Approximation by nonlinear sampling Durrmeyer operators and medical applications

This PhD thesis explores advancements in Approximation Theory, focusing on sampling theory and signal/image processing. Starting from an overiview of the classical sampling theory, the study further involves Kantorovich and Durrmeyer sampling-type operators, which are studied in several function spaces. A key contribution is the study of the approximation properties of the nonlinear version of the sampling Durrmeyer operators, which include, as particular cases, other well known families of nonlinear sampling-type operators. Both pointwise and uniform convergence results for the above family of operators, assuming that f is bounded and continuous or uniformly continuous, are estabilished. Moreover, the convergence of nonlinear sampling Durrmeyer operators within the general framework of Orlicz spaces is also analyzed. Furthermore, the thesis presents practical applications in digital image processing. The Sampling Kantorovich (SK) algorithm is implemented in medical imaging, enhancing CT and MRI scans for segmentation and biomarker analysis in conditions like aortic aneurysm and Alzheimer's disease. The SK algorithm also proves effective in retinal image analysis, aiding in diabetic retinopathy research. These results highlight the potential of advanced sampling operators in both theoretical mathematics and real-world applications, paving the way for improved signal and image processing techniques.