Search Results (15)

Integral transforms play a fundamental role in science and engineering. Above all, the Fourier transform is the most vital, which has some specifications—Laplace transform, Mellin transform, etc., with their inverse transforms. In this paper, we restrict ourselves to the use of a few versions of the Mellin transform, which are best suited to the treatment of zeta functions as Dirichlet series. In particular, we shall manifest the underlying principle that automorphy (which is a modular relation, an equivalent to the functional equation) is intrinsic to lattice (or Epstein) zeta functions by considering some generalizations of the holomorphic and non-holomorphic Eisenstein series as the Epstein-type Eisenstein series, which have been treated as totally foreign subjects to each other. We restrict to the modular relations with one gamma factor and the resulting integrals reduce to a form of the modified Bessel function. In the H-function hierarchy, what we work with is the second simplest $H_{1,1}^{1,1}\leftrightarrow H_{0,2}^{2,0}$, with H denoting the Fox H-function. Full article

We study an integral transform—here called the Master Integral Transform—in which the kernel is an arbitrary entire function of finite order. When the nonzero Taylor coefficients of the kernel have positive Beurling–Malliavin density, we prove completeness and global injectivity in a Cauchy-weighted Hilbert space, and we furnish explicit Mellin–Fourier inversion formulae with exponentially decaying integrands. Classical Fourier, Laplace, and Mellin transforms appear only as strict special cases. Beyond these, we establish structural properties (multiplier/composition law, dilation covariance, parameter regularity) and present applications not captured by fixed-kernel frameworks, including inverse-kernel identification and hybrid boundary value models, e.g., the Poisson–Airy pair produces a closed-form transformed Green’s function and a solvable variable-coefficient PDE, illustrating capabilities unavailable to fixed-kernel frameworks. Full article

The aim of this work was to construct explicit expressions for the summation of Dirichlet Beta functions with odd arguments and Zeta functions with even arguments. In the established literature, this is typically done using Fourier series expansions or Bernoulli numbers and polynomials. Here, instead, we achieve our goal by employing tools from probability: specifically, we introduce a generalisation of a technique based on multiple integrals and the algebra of random variables. This also allows us to increase the number of nested integrals and Cauchy random variables involved. Another key contribution is that, by generalising the exponent of Cauchy random variables, we obtain an original proof of the reflection formulae for the Digamma and Trigamma functions. These probabilistic proofs crucially utilise the Mellin transform to compute the integrals needed to determine probability density functions. It is noteworthy that, while understanding the presented topic requires knowledge of the rules for calculating multiple integrals (Fubini’s Theorem) and the algebra of continuous random variables, these are concepts commonly acquired by second-year university students in STEM disciplines. Our study thus offers new perspectives on how the mathematical functions considered relate and shows the significant role of probabilistic methods in promoting comprehension of this research area, in a way accessible to a broad and non-specialist audience. Full article

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

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

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 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

The real-time nondestructive monitoring of plant water content can enable operators to understand the water demands of crops in a timely manner and provide a reliable basis for precise irrigation. In this study, a method for rapid estimation of water content in Aquilaria sinensis using multispectral imaging was proposed. First, image registration and segmentation were performed using the Fourier–Mellin transform (FFT) and the fuzzy local information c-means clustering algorithm (FLICM). Second, the spectral features (SFs), texture features (TFs), and comprehensive features (CFs) of the image were extracted. Third, using the eigenvectors of the SFs, TFs, and CFs as input, a random forest regression model for estimating the water content of A. sinensis was constructed, respectively. Finally, the monarch butterfly optimization (MBO), Harris hawks optimization (HHO), and sparrow search algorithm (SSA) were used to optimize all models to determine the best estimation model. The results showed that: (1) 60%–80% soil water content is the most suitable for A. sinensis growth. Compared with waterlogging, drought inhibited A. sinensis growth more significantly. (2) FMT + FLICM could achieve rapid segmentation of discrete A. sinensis multispectral images on the basis of guaranteed accuracy. (3) The prediction effect of TFs was basically the same as that of SFs, and the prediction effect of CFs was higher than that of SFs and TFs, but this difference would decrease with the optimization of the RFR model. (4) Among all models, SSA-RFR_CFs had the highest accuracy, with an R² of 0.8282. These results confirmed the feasibility and accuracy of applying multispectral imaging technology to estimate the water content of A. sinensis and provide a reference for the protection and cultivation of endangered precious tree species. Full article

The Voronoĭ summation formula is known to be equivalent to the functional equation for the square of the Riemann zeta function in case the function in question is the Mellin tranform of a suitable function. There are some other famous summation formulas which are treated as independent of the modular relation. In this paper, we shall establish a far-reaching principle which furnishes the following. Given a zeta function $Z\left(s\right)$ satisfying a suitable functional equation, one can generalize it to $Z_f\left(s\right)$ in the form of an integral involving the Mellin transform $F\left(s\right)$ of a certain suitable function $f\left(x\right)$ and process it further as ${\tilde{Z}}_f\left(s\right)$. Under the condition that $F\left(s\right)$ is expressed as an integral, and the order of two integrals is interchangeable, one can obtain a closed form for ${\tilde{Z}}_f\left(s\right)$. Ample examples are given: the Lipschitz summation formula, Koshlyakov’s generalized Dedekind zeta function and the Plana summation formula. In the final section, we shall elucidate Hamburger’s results in light of RHBM correspondence (i.e., through Fourier–Whittaker expansion). Full article

