosis of such disease is generally done by analyzing the Magnetic Resonance Imaging (MRI) scan of the lumbar spine region. MRI analysis is done by well experienced medical professionals (radiologists and orthopedists). The flip side to this inspection is that it is time consuming and may be subjected to a lack of accuracy. The manual segmentation of MRI scans from a large number of scan images is a tedious and time - consuming process. Thus, there is a need for automatic segmentation and analysis of spine MRI scans to improve clinical outputs and the accuracy of spinal measurements. In recent years, the rise of deep learning technologies is making a revolution in medical systems. It is capable to examine a big amount of data thus yielding a better accuracy. So, deep learning approaches can be efficiently used for the automatic segmentation of MRI scans. For Disc degeneration detection we trained two models namely normal CNN Model and Densenet121 model. Out of these two models the Densenet121 model performed the best against our standards. It achieved training Accuracy of 99.75% , validation accuracy of 93.74% , testing Accuracy of 92.74% and hence was chosen as the final model for Disc degeneration detection