MDPI - Publisher of Open Access Journals

15 pages, 2194 KiB

Open AccessArticle

Hybrid Transform-Based Feature Extraction for Skin Lesion Classification Using RGB and Grayscale Analysis

by Luis Felipe López-Ávila and Josué Álvarez-Borrego

Appl. Sci. 2025, 15(11), 5860; https://doi.org/10.3390/app15115860 - 23 May 2025

Viewed by 649

Automated skin lesion classification using machine learning techniques is crucial for early and accurate skin cancer detection. This study proposes a hybrid method combining the Hermite, Radial Fourier–Mellin, and Hilbert transform to extract comprehensive features from skin lesion images. By separating the images [...] Read more.

Automated skin lesion classification using machine learning techniques is crucial for early and accurate skin cancer detection. This study proposes a hybrid method combining the Hermite, Radial Fourier–Mellin, and Hilbert transform to extract comprehensive features from skin lesion images. By separating the images into red, green, and blue (RGB) channels and grayscale, unique textural and structural information specific to each channel is analyzed. The Hermite transform captures localized spatial features, while the Radial Fourier–Mellin and Hilbert transforms ensure global invariance to scale, translation, and rotation. Texture information for each channel is also obtained based on the Local Binary Pattern (LBP) technique. The proposed hybrid transform-based feature extraction was applied to multiple lesion classes using the International Skin Imaging Collaboration (ISIC) 2019 dataset, preprocessed with data augmentation. Experimental results demonstrate that the proposed method improves classification accuracy and robustness, highlighting its potential as a non-invasive AI-based tool for dermatological diagnosis. Full article

(This article belongs to the Special Issue Recent Advances in Biomedical Data Analysis)

► Show Figures

Figure 1

16 pages, 942 KiB

Open AccessFeature PaperArticle

Deformed Boson Algebras and W_α,β,ν-Coherent States: A New Quantum Framework

by Riccardo Droghei

Mathematics 2025, 13(5), 759; https://doi.org/10.3390/math13050759 - 25 Feb 2025

Cited by 1 | Viewed by 577

Abstract

We introduce a novel class of coherent states, termed

W_{α, β, ν}

-coherent states, constructed using a deformed boson algebra based on the generalised factorial

{[n]}_{α, β, ν}!

. This algebra extends conventional factorials, [...] Read more.

We introduce a novel class of coherent states, termed

W_{α, β, ν}

-coherent states, constructed using a deformed boson algebra based on the generalised factorial

{[n]}_{α, β, ν}!

. This algebra extends conventional factorials, incorporating advanced special functions such as the Mittag-Leffler and Wright functions, enabling the exploration of a broader class of quantum states. The mathematical properties of these states, including their continuity, completeness, and quantum fluctuations, are analysed. A key aspect of this work is the resolution of the Stieltjes moment problem associated with these states, achieved through the inverse Mellin transformation method. The framework provides insights into the interplay between the classical and quantum regimes, with potential applications in quantum optics and fractional quantum mechanics. By extending the theoretical landscape of coherent states, this study opens avenues for further exploration in mathematical physics and quantum technologies. Full article

(This article belongs to the Section E4: Mathematical Physics)

► Show Figures

Figure 1

14 pages, 266 KiB

Open AccessFeature PaperArticle

Mellin Transform of Weierstrass Zeta Function and Integral Representations of Some Lambert Series

by Namhoon Kim

Mathematics 2025, 13(4), 582; https://doi.org/10.3390/math13040582 - 10 Feb 2025

Cited by 1 | Viewed by 514

Abstract

We consider a series which combines two Dirichlet series constructed from the coefficients of a Laurent series and derive a general integral representation of the series as a Mellin transform. As an application, we obtain a family of Mellin integral identities involving the [...] Read more.