A tensor representation structure based on the multilayer tensor spectrum pyramid (MLTSP) is introduced in this work. The structure is “truncated”, i.e., part of the high-frequency spectrum coefficients is cut-off, and on the retained low-frequency coefficients, obtained at the output of each pyramid layer, a hierarchical tensor SVD (HTSVD) is applied. This ensures a high concentration of the input tensor energy into a small number of decomposition components of the tensors obtained at the coder output. The implementation of this idea is based on a symmetrical coder/decoder. An example structure for a cubical tensor of size 8 × 8 × 8, which is represented as a two-layer tensor spectrum pyramid, where 3D frequency-ordered fast Walsh–Hadamard transform and HTSVD are used, is given in this paper. The analysis of the needed mathematical operations proved the low computational complexity of the new approach, due to a lack of iterative calculations. The high flexibility of the structure in respect to the number of pyramid layers, the kind of used orthogonal transforms, the number of retained spectrum coefficients, and HTSVD components, permits us to achieve the desired accuracy of the restored output tensor, imposed by the application. Furthermore, this paper presents one possible application for 3D object searches in a tensor database. In this case, to obtain the invariant representation of the 3D objects, in the spectrum pyramid, the 3D modified Mellin–Fourier transform is embedded, and the corresponding algorithm is shown. Full article

Fourier-Mellin-SOFT (FMS) is a rigid 3D registration method, which allows the robust registration of 3 degrees-of-freedom (dof) rotation, 1-dof scale, and 3-dof translation between scans on discrete grids. FMS is based on a spectral decomposition of these 7-dof. This complete spectral representation of the input data enables an adaption to certain frequency ranges. This special property is used here to focus on relevant mutual 3D information between bone structures with a Low Frequency adaptation of FMS (LF-FMS), that is, it is utilized for matching and concurrently determining corresponding transformation parameters. This process is applied on a set of Magnetic Resonance Tomography (MRT) data representing the hand region, in particular the carpal bone area, in a sequence of different hand positions. This data set is available for different probands, which allows a comparison of resulting parameter plots and furthermore matching in between bone structures. Full article

Remote sensing and robotics often rely on visual odometry (VO) for localization. Many standard approaches for VO use feature detection. However, these methods will meet challenges if the environments are feature-deprived or highly repetitive. Fourier-Mellin Transform (FMT) is an alternative VO approach that has been shown to show superior performance in these scenarios and is often used in remote sensing. One limitation of FMT is that it requires an environment that is equidistant to the camera, i.e., single-depth. To extend the applications of FMT to multi-depth environments, this paper presents the extended Fourier-Mellin Transform (eFMT), which maintains the advantages of FMT with respect to feature-deprived scenarios. To show the robustness and accuracy of eFMT, we implement an eFMT-based visual odometry framework and test it in toy examples and a large-scale drone dataset. All these experiments are performed on data collected in challenging scenarios, such as, trees, wooden boards and featureless roofs. The results show that eFMT performs better than FMT in the multi-depth settings. Moreover, eFMT also outperforms state-of-the-art VO algorithms, such as ORB-SLAM3, SVO and DSO, in our experiments. Full article

In this paper, we introduce a new type of integral transforms, called the ARA integral transform that is defined as: ${\mathcal{G}}_n\left[g\left(t\right)\right]\left(s\right)=G\left(n,s\right)=s\underset{0}{\overset{\infty }{{\displaystyle \int }}}{t}^{n-1}{e}^{-st}g\left(t\right)dt,\hspace{1em}s>0$ . We prove some properties of ARA transform and give some examples. Also, some applications of the ARA transform are given. Full article

Given a query shoeprint image, shoeprint retrieval aims to retrieve the most similar shoeprints available from a large set of shoeprint images. Most of the existing approaches focus on designing single low-level features to highlight the most similar aspects of shoeprints, but their retrieval precision may vary dramatically with the quality and the content of the images. Therefore, in this paper, we proposed a shoeprint retrieval method to enhance the retrieval precision from two perspectives: (i) integrate the strengths of three kinds of low-level features to yield more satisfactory retrieval results; and (ii) enhance the traditional distance-based similarity by leveraging the information embedded in the neighboring shoeprints. The experiments were conducted on a crime scene shoeprint image dataset, that is, the MUES-SR10KS2S dataset. The proposed method achieved a competitive performance, and the cumulative match score for the proposed method exceeded 92.5% in the top 2% of the dataset, which was composed of 10,096 crime scene shoeprints. Full article

Mellin Transform occurs in many branches of Applied Mathematics and Engineering. The Mellin transform is very much related to the Laplace and Fourier transforms and the theory for the ordinary functions is well established. In the distributional sense, first it was studied by Zemanian in [2]. In this work we try to extend to the wider class of distributions. Full article