A comparative study of some wavelet and sampling operators on various features of an image

TL;DR

This paper compares wavelet and sampling operators in image feature analysis, examining their approximation properties and effectiveness under various conditions through theoretical analysis and numerical examples.

Contribution

It provides a comprehensive comparison of sampling Kantorovich and wavelet-based operators for image feature approximation, including convergence analysis and practical performance evaluation.

Findings

Different operators have varying effectiveness depending on image features.

Wavelet and sampling operators show distinct approximation behaviors under ideal and non-ideal conditions.

Numerical examples validate the theoretical convergence and feature preservation properties.

Abstract

This research includes the study of some positive sampling Kantorovich operators (SK operators) and their convergence properties. A comprehensive analysis of both local and global approximation properties is presented using sampling Kantorovich (SK), Gaussian, Bilateral and the thresholding wavelet-based operators in the framework of SK-operators. Explicitly, we start the article by introducing the basic terminology and state the fundamental theorem of approximation (FTA) by imposing the various required conditions corresponding to the various defined operators. We measure the error and study the other mathematical parameters such as the mean square error (MSE), the speckle index (SI), the speckle suppression index (SSI), the speckle mean preservation index (SMPI), and the equivalent number of looks (ENL) at various levels of resolution parameters. The nature of these operators are demonstrated via an example under ideal conditions in tabulated form at a certain level of samples. Eventually, another numerical example is illustrated to discuss the region of interest (ROI) via SI, SSI and SMPI of 2D Shepp-Logan Phantom taken slice from the 3D image, which gives the justification of the fundamental theorem of approximation (FTA). At the end of the derivation and illustrations we observe that the various operators have their own significance while studying the various features of the image because of the uneven nature of an image (non-ideal condition). Therefore, to some extent, some operators work well and some do not for some specific features of the image.