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Journal articles on the topic 'Mellin and scale transform'

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Author: Grafiati

Published: 11 March 2023

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1

Pivarčiová, Elena, Daynier Rolando Delgado Sobrino, Yury Rafailovich Nikitin, Radovan Holubek, and Roman Ružarovský. "Measuring and evaluating the differences of compared images for a correct car silhouette categorization using integral transforms." Measurement Science Review 18, no. 4 (August 1, 2018): 168–74. http://dx.doi.org/10.1515/msr-2018-0024.

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Abstract The present paper focuses on the analysis of the possibilities of using integral transforms for measuring and evaluating the differences of compared images (car silhouettes) with the purpose of a correct car body categorization. Approaches such as the light intensities frequency change, the application of discrete integral transforms without the use of further supplementary information enabling automated data processing using the Fourier-Mellin transforms are used within this work. The calculation of the several metrics was verified through different combinations that implied using and not using the Hamming window and a low-pass filter. The paper introduced a method for measuring and evaluating the differences in the compared images (car silhouettes). The proposed method relies on the fact that the integral transforms have their own transformants in the case of translation, scaling and rotation, in the frequency area. Besides, the Fourier-Mellin transform was to offer image transformation that is resistant to the translation, rotation and scale.

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2

Xu, Qingwen, Haofei Kuang, Laurent Kneip, and Sören Schwertfeger. "Rethinking the Fourier-Mellin Transform: Multiple Depths in the Camera’s View." Remote Sensing 13, no. 5 (March 5, 2021): 1000. http://dx.doi.org/10.3390/rs13051000.

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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.

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3

Wang, Baocheng, Dandan Qu, Qing Tian, and Liping Pang. "Mellin Transform-Based Correction Method for Linear Scale Inconsistency of Intrusion Events Identification in OFPS." Photonic Sensors 8, no. 3 (May 22, 2018): 220–27. http://dx.doi.org/10.1007/s13320-018-0486-9.

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4

VYAS, VIBHA S., and PRITI P. REGE. "GEOMETRIC TRANSFORM INVARIANT TEXTURE ANALYSIS WITH MODIFIED CHEBYSHEV MOMENTS BASED ALGORITHM." International Journal of Image and Graphics 09, no. 04 (October 2009): 559–74. http://dx.doi.org/10.1142/s0219467809003587.

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Texture based Geometric invariance, which comprises of rotation scale and translation (RST) invariant is finding application in various areas including industrial inspection, estimation of object range and orientation, shape analysis, satellite imaging, and medical diagnosis. Moments based techniques, apart from being computationally simple as compared to other RST invariant texture operators, are also robust in presence of noise. Zernike moments (ZM) based techniques are one of the well-established methods used for texture identification. As ZM are continuous moments, when discretization is done for implementation, errors are introduced. Error, calculated as difference between theoretically computed values and simulated values is proved to be prominent for fine textures. Therefore, a novel approach to detect RST invariant textures present in image is presented in this paper. This approach calculates discrete Chebyshev moments (CM) of log-polar transformed images to achieve rotation and scale invariance. The image is made translation invariant by shifting it to its centroid. The data is collected as samples from Brodatz and Vistex data sets. Zernike moments and its modifications, along with proposed scheme are applied to the same and Performance evaluation apart from RST invariance is noise sensitivity and redundancy. The performance is also compared with circular Mellin Feature extractors.

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5

Monakov, A. A. "An Algorithm for Estimating the Velocity of a Moving Target Based on Mellin Matched Filter." Journal of the Russian Universities. Radioelectronics 25, no. 3 (June 27, 2022): 22–38. http://dx.doi.org/10.32603/1993-8985-2022-25-3-22-38.

