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Over the last decade or so, wavelets have had a growing impact on signal processing theory and practice, both because of their unifying role and their successes in applications. Filter banks, which lie at the heart of wavelet-based algorithms, have become standard signal processing operators, used routinely in applications ranging from compression to modems. The contributions of wavelets have often been in the subtle interplay between discrete-time and continuous-time signal processing. The purpose of this article is to look at recent wavelet advances from a signal processing perspective. In particular, approximation results are reviewed, and the implication on compression algorithms is discussed. New constructions and open problems are also addressed. Finding a good basis to solve aproblem dates at least back to Fourier and his investigation of the heat equation. The series proposed by Fourier has several distinguishing features: The series is able to represent any finite energy function on an interval. The basis functions are eigenfunctions of linear shift invariant systems or, in other words, Fourier series diagonalize linear, shift invariant operators.

Abstract Background This study aimed to evaluate the effectiveness of DeepLabv3+with Squeeze-and-Excitation (DeepLabv3+SE) architectures for segmenting the choroid in optical coherence tomography (OCT) images of patients with diabetic retinopathy. Methods A total of 300 B-scans were selected from 21 patients with mild to moderate diabetic retinopathy. Six DeepLabv3+SE variants, each utilizing a different pre-trained convolutional neural network (CNN) for feature extraction, were compared. Segmentation performance was assessed using the Jaccard index, Dice score (DSC), precision, recall, and F1-score. Binarization and Bland-Altman analysis were employed to evaluate the agreement between automated and manual measurements of choroidal area, luminal area (LA), and Choroidal Vascularity Index (CVI). Results DeepLabv3+SE with EfficientNetB0 achieved the highest segmentation performance, with a Jaccard index of 95.47, DSC of 98.29, precision of 98.80, recall of 97.41, and F1-score of 98.10 on the validation set. Bland-Altman analysis indicated good agreement between automated and manual measurements of LA and CVI. Conclusions DeepLabv3+SE with EfficientNetB0 demonstrates promise for accurate choroid segmentation in OCT images. This approach offers a potential solution for automated CVI calculation in diabetic retinopathy patients. Further evaluation of the proposed method on a larger and more diverse dataset can strengthen its generalizability and clinical applicability.

Abstract Diabetes retinopathy prevention necessitates early detection, monitoring, and treatment. Non-invasive optical coherence tomography (OCT) shows structural changes in the retinal layer. OCT image evaluation necessitates retinal layer segmentation. The ability of our automated retinal layer segmentation to distinguish between normal, non-proliferative (NPDR), and proliferative diabetic retinopathy (PDR) was investigated in this study using quantifiable biomarkers such as retina layer smoothness index (SI) and area (S) in horizontal and vertical OCT images for each zone (fovea, superior, inferior, nasal, and temporal). This research includes 84 eyes from 57 individuals. The study shows a significant difference in the Area (S) of inner nuclear layer (INL) and outer nuclear layer (ONL) in the horizontal foveal zone across the three groups (p

Abstract Quantifying the smoothness of different layers of the retina can potentially be an important and practical biomarker in various pathologic conditions like diabetic retinopathy. The purpose of this study is to develop an automated machine learning algorithm which uses support vector regression method with wavelet kernel and automatically segments two hyperreflective retinal layers (inner plexiform layer (IPL) and outer plexiform layer (OPL)) in 50 optical coherence tomography (OCT) slabs and calculates the smoothness index (SI). The Bland–Altman plots, mean absolute error, root mean square error and signed error calculations revealed a modest discrepancy between the manual approach, used as the ground truth, and the corresponding automated segmentation of IPL/ OPL, as well as SI measurements in OCT slabs. It was concluded that the constructed algorithm may be employed as a reliable, rapid and convenient approach for segmenting IPL/OPL and calculating SI in the appropriate layers.

This paper introduces two novel methods for solving multi-order fractional differential equations using Bernstein polynomials. The first method, referred to as the fractional operational matrix of differentiation of Bernstein polynomials, is employed to solve fractional differential equations with high precision. The second method merges the development of the collocation method with Bernstein polynomials, ensuring an even higher accuracy than previously utilized numerical techniques. The Caputo fractional derivative of Bernstein polynomials is calculated using a specialized method, that combines Gauss-Jacobi quadrature rules and three-term recurrence formulas, resulting in a matrix with a notably small condition number. This allows for the solution of the algebraic system through standard linear algebra techniques, and the method provides substantial accuracy, particuarly for problems with smooth solutions. Through a comparison of numerical examples with existing methods and a physical test example predicting the sphere’s path and behavior within a fluid the practical applicability and accuracy of our novel methods were demonstrated. Utilizing MATLAB 2021, the efficacy of the presented methods was showcased, revealing superior performance in terms of accuracy and reliability in solving fractional differential equations.

This article aims to introduce a new linear positive operator with a parameter. Our focus lies in analyzing the distinct characteristics and inherent properties exhibited by this operator. Additionally, we provide a proof of the convergence rate and present a revised version of the Voronovskaja theorem specifically tailored for this newly defined operator. Furthermore, we provide an upper bound for the error according to the modulus of continuity. Finally, the preservation of monotonicity and convexity by the operator is being investigated.