We consider a series which combines two Dirichlet series constructed from the coefficients of a Laurent series and derive a general integral representation of the series as a Mellin transform. As an application, we obtain a family of Mellin integral identities involving the Weierstrass elliptic functions and some Lambert series. These identities are used to derive some of the properties of the Lambert series. Full article

(This article belongs to the Special Issue Analytic Methods in Number Theory and Allied Fields)

12 pages, 261 KiB

Open AccessArticle

Mellin and Widder–Lambert Transforms with Applications in the Salem Equivalence to the Riemann Hypothesis

by Emilio R. Negrín, Jeetendrasingh Maan and Benito J. González

Axioms 2025, 14(2), 129; https://doi.org/10.3390/axioms14020129 - 10 Feb 2025

Cited by 1 | Viewed by 607

Abstract

This paper presents a comprehensive study of Plancherel’s theorem and inversion formulae for the Widder–Lambert transform, extending its scope to Lebesgue integrable functions, compactly supported distributions, and regular distributions with compact support. By employing the Plancherel theorem for the classical Mellin transform, we [...] Read more.

This paper presents a comprehensive study of Plancherel’s theorem and inversion formulae for the Widder–Lambert transform, extending its scope to Lebesgue integrable functions, compactly supported distributions, and regular distributions with compact support. By employing the Plancherel theorem for the classical Mellin transform, we derive a corresponding Plancherel’s theorem specific to the Widder–Lambert transform. This novel approach highlights an intriguing connection between these integral transforms, offering new insights into their role in harmonic analysis. Additionally, we explore a class of functions that satisfy Salem’s equivalence to the Riemann hypothesis, providing a deeper understanding of the interplay between such equivalences and integral transforms. These findings open new avenues for further research on the Riemann hypothesis within the framework of integral transforms. Full article

(This article belongs to the Special Issue Elliptic Curves, Modular Forms, L-Functions and Applications)

16 pages, 2897 KiB

Open AccessArticle

Adaptive Invariant Object Representation

by Roumiana Kountcheva and Rumen Mironov

Symmetry 2025, 17(2), 234; https://doi.org/10.3390/sym17020234 - 6 Feb 2025

Viewed by 673

Abstract

In this work, one new approach for RSTB-invariant object representation is presented based on the modified Mellin–Fourier Transform (MFT). For this, in the well-known steps of MFT, the logarithm operation in the log-polar transform is replaced by the operation “rising on a power”. [...] Read more.

In this work, one new approach for RSTB-invariant object representation is presented based on the modified Mellin–Fourier Transform (MFT). For this, in the well-known steps of MFT, the logarithm operation in the log-polar transform is replaced by the operation “rising on a power”. As a result, the central part of the processed area is represented by a significantly larger number of points (transform coefficients), which permits us to give a more accurate description of the main part of the object. The symmetrical properties of the complex conjugated transform coefficients were used, and as a result, the number of coefficients participating in the object representation can be halved without deteriorating the quality of the restored image. The invariant representation is particularly suitable when searching for objects in large databases, which comprise different classes of objects. To verify the performance of the algorithm, object search experiments using the K-Nearest Neighbors (KNN) algorithm were performed, which confirmed this idea. As a result of the analysis, it can be concluded that the complexity of the solutions based on the proposed method depends on the applications, and the inclusion of neural networks is suggested. The neural networks have no conflict with the proposed idea and can only support decision making. Full article

(This article belongs to the Section Computer)

► Show Figures

Figure 1

37 pages, 398 KiB

Open AccessArticle

Explicit Formulas for Hedging Parameters of Perpetual American Options with General Payoffs: A Mellin Transform Approach

by Stefan Zecevic and Mariano Rodrigo

Mathematics 2025, 13(3), 479; https://doi.org/10.3390/math13030479 - 31 Jan 2025

Viewed by 667

Abstract

Risk is often the most concerning factor in financial transactions. Option Greeks provide valuable insights into the risks inherent in option trading and serve as tools for risk mitigation. Traditionally, scholars compute option Greeks through extensive calculations. This article introduces an alternative method [...] Read more.

Risk is often the most concerning factor in financial transactions. Option Greeks provide valuable insights into the risks inherent in option trading and serve as tools for risk mitigation. Traditionally, scholars compute option Greeks through extensive calculations. This article introduces an alternative method that bypasses conventional derivative computation using the Mellin transform and its properties. Specifically, we derive Greeks for perpetual American options with general payoffs, represented as piecewise linear functions, and examine higher-order risk metrics for practical implications. Examples are provided to illustrate the effectiveness of our approach, offering a novel perspective on calculating and interpreting option Greeks. Full article

21 pages, 353 KiB

Open AccessArticle

On Value Distribution for the Mellin Transform of the Fourth Power of the Riemann Zeta Function

by Virginija Garbaliauskienė, Audronė Rimkevičienė, Mindaugas Stoncelis and Darius Šiaučiūnas

Axioms 2025, 14(1), 34; https://doi.org/10.3390/axioms14010034 - 3 Jan 2025

Viewed by 583

Abstract

In this paper, the asymptotic behavior of the modified Mellin transform

Z_{2} (s)

,

s = σ + i t

, of the fourth power of the Riemann zeta function is characterized by weak convergence of probability measures in the [...] Read more.