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Introduction. Construction of the radar image of a moving target and estimation of its velocity in synthetic aperture radars (SAR) presents a relevant research problem. The low quality of radar imaging is frequently related to the phenomenon of range cell migration (RCM). Conventional methods for RCM compensation, which are successfully used to obtain radar images of stationary targets, fail to provide the required quality when applied to moving targets. At present, a number of algorithms are used to solve this problem. However, the majority of them employ optimization procedures when searching for estimates of unknown parameters, which fact greatly complicates their implementation. An exception is the LvD algorithm, which implements double keystone transform to construct a radar image without using complex estimate search procedures. Radar images are constructed in the coordinates "longitudinal velocity - lateral velocity", which facilitates estimation of the target velocity components.Aim. Development of an alternative algorithm based on the Mellin matched filter (MMF) for estimating the velocity and constructing the radar image of a moving target in a side-looking SAR.Materials and methods. The derived algorithm is based on the invariance of the integral Mellin transform to the signal scale and uses the MMF to estimate the target velocity components.Results. An algorithm for constructing the radar image of a moving target based on the MMF was synthesized. An analysis of the LvD algorithm showed its capacity for selecting the optimum scale factor when implementing a second KT. The conducted computer simulation of the MMF and LvD algorithms showed their equal accuracy. Under the same simulation scenarios, both algorithms yield effective estimates of the velocity components of a moving target when the signal-to-noise ratio is greater than -10 dB.Conclusion. The proposed algorithm for constructing a radar image can be used in SAR systems designed for detection and velocity estimation of a moving target.

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6

Chen, Hui-Chi. "Improved performance of scale-invariant pattern recognition using a combined Mellin radial harmonic function and wavelet transform." Optical Engineering 46, no. 10 (October 1, 2007): 107204. http://dx.doi.org/10.1117/1.2799534.

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7

Monjur, Mehjabin Sultana, Shih Tseng, Renu Tripathi, and M. S. Shahriar. "Incorporation of polar Mellin transform in a hybrid optoelectronic correlator for scale and rotation invariant target recognition." Journal of the Optical Society of America A 31, no. 6 (May 16, 2014): 1259. http://dx.doi.org/10.1364/josaa.31.001259.

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8

Lizzi, Fedele, Mattia Manfredonia, and Flavio Mercati. "Localizability in κ-Minkowski spacetime." International Journal of Geometric Methods in Modern Physics 17, supp01 (June 23, 2020): 2040010. http://dx.doi.org/10.1142/s0219887820400101.

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Using the methods of ordinary quantum mechanics, we study [Formula: see text]-Minkowski space as a quantum space described by noncommuting self-adjoint operators, following and enlarging T. Poulain and J.-C. Wallet, “[Formula: see text]-Poincaré invariant orientable field theories with at 1-loop: Scale-invariant couplings, preprint (2018), arXiv:1808.00350 [hep-th]. We see how the role of Fourier transforms is played in this case by Mellin transforms. We briefly discuss the role of transformations and observers.

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9

Padovese, L. R., N. Martin, and F. Millioz. "Time—frequency and time-scale analysis of Barkhausen noise signals." Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering 223, no. 5 (April 30, 2009): 577–88. http://dx.doi.org/10.1243/09544100jaero436.

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Carrying out information about the microstructure and stress behaviour of ferromagnetic steels, magnetic Barkhausen noise (MBN) has been used as a basis for effective non-destructive testing methods, opening new areas in industrial applications. One of the factors that determines the quality and reliability of the MBN analysis is the way information is extracted from the signal. Commonly, simple scalar parameters are used to characterize the information content, such as amplitude maxima and signal root mean square. This paper presents a new approach based on the time—frequency analysis. The experimental test case relates the use of MBN signals to characterize hardness gradients in a AISI4140 steel. To that purpose different time—frequency (TFR) and time-scale (TSR) representations such as the spectrogram, the Wigner-Ville distribution, the Capongram, the ARgram obtained from an AutoRegressive model, the scalogram, and the Mellingram obtained from a Mellin transform are assessed. It is shown that, due to nonstationary characteristics of the MBN, TFRs can provide a rich and new panorama of these signals. Extraction techniques of some time—frequency parameters are used to allow a diagnostic process. Comparison with results obtained by the classical method highlights the improvement on the diagnosis provided by the method proposed.

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10

Sasiela, Richard J., and John D. Shelton. "Transverse spectral filtering and Mellin transform techniques applied to the effect of outer scale on tilt and tilt anisoplanatism." Journal of the Optical Society of America A 10, no. 4 (April 1, 1993): 646. http://dx.doi.org/10.1364/josaa.10.000646.