In this paper, the asymptotic behavior of the modified Mellin transform

Z_{2} (s)

,

s = σ + i t

, of the fourth power of the Riemann zeta function is characterized by weak convergence of probability measures in the space of analytic functions. The main results are devoted to probability measures defined by generalized shifts

Z_{2} (s + i φ (τ))

with a real increasing to

+ \infty

differentiable functions connected to the growth of the second moment of

Z_{2} (s)

. It is proven that the mass of the limit measure is concentrated at the point expressed as

h (s) \equiv 0

. This is used for approximation of

h (s)

by

Z_{2} (s + i φ (τ))

. Full article

15 pages, 277 KiB

Open AccessArticle

Exchange Formulae for the Stieltjes–Poisson Transform over Weighted Lebesgue Spaces

by Hari M. Srivastava, Emilio Ramón Negrín and Jeetendrasingh Maan

Axioms 2024, 13(11), 748; https://doi.org/10.3390/axioms13110748 - 30 Oct 2024

Viewed by 833

Abstract

This paper aims to develop exchange formulae for the Stieltjes–Poisson transform by using Mellin-type convolutions in the context of weighted Lebesgue spaces. A key result is the introduction of bilinear and continuous Mellin-type convolutions, expanding the scope of the analysis to include the [...] Read more.

This paper aims to develop exchange formulae for the Stieltjes–Poisson transform by using Mellin-type convolutions in the context of weighted Lebesgue spaces. A key result is the introduction of bilinear and continuous Mellin-type convolutions, expanding the scope of the analysis to include the space of weighted

L^{1}

functions and the space of continuous functions vanishing at infinity. Full article

(This article belongs to the Special Issue Advanced Approximation Techniques and Their Applications, 2nd Edition)

10 pages, 288 KiB

Open AccessArticle

Multi-Dimensional Integral Transform with Fox Function in Kernel in Lebesgue-Type Spaces

by Sergey Sitnik and Oksana Skoromnik

Mathematics 2024, 12(12), 1829; https://doi.org/10.3390/math12121829 - 12 Jun 2024

Cited by 1 | Viewed by 894

Abstract

This paper is devoted to the study of the multi-dimensional integral transform with the Fox H-function in the kernel in weighted spaces with integrable functions in the domain

R_{+}^{n}

with positive coordinates. Due to the generality of the Fox H [...] Read more.

This paper is devoted to the study of the multi-dimensional integral transform with the Fox H-function in the kernel in weighted spaces with integrable functions in the domain

R_{+}^{n}

with positive coordinates. Due to the generality of the Fox H-function, many special integral transforms have the form studied in this paper, including operators with such kernels as generalized hypergeometric functions, classical hypergeometric functions, Bessel and modified Bessel functions and so on. Moreover, most important fractional integral operators, such as the Riemann–Liouville type, are covered by the class under consideration. The mapping properties in Lebesgue-weighted spaces, such as the boundedness, the range and the representations of the considered transformation, are established. In special cases, it is applied to the specific integral transforms mentioned above. We use a modern technique based on the extensive use of the Mellin transform and its properties. Moreover, we generalize our own previous results from the one-dimensional case to the multi-dimensional one. The multi-dimensional case is more complex and needs more delicate techniques. Full article

(This article belongs to the Special Issue Direct and Inverse Spectral Problems for Ordinary Differential and Functional-Differential Operators)

17 pages, 303 KiB

Open AccessArticle

Generalized Limit Theorem for Mellin Transform of the Riemann Zeta-Function

by Antanas Laurinčikas and Darius Šiaučiūnas

Axioms 2024, 13(4), 251; https://doi.org/10.3390/axioms13040251 - 10 Apr 2024

Cited by 3 | Viewed by 1409

Abstract

In the paper, we prove a limit theorem in the sense of the weak convergence of probability measures for the modified Mellin transform

Z (s)

,

s = σ + i t

, with fixed [...] Read more.