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11

Kudryavtsev, Alexey, and Oleg Shestakov. "The Estimators of the Bent, Shape and Scale Parameters of the Gamma-Exponential Distribution and Their Asymptotic Normality." Mathematics 10, no. 4 (February 17, 2022): 619. http://dx.doi.org/10.3390/math10040619.

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When modeling real phenomena, special cases of the generalized gamma distribution and the generalized beta distribution of the second kind play an important role. The paper discusses the gamma-exponential distribution, which is closely related to the listed ones. The asymptotic normality of the previously obtained strongly consistent estimators for the bent, shape, and scale parameters of the gamma-exponential distribution at fixed concentration parameters is proved. Based on these results, asymptotic confidence intervals for the estimated parameters are constructed. The statements are based on the method of logarithmic cumulants obtained using the Mellin transform of the considered distribution. An algorithm for filtering out unnecessary solutions of the system of equations for logarithmic cumulants and a number of examples illustrating the results obtained using simulated samples are presented. The difficulties arising from the theoretical study of the estimates of concentration parameters associated with the inversion of polygamma functions are also discussed. The results of the paper can be used in the study of probabilistic models based on continuous distributions with unbounded non-negative support.

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12

MIRJALILI, A., and K. KESHAVARZIAN. "THE NLO QCD CALCULATION OF SEA QUARK DISTRIBUTIONS IN THE CORGI APPROACH, BASED ON THE CONSTITUENT QUARK MODEL." International Journal of Modern Physics A 22, no. 24 (September 30, 2007): 4519–35. http://dx.doi.org/10.1142/s0217751x07037056.

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Sea quark distributions in the NLO approximation, based on the phenomenological valon model or constituent quark model are analyzed. We use the parametrized inverse Mellin transform technique to perform a direct fit with available experimental data and obtain the unknown parameters of the distributions. We try to extend the calculation to the NLO approximation for the singlet and nonsinglet cases in DIS phenomena. We do also the same calculation for electron–positron annihilation. The resulting sea distributions are effectively independent of the process used. The approach of complete RG improvement (CORGI) is employed and the results are compared with the standard approach of perturbative QCD in the [Formula: see text] scheme with a physical scale. The comparisons with data are in good agreement. As is expected, the results in the CORGI approach indicate a better agreement to the data than the NLO calculation in the standard approach.

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13

Bülow, Heiko, and Andreas Birk. "Scale-Free Registrations in 3D: 7 Degrees of Freedom with Fourier Mellin SOFT Transforms." International Journal of Computer Vision 126, no. 7 (February 23, 2018): 731–50. http://dx.doi.org/10.1007/s11263-018-1067-5.

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14

Lu, Huimin, Dan Xiong, Junhao Xiao, and Zhiqiang Zheng. "Robust long-term object tracking with adaptive scale and rotation estimation." International Journal of Advanced Robotic Systems 17, no. 2 (March 1, 2020): 172988142090973. http://dx.doi.org/10.1177/1729881420909736.

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In this article, a robust long-term object tracking algorithm is proposed. It can tackle the challenges of scale and rotation changes during the long-term object tracking for security robots. Firstly, a robust scale and rotation estimation method is proposed to deal with scale changes and rotation motion of the object. It is based on the Fourier–Mellin transform and the kernelized correlation filter. The object’s scale and rotation can be estimated in the continuous space, and the kernelized correlation filter is used to improve the estimation accuracy and robustness. Then a weighted object searching method based on the histogram and the variance is introduced to handle the problem that trackers may fail in the long-term object tracking (due to semi-occlusion or full occlusion). When the tracked object is lost, the object can be relocated in the whole image using the searching method, so the tracker can be recovered from failures. Moreover, two other kernelized correlation filters are learned to estimate the object’s translation and the confidence of tracking results, respectively. The estimated confidence is more accurate and robust using the dedicatedly designed kernelized correlation filter, which is employed to activate the weighted object searching module, and helps to determine whether the searching windows contain objects. We compare the proposed algorithm with state-of-the-art tracking algorithms on the online object tracking benchmark. The experimental results validate the effectiveness and superiority of our tracking algorithm.

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15

Zhao, Zeng-Shun, Xiang Feng, Sheng-Hua Teng, Yi-Bin Li, and Chang-Shui Zhang. "Multiscale Point Correspondence Using Feature Distribution and Frequency Domain Alignment." Mathematical Problems in Engineering 2012 (2012): 1–14. http://dx.doi.org/10.1155/2012/382369.

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In this paper, a hybrid scheme is proposed to find the reliable point-correspondences between two images, which combines the distribution of invariant spatial feature description and frequency domain alignment based on two-stage coarse to fine refinement strategy. Firstly, the source and the target images are both down-sampled by the image pyramid algorithm in a hierarchical multi-scale way. The Fourier-Mellin transform is applied to obtain the transformation parameters at the coarse level between the image pairs; then, the parameters can serve as the initial coarse guess, to guide the following feature matching step at the original scale, where the correspondences are restricted in a search window determined by the deformation between the reference image and the current image; Finally, a novel matching strategy is developed to reject the false matches by validating geometrical relationships between candidate matching points. By doing so, the alignment parameters are refined, which is more accurate and more flexible than a robust fitting technique. This in return can provide a more accurate result for feature correspondence. Experiments on real and synthetic image-pairs show that our approach provides satisfactory feature matching performance.

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16

Xiao, Bin, Jian-Feng Ma, and Jiang-Tao Cui. "Combined blur, translation, scale and rotation invariant image recognition by Radon and pseudo-Fourier–Mellin transforms." Pattern Recognition 45, no. 1 (January 2012): 314–21. http://dx.doi.org/10.1016/j.patcog.2011.06.017.

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17

Meena, Kunj Bihari, and Vipin Tyagi. "A hybrid copy-move image forgery detection technique based on Fourier-Mellin and scale invariant feature transforms." Multimedia Tools and Applications 79, no. 11-12 (January 3, 2020): 8197–212. http://dx.doi.org/10.1007/s11042-019-08343-0.

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18

Shimizu, Sota, and Joel W. Burdick. "Eccentricity Estimator for Wide-Angle Fovea Vision Sensor." Journal of Robotics and Mechatronics 21, no. 1 (February 20, 2009): 128–34. http://dx.doi.org/10.20965/jrm.2009.p0128.

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This paper proposes a method for estimating the eccentricity that corresponds to the incident angle to a fovea sensor. The proposed method uses the Fourier-Mellin Invariant descriptor to estimate rotation, scale, and translation, by taking both geometrical distortion and non-uniform resolution of a space-variant image from the fovea sensor into account. The following 2 points are focused on in this paper. One is to use multi-resolution images by Discrete Wavelet Transform to properly reduce noise caused by foveation. Another is to use a variable window function (although the window function is generally used for reducing DFT leakage caused by both ends of a signal) to change the effective field of view (FOV) so as not to sacrifice high accuracy. The simulation compares the root mean square (RMS) of the foveation noise between uniform and non-uniform resolutions when a resolution level and a FOV level are changed, respectively. The result shows the proposed method is suitable for the wide-angle space-variant image from the fovea sensor, and, moreover, it does not sacrifice the high accuracy in the central FOV. Another simulation is done to determine a reliable resolution level. This paper is the full translation from the transactions of JSME Vol.74, No.744.

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19

Rajagopal, A. K. "Bivariate averaging functions, translation and scale autocorrelations, Fourier and Mellin transforms, the Wiener-Khinchine theorem and their inter-relationships." Pramana 38, no. 3 (March 1992): 233–47. http://dx.doi.org/10.1007/bf02875370.

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20

Elmasry, Amr, and Hosam Mahmoud. "Analysis of swaps in radix selection." Advances in Applied Probability 43, no. 2 (June 2011): 524–44. http://dx.doi.org/10.1239/aap/1308662491.

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Radix Sort is a sorting algorithm based on analyzing digital data. We study the number of swaps made by Radix Select (a one-sided version of Radix Sort) to find an element with a randomly selected rank. This kind of grand average provides a smoothing over all individual distributions for specific fixed-order statistics. We give an exact analysis for the grand mean and an asymptotic analysis for the grand variance, obtained by poissonization, the Mellin transform, and depoissonization. The digital data model considered is the Bernoulli(p). The distributions involved in the swaps experience a phase change between the biased cases (p ≠ ½) and the unbiased case (p = ½). In the biased cases, the grand distribution for the number of swaps (when suitably scaled) converges to that of a perpetuity built from a two-point distribution. The tool for this proof is contraction in the Wasserstein metric space, and identifying the limit as the fixed-point solution of a distributional equation. In the unbiased case the same scaling for the number of swaps gives a limiting constant in probability.

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Elmasry, Amr, and Hosam Mahmoud. "Analysis of swaps in radix selection." Advances in Applied Probability 43, no. 02 (June 2011): 524–44. http://dx.doi.org/10.1017/s0001867800004973.

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Radix Sort is a sorting algorithm based on analyzing digital data. We study the number of swaps made by Radix Select (a one-sided version of Radix Sort) to find an element with a randomly selected rank. This kind of grand average provides a smoothing over all individual distributions for specific fixed-order statistics. We give an exact analysis for the grand mean and an asymptotic analysis for the grand variance, obtained by poissonization, the Mellin transform, and depoissonization. The digital data model considered is the Bernoulli(p). The distributions involved in the swaps experience a phase change between the biased cases (p≠ ½) and the unbiased case (p= ½). In the biased cases, the grand distribution for the number of swaps (when suitably scaled) converges to that of a perpetuity built from a two-point distribution. The tool for this proof is contraction in the Wasserstein metric space, and identifying the limit as the fixed-point solution of a distributional equation. In the unbiased case the same scaling for the number of swaps gives a limiting constant in probability.

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22

Butzer, P. L., and S. Jansche. "Mellin-Fourier series and the classical Mellin transform." Computers & Mathematics with Applications 40, no. 1 (July 2000): 49–62. http://dx.doi.org/10.1016/s0898-1221(00)00139-5.

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23

Twamley, J., and G. J. Milburn. "The quantum Mellin transform." New Journal of Physics 8, no. 12 (December 20, 2006): 328. http://dx.doi.org/10.1088/1367-2630/8/12/328.

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24

YANG, JIANWEI, LIANG ZHANG, and ZHENGDA LU. "THE MELLIN CENTRAL PROJECTION TRANSFORM." ANZIAM Journal 58, no. 3-4 (March 7, 2017): 256–64. http://dx.doi.org/10.1017/s1446181116000341.

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The central projection transform can be employed to extract invariant features by combining contour-based and region-based methods. However, the central projection transform only considers the accumulation of the pixels along the radial direction. Consequently, information along the radial direction is inevitably lost. In this paper, we propose the Mellin central projection transform to extract affine invariant features. The radial factor introduced by the Mellin transform, makes up for the loss of information along the radial direction by the central projection transform. The Mellin central projection transform can convert any object into a closed curve as a central projection transform, so the central projection transform is only a special case of the Mellin central projection transform. We prove that closed curves extracted from the original image and the affine transformed image by the Mellin central projection transform satisfy the same affine transform relationship. A method is provided for the extraction of affine invariants by employing the area of closed curves derived by the Mellin central projection transform. Experiments have been conducted on some printed Chinese characters and the results establish the invariance and robustness of the extracted features.

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25

Čučković, Željko, and Bo Li. "Berezin Transform, Mellin Transform and Toeplitz Operators." Complex Analysis and Operator Theory 6, no. 1 (March 2, 2010): 189–218. http://dx.doi.org/10.1007/s11785-010-0051-z.

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26

Yonggong Peng, Yixian Wang, Xiangwu Zuo, and Lihua Gong. "Properties of Fractional Mellin Transform." INTERNATIONAL JOURNAL ON Advances in Information Sciences and Service Sciences 5, no. 5 (March 15, 2013): 90–96. http://dx.doi.org/10.4156/aiss.vol5.issue5.11.

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27

Yang, Jianwei, Liang Zhang, and Zhengda Lu. "The Mellin central projection transform." ANZIAM Journal 58 (July 20, 2017): 256. http://dx.doi.org/10.21914/anziamj.v58i0.10980.

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28

Lamb, George, and O. P. Lossers. "Integral by Mellin Transform: 11067." American Mathematical Monthly 112, no. 9 (November 1, 2005): 843. http://dx.doi.org/10.2307/30037618.

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29

Birmingham, D., and S. Sen. "A Mellin transform summation technique." Journal of Physics A: Mathematical and General 20, no. 13 (September 11, 1987): 4557–60. http://dx.doi.org/10.1088/0305-4470/20/13/054.

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30

Venkataramanan, Lalitha, Fred K. Gruber, Tarek M. Habashy, and Denise E. Freed. "Mellin transform of CPMG data." Journal of Magnetic Resonance 206, no. 1 (September 2010): 20–31. http://dx.doi.org/10.1016/j.jmr.2010.05.015.

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31

Rogov, V. B. K. "The mellin-whittaker integral transform." Mathematical Notes of the Academy of Sciences of the USSR 39, no. 6 (June 1986): 434–37. http://dx.doi.org/10.1007/bf01157027.

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32

ZHENG SHI-HAI, CHEN YAN-SONG, and LI DE-HUA. "REALIZATION OF TWO-DIMENSIONAL MELLIN TRANSFORM." Acta Physica Sinica 39, no. 5 (1990): 749. http://dx.doi.org/10.7498/aps.39.749.

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33

Mohan, N. L., L. Anandababu, and S. V. Seshagiri Rao. "Gravity interpretation using the Mellin transform." GEOPHYSICS 51, no. 1 (January 1986): 114–22. http://dx.doi.org/10.1190/1.1442024.

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The Mellin transform of the gravity effect of a buried sphere and two‐dimensional horizontal circular cylinder, and the first horizontal derivative of the gravity effect of a two‐dimensional thin fault layer are derived. The transformed functions are bounded by two asymptotes. They are analyzed and procedures are formulated excluding the asymptotic regions for the extraction of the body parameters. The application of the Mellin transform is tested on simulated models as well as on two field examples: (1) the Humble Dome gravity anomaly near Houston, USA; and (2) the Louga gravity anomaly, USA.

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34

Kaiser, G. "Wavelet filtering with the Mellin transform." Applied Mathematics Letters 9, no. 5 (September 1996): 69–74. http://dx.doi.org/10.1016/0893-9659(96)00075-4.

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35

Fitouhi, A., N. Bettaibi, and K. Brahim. "The Mellin Transform in Quantum Calculus." Constructive Approximation 23, no. 3 (June 3, 2005): 305–23. http://dx.doi.org/10.1007/s00365-005-0597-6.

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36

Passare, Mikael, and August Tsikh. "Residue Integrals and their Mellin Transforms." Canadian Journal of Mathematics 47, no. 5 (October 1, 1995): 1037–50. http://dx.doi.org/10.4153/cjm-1995-055-4.

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AbstractGiven an almost arbitrary holomorphic map we study the structure of the associated residue integral and its Mellin transform, and the relation between these two objects. More precisely, we relate the limit behaviour of the residue integral to the polar structure of the Mellin transform. We consider also ideals connected to nonisolated singularities.

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Zhou, Zhibiao, Wei Xiao, and Yongshun Liang. "Partially Explore the Differences and Similarities between Riemann-Liouville Integral and Mellin Transform." Fractal and Fractional 6, no. 11 (November 1, 2022): 638. http://dx.doi.org/10.3390/fractalfract6110638.

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At present many researchers devote themselves to studying the relationship between continuous fractal functions and their fractional integral. But little attention is paid to the relationship between Mellin transform and fractional integral. This paper aims to partially explore the differences and similarities between Riemann-Liouville integral and Mellin transform, then a 1-dimensional continuous and unbounded variational function defined on the closed interval [0,1] needs to be constructed. Through describing the image of the constructed function and its transformed function and proving the relevant properties, we obtain that Box dimension of its Riemann–Liouville integral of arbitrary order and its Mellin transformed function are also one. The smoothness of its Riemann–Liouville integral can only be improved, and its Mellin transformed function is differentiable.

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38

Yang, Jianwei, Liang Zhang, and Peiyao Li. "Radon–Fourier descriptor for invariant pattern recognition." International Journal of Wavelets, Multiresolution and Information Processing 17, no. 02 (March 2019): 1940004. http://dx.doi.org/10.1142/s0219691319400046.

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Radon transform is not only robust to noise, but also independent on the calculation of pattern centroid. In this paper, Radon–Mellin transform (RMT), which is a combination of Radon transform and Mellin transform, is proposed to extract invariant features. RMT converts any object into a closed curve. Radon–Fourier descriptor (RFD) is derived by applying Fourier descriptor to the obtained closed curve. The obtained RFD is invariant to scaling and rotation. (Generic) R-transform and some other Radon-based methods can be viewed as special cases of the proposed method. Experiments are conducted on some binary images and gray images.

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39

A. S. Gudadhe, A. S. Gudadhe. "Abelian Theorem for Generalized Mellin-Whittaker Transform." IOSR Journal of Mathematics 1, no. 4 (2012): 19–23. http://dx.doi.org/10.9790/5728-0141923.

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40

Kiliçman, Adem. "A Note on Mellin Transform and Distributions." Mathematical and Computational Applications 9, no. 1 (April 1, 2004): 65–72. http://dx.doi.org/10.3390/mca9010065.

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41

Deiermann, Paul, Cape Girardeau, Joseph Chalmers, and George L. Lamb. "An Application of the Mellin Transform: 10963." American Mathematical Monthly 111, no. 5 (May 2004): 441. http://dx.doi.org/10.2307/4145282.

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Al-Omari, S. K. Q., and Adem Kılıçman. "On Modified Mellin Transform of Generalized Functions." Abstract and Applied Analysis 2013 (2013): 1–6. http://dx.doi.org/10.1155/2013/539240.

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Rajan, Arvind, Melanie Po-Leen Ooi, Ye Chow Kuang, and Serge N. Demidenko. "Analytical Standard Uncertainty Evaluation Using Mellin Transform." IEEE Access 3 (2015): 209–22. http://dx.doi.org/10.1109/access.2015.2415592.

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Wu, C. Y., A. R. D. Somervell, T. G. Haskell, and T. H. Barnes. "Optical Mellin transform through Haar wavelet transformation." Optics Communications 227, no. 1-3 (November 2003): 75–82. http://dx.doi.org/10.1016/j.optcom.2003.09.040.

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Butzer, Paul L., and Stefan Jansche. "A direct approach to the mellin transform." Journal of Fourier Analysis and Applications 3, no. 4 (July 1997): 325–76. http://dx.doi.org/10.1007/bf02649101.

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Ilie, Mousa, Jafar Biazar, and Zainab Ayati. "Mellin transform and conformable fractional operator: applications." SeMA Journal 76, no. 2 (August 30, 2018): 203–15. http://dx.doi.org/10.1007/s40324-018-0171-3.

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Ostrovsky, Dmitry. "Mellin Transform of the Limit Lognormal Distribution." Communications in Mathematical Physics 288, no. 1 (March 11, 2009): 287–310. http://dx.doi.org/10.1007/s00220-009-0771-y.

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Prudnikov, A. P., Yu A. Brychkov, and O. I. Marichev. "Evaluation of integrals and the mellin transform." Journal of Soviet Mathematics 54, no. 6 (May 1991): 1239–341. http://dx.doi.org/10.1007/bf01373648.

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BRAHIM, Kamel, and Latifa RIAHI. "Two dimensional Mellin transform in Quantum Calculus." Acta Mathematica Scientia 38, no. 2 (March 2018): 546–60. http://dx.doi.org/10.1016/s0252-9602(18)30765-3.

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Alotta, Gioacchino, Mario Di Paola, and Giuseppe Failla. "A Mellin transform approach to wavelet analysis." Communications in Nonlinear Science and Numerical Simulation 28, no. 1-3 (November 2015): 175–93. http://dx.doi.org/10.1016/j.cnsns.2015.04.001.

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