In the paper, we prove a limit theorem in the sense of the weak convergence of probability measures for the modified Mellin transform

Z (s)

,

s = σ + i t

, with fixed

1 / 2 < σ < 1

, of the square

{| ζ (1 / 2 + i t) |}^{2}

of the Riemann zeta-function. We consider probability measures defined by means of

Z (σ + i φ (t))

, where

φ (t)

,

t ⩾ t_{0} > 0

, is an increasing to

+ \infty

differentiable function with monotonically decreasing derivative

φ^{'} (t)

satisfying a certain normalizing estimate related to the mean square of the function

Z (σ + i φ (t))

. This allows us to extend the distribution laws for

Z (s)

. Full article

(This article belongs to the Special Issue Advances in Generalized Hypergeometric Functions, Integral Transforms and Number Theory)

19 pages, 4278 KiB

Open AccessArticle

Remaining Useful Life Prediction of Lithium-Ion Battery Based on Adaptive Fractional Lévy Stable Motion with Capacity Regeneration and Random Fluctuation Phenomenon

by Wanqing Song, Jianxue Chen, Zhen Wang, Aleksey Kudreyko, Deyu Qi and Enrico Zio

Fractal Fract. 2023, 7(11), 827; https://doi.org/10.3390/fractalfract7110827 - 17 Nov 2023

Cited by 5 | Viewed by 2164

Abstract

The capacity regeneration phenomenon is often overlooked in terms of prediction of the remaining useful life (RUL) of LIBs for acceptable fitting between real and predicted results. In this study, we suggest a novel method for quantitative estimation of the associated uncertainty with [...] Read more.

The capacity regeneration phenomenon is often overlooked in terms of prediction of the remaining useful life (RUL) of LIBs for acceptable fitting between real and predicted results. In this study, we suggest a novel method for quantitative estimation of the associated uncertainty with the RUL, which is based on adaptive fractional Lévy stable motion (AfLSM) and integrated with the Mellin–Stieltjes transform and Monte Carlo simulation. The proposed degradation model exhibits flexibility for capturing long-range dependence, has a non-Gaussian distribution, and accurately describes heavy-tailed properties. Additionally, the nonlinear drift coefficients of the model can be adaptively updated on the basis of the degradation trajectory. The performance of the proposed RUL prediction model was verified by using the University of Maryland CALEC dataset. Our forecasting results demonstrate the high accuracy of the method and its superiority over other state-of-the-art methods. Full article

(This article belongs to the Special Issue Fractal Dimension and Fractional Calculus in Mechanical Signal Processing)

► Show Figures

Figure 1

11 pages, 779 KiB

Open AccessArticle

The Fractional Dunkl Laplacian: Definition and Harmonization via the Mellin Transform

by Fethi Bouzeffour

Mathematics 2023, 11(22), 4668; https://doi.org/10.3390/math11224668 - 16 Nov 2023

Viewed by 1413

Abstract

In this paper, we extend the scope of the Tate and Ormerod Lemmas to the Dunkl setting, revealing a profound interconnection that intricately links the Dunkl transform and the Mellin transform. This illumination underscores the pivotal significance of the Mellin integral transform in [...] Read more.

In this paper, we extend the scope of the Tate and Ormerod Lemmas to the Dunkl setting, revealing a profound interconnection that intricately links the Dunkl transform and the Mellin transform. This illumination underscores the pivotal significance of the Mellin integral transform in the realm of fractional calculus associated with differential-difference operators. Our primary focus centers on the Dunkl–Laplace operator, which serves as a prototype of a differential-difference second-order operator within an unbounded domain. Following influential research by Pagnini and Runfola, we embark on an innovative exploration employing Bochner subordination approaches tailored for the fractional Dunkl Laplacian (FDL). Notably, the Mellin transform emerges as a robust and enlightening tool, particularly in its application to the FDL. Full article

(This article belongs to the Special Issue Fractional Calculus and Mathematical Applications, 2nd Edition)

15 pages, 307 KiB

Open AccessArticle

On the Fractional Derivative Duality in Some Transforms

by Manuel Duarte Ortigueira and Gabriel Bengochea

Mathematics 2023, 11(21), 4464; https://doi.org/10.3390/math11214464 - 27 Oct 2023

Cited by 3 | Viewed by 1529

Abstract

Duality is one of the most interesting properties of the Laplace and Fourier transforms associated with the integer-order derivative. Here, we will generalize it for fractional derivatives and extend the results to the Mellin, Z and discrete-time Fourier transforms. The scale and nabla [...] Read more.

Duality is one of the most interesting properties of the Laplace and Fourier transforms associated with the integer-order derivative. Here, we will generalize it for fractional derivatives and extend the results to the Mellin, Z and discrete-time Fourier transforms. The scale and nabla derivatives are used. Some consequences are described. Full article

(This article belongs to the Special Issue Recent Research on Fractional Calculus: Theory and Applications)

17 pages, 401 KiB

Open AccessArticle

Pricing Path-Dependent Options under Stochastic Volatility via Mellin Transform

by Jiling Cao, Xi Li and Wenjun Zhang

J. Risk Financial Manag. 2023, 16(10), 456; https://doi.org/10.3390/jrfm16100456 - 20 Oct 2023

Viewed by 1987

Abstract

In this paper, we derive closed-form formulas of first-order approximation for down-and-out barrier and floating strike lookback put option prices under a stochastic volatility model using an asymptotic approach. To find the explicit closed-form formulas for the zero-order term and the first-order correction [...] Read more.

In this paper, we derive closed-form formulas of first-order approximation for down-and-out barrier and floating strike lookback put option prices under a stochastic volatility model using an asymptotic approach. To find the explicit closed-form formulas for the zero-order term and the first-order correction term, we use Mellin transform. We also conduct a sensitivity analysis on these formulas, and compare the option prices calculated by them with those generated by Monte-Carlo simulation. Full article

(This article belongs to the Special Issue Featured Papers in Mathematics and Finance)

► Show Figures

Figure 1

20 pages, 2506 KiB

Open AccessArticle

Classification of Skin Lesion Images Using Artificial Intelligence Methodologies through Radial Fourier–Mellin and Hilbert Transform Signatures

by Esperanza Guerra-Rosas, Luis Felipe López-Ávila, Esbanyely Garza-Flores, Claudia Andrea Vidales-Basurto and Josué Álvarez-Borrego

Appl. Sci. 2023, 13(20), 11425; https://doi.org/10.3390/app132011425 - 18 Oct 2023

Cited by 2 | Viewed by 2123

Abstract

This manuscript proposes the possibility of concatenated signatures (instead of images) obtained from different integral transforms, such as Fourier, Mellin, and Hilbert, to classify skin lesions. Eight lesions were analyzed using some algorithms of artificial intelligence: basal cell carcinoma (BCC), squamous cell carcinoma [...] Read more.

This manuscript proposes the possibility of concatenated signatures (instead of images) obtained from different integral transforms, such as Fourier, Mellin, and Hilbert, to classify skin lesions. Eight lesions were analyzed using some algorithms of artificial intelligence: basal cell carcinoma (BCC), squamous cell carcinoma (SCC), melanoma (MEL), actinic keratosis (AK), benign keratosis (BKL), dermatofibromas (DF), melanocytic nevi (NV), and vascular lesions (VASCs). Eleven artificial intelligence models were applied so that eight skin lesions could be classified by analyzing the signatures of each lesion. The database was randomly divided into 80% and 20% for the training and test dataset images, respectively. The metrics that are reported are accuracy, sensitivity, specificity, and precision. Each process was repeated 30 times to avoid bias, according to the central limit theorem in this work, and the averages and ± standard deviations were reported for each metric. Although all the results were very satisfactory, the highest average score for the eight lesions analyzed was obtained using the subspace k-NN model, where the test metrics were 99.98% accuracy, 99.96% sensitivity, 99.99% specificity, and 99.95% precision. Full article

(This article belongs to the Special Issue New Trends in Machine Learning for Biomedical Data Analysis)

► Show Figures

Figure 1

Search Results (72)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (72)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI