New type of Gegenbauer-Hermite monogenic polynomials and associated Clifford wavelets. (English) Zbl 1455.94005
Summary: Nowadays, 3D images processing constitutes a challenging topic in many scientific fields such as medicine, computational physics, informatics. Therefore, the construction of suitable functional bases that allow computational aspects to be easily done is a necessity. Wavelets and Clifford algebras are ones of the most import mathematical tools for achieving such necessities. In the present work, new classes of wavelet functions are presented in the framework of Clifford analysis. Firstly, some classes of new monogenic polynomials are provided based on two-parameter weight functions. Such classes englobe the well-known Gegenbauer and Hermite ones. The discovered polynomial sets are next applied to introduce new wavelet functions. Reconstruction formula and Fourier-Plancherel rules have been proved. Computational concrete examples are developed by means of some illustrative examples with graphical representations of the Clifford mother wavelets in some cases. These discovered wavelets have been applied by the next for modeling some biomedical signals such as EEG, ECG and proteins.
MSC:
| 94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |
| 42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |
| 42B10 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
| 92C55 | Biomedical imaging and signal processing |
Keywords:
continuous wavelet transform; Clifford analysis; Clifford Fourier transform; Fourier-Plancherel; monogenic polynomials; biosignalsSoftware:
FHTReferences:
| [1] | Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions with formulas, graphs, and mathematical tables. National Bureau of Standards, Applied Mathematics Series, 55, USA (1964) · Zbl 0171.38503 |
| [2] | Abreu-Blaya, R., Bory-Reyes, J., Bosch, P.: Extension Theorem for Complex Clifford Algebras-Valued Functions on Fractal Domains, Boundary Value Problems, Article ID 513186, 9 pages (2010) · Zbl 1196.30044 |
| [3] | Almeida, J.B.: Can physics laws be derived from monogenic functions? arXiv:physics/0601194v1 (2006) |
| [4] | Altaiski, M.; Mornev, M.; Polozov, O., Wavelet analysis of DNA sequence, Genet. Anal., 12, 1996, 165-168 (1996) |
| [5] | Antoine, J-P; Murenzi, R.; Vandergheynst, P., Two-dimensional directional wavelets in image processing, Int. J. Imaging Syst. Technol., 7, 3, 152-165 (1996) |
| [6] | Arfaoui, S.: New Monogenic Gegenbauer Jacobi Polynomials And Associated Spheroidal Wavelets, Doctoral Thesis in Mathematics, University of Monastir, Tunisia, Faculty of Sciences, July 26 (2017) · Zbl 1409.42024 |
| [7] | Arfaoui, S.; Mabrouk, Aben, Some ultraspheroidal monogenic Clifford Gegenbauer Jacobi polynomials and associated wavelets, Adv. Appl. Clifford Algebra, 27, 3, 2287-2306 (2017) · Zbl 1409.42024 |
| [8] | Arfaoui, S.; Mabrouk, A., Some old orthogonal polynomials revisited and associated wavelets: two-parameters Clifford-Jacobi polynomials and associated spheroidal wavelets, Acta Applicandae Mathematicae, 155, 1, 177-195 (2018) · Zbl 1407.42028 |
| [9] | Arfaoui, S., Mabrouk, A.B.: Some Generalized Clifford-Jacobi Polynomials and Associated Spheroidal Wavelets. arXiv:1704.03513 (2017) · Zbl 1409.42024 |
| [10] | Arfaoui, S., Rezgui, I., Mabrouk, A.B.: Harmonic wavelet analysis on the sphere, spheroidal wavelets, Degryuter (2016). ISBN 978-11-048188-4 · Zbl 1369.42026 |
| [11] | Baleanu, D.: Wavelet Transforms and Their Recent Applications in Biology and Geoscience. ISBN 978-953-51-0212-0, 310 pages, InTech Publisher (2012) |
| [12] | Baspinar, E.; Citti, G.; Sarti, A., A geometric model of multi-scale orientation preference maps via Gabor functions, J. Math. Imaging Vis., 60, 900-912 (2018) · Zbl 1437.94006 · doi:10.1007/s10851-018-0803-3 |
| [13] | Bekkers, E.; Duits, R.; Berendschot, T.; Romeny, Bh, A multi-orientation analysis approach to retinal vessel tracking, J. Math. Imaging Vis., 49, 3, 583-610 (2014) · Zbl 1291.92075 |
| [14] | Mabrouk, A. Ben, Rabbouch, B., Saadaoui, F.: A wavelet based methodology for predicting transmembrane segments. Poster Session, the International Conference of Engineering Sciences for Biology and Medicine, 1-3 May Monastir, Tunisia (2015) |
| [15] | Bernstein, S., Clifford continuous wavelet transforms in \({\cal{L}}0,2\) and \({\cal{L}}0,3\), AIP Conf. Proc., 1048, 634 (2008) · Zbl 1163.42308 · doi:10.1063/1.2991006 |
| [16] | Bin, Y.; Zhang, Y., A simple method for predicting transmembrane proteins based on wavelet transform, Int. J. Biol. Sci., 9, 1, 22-33 (2013) |
| [17] | Bin, Y.; Zhang, Y., On the prediction of transmembrane helical segments in membrane proteins, World Acad. Sci. Eng. Technol., 37, 554-557 (2010) |
| [18] | Bodmann, Bg; Hoffman, Dk; Kouri, Dj; Papadakis, M., Hermite distributed approximating functionals as almost-ideal low-pass filters, Sampl. Theory Signal Image Process., 1, 1, 0-50 (2002) |
| [19] | Brackx, F.; Chisholm, Jsr; Soucek, V., Clifford Analysis and Its Applications NATO Science Series, Series II: Mathematics, Physics and Chemistry (2000), Berlin: Springer, Berlin |
| [20] | Brackx, F.; Delanghe, R.; Sommen, F., Clifford Analysis (1982), Trowbridge: Pitman Publication, Trowbridge · Zbl 0529.30001 |
| [21] | Brackx, F.; De Schepper, N.; Sommen, F., The Clifford-Gegenbauer polynomials and the associated continuous wavelet transform, Integral Transforms and Special Functions, 15, 5, 387-404 (2004) · Zbl 1062.30057 |
| [22] | Brackx, F.; De Schepper, N.; Sommen, F., The two-dimensional Clifford-Fourier transform, J. Math. Imaging, 26, 5-18 (2006) · Zbl 1122.42016 |
| [23] | Brackx, F.; De Schepper, N.; Sommen, F., The Fourier Transform in Clifford analysis, Adv. Imaging Electron Phys., 156, 55-201 (2009) |
| [24] | Brackx, F.; De Schepper, N.; Sommen, F.; Qian, T.; Vai, Mi; Xu, Y., Clifford-Jacobi Polynomials and the associated continuous wavelet transform in Eucllidean space, Wavelet Analysis and Applications. Applied and Numerical Harmonic Analysis, 185-198 (2006), Basel: Birkhäuser, Basel · Zbl 1110.42008 |
| [25] | Brackx, F.; Schepper, N. De; Sommen, F., The Clifford-Laguerre continuous wavelet transform, Bull. te Belgian Math. Soc.-Simon Stvin, 11, 2, 201-215 (2004) · Zbl 1077.42026 |
| [26] | Brackx, F.; Sommen, F., Clifford-Hermite wavelets in Euclidean space, J. Fourier Anal. Appl., 6, 3, 299-310 (2000) · Zbl 0961.42021 |
| [27] | Brackx, F.; Sommen, F., The generalized Clifford-Hermite continuous wavelet transform, Adv. Appl. Clifford Algebras, 11, 51, 219-231 (2001) · Zbl 1221.42058 |
| [28] | Brackx, F., Schepper, N. De, Sommen, F.: Clifford-Hermite and two-dimensional Clifford-Gabor filters for early vision. In (digital) Proceedings 17th International Conference on the Application of Computer Science and Mathematics in Architecture and Civil Engineering (K. Gürlebeck and C. Könke, eds.), July 12-14, Bauhaus-Universität Weimar (2006) · Zbl 1164.47336 |
| [29] | Brackx, F.; Hitzer, E.; Sangwine, Sj, History of quaternion and Clifford-fourier transforms and wavelets, in Quaternion and Clifford fourier transforms and wavelets, Trends Math., 27, 11-27 (2013) · Zbl 1273.42007 |
| [30] | Bujack, R.; De Bie, H.; De Schepper, N.; Scheuermann, G., Products, convolution, for Hypercomplex Fourier Transforms, J. Math. Imaging Vis., 48, 606-624 (2013) · Zbl 1292.42007 · doi:10.1007/s10851-013-0430-y |
| [31] | Bujack, R.; Scheuermann, G.; Hitzer, E.; Hitzer, E.; Sangwine, Sj, A General Geometric Fourier Transform, Quaternion and Clifford Fourier transforms and wavelets, Trends in Mathematics 27, 155-176 (2013), Basel: Birkhauser, Basel · Zbl 1273.15020 |
| [32] | Bujack, R.; Scheuermann, G.; Hitzer, E., A general geometric Fourier transform convolution theorem, Adv. Appl. Clifford Algebras, 23, 1, 15-38 (2013) · Zbl 1263.15021 · doi:10.1007/s00006-012-0338-4 |
| [33] | Carré, P.; Berthier, M.; Fernandez-Maloigne, C., Chapter 6, Color Representation and Processes with Clifford Algebra, Advanced Color Image Processing and Analysis, 147-179 (2013), New York: Springer, New York |
| [34] | Carré, P.; Denis, P.; Fernandez-Maloigne, C., Spatial color image processing using Clifford algebras: application to color active contour, Signal, Image Video Process., 8, 7, 1357-1372 (2012) |
| [35] | Cattani, C.; Kudreyko, A., Harmonic wavelet method towards solution of the Fredholm type integral equations of the second kind, Appl. Math. Comput., 215, 12, 4164-4171 (2010) · Zbl 1186.65160 |
| [36] | Cattani, C.: Signorini Cylindrical Waves and Shannon Wavelets, Advances in Numerical Analysis, Volume 2012, Article ID 731591, 24 pages · Zbl 1329.74105 |
| [37] | Cattani, C., Harmonic wavelet approximation of random, fractal and high frequency signals, Telecommun. Syst., 43, 3-4, 207-217 (2010) |
| [38] | Cattani, Carlo, Fractals and Hidden Symmetries in DNA, Mathematical Problems in Engineering, 2010, 1-31 (2010) · Zbl 1189.92015 |
| [39] | Cattani, C.: Wavelet algorithms for DNA analysis. In Algorithms in Computational Molecular Biology: Techniques, Approaches and Applications. M. Elloumi and A. Y. Zomaya, (Eds.), Wiley Series in Bioinformatics, John Wiley & Sons, pp. 799-842 |
| [40] | Cattani, Carlo, On the Existence of Wavelet Symmetries in Archaea DNA, Computational and Mathematical Methods in Medicine, 2012, 1-21 (2012) · Zbl 1234.92014 |
| [41] | Cattani C.: Complexity and symmetries in DNA sequences. In Handbook of Biological Discovery, Elloumi, M. & Zomaya, A. Y. (Eds.), Wiley Series in Bioinformatics, John Wiley & Sons, Hoboken, pp. 700-742 |
| [42] | Cattani, Carlo; Pierro, Gaetano, Complexity on Acute Myeloid Leukemia mRNA Transcript Variant, Mathematical Problems in Engineering, 2011, 1-16 (2011) · Zbl 1235.92025 |
| [43] | Cattani, C.; Pierro, G., On the fractal geometry of DNA by the binary image analysis, Bull. Math. Biol., 75, 9, 28 (2013) · Zbl 1272.92013 |
| [44] | Cattani, C.; Bellucci, M.; Scalia, M.; Mattioli, G., Wavelet analysis of correlation in DNA sequences, Izvestia Vysshikh Uchebnykh Zavedenii Radioelektronika, 29, 50-58 (2006) |
| [45] | Chen, Cp; Rost, B., State-of-the-art in membrane protein prediction, Appl. Bioinform., 1, 21-35 (2002) |
| [46] | Chen, Y.; Yang, B.; Dong, J., Time-series prediction using a local wavelet neural network, Neurocomputing, 69, 449-465 (2006) |
| [47] | Colombo, F.; Sabadini, I.; Sommen, F.; Struppa, Dc, Analysis of Dirac Systems and Computational Algebra, Progress in Mathematical Physics (2004), Boston: Birkhäuser, Boston · Zbl 1064.30049 |
| [48] | Cserzäo, M.; Wallin, E.; Simon, I.; Von Heijne, G.; Elofsson, A., Prediction of transmembrane a-helices in prokaryotic membrane proteins: the dense alignment surface method, Protein Eng., 10, 673-676 (1997) |
| [49] | Bie, H. De: Clifford algebras, Fourier transforms and quantum mechanics. arXiv:1209.6434v1 (2012) · Zbl 1259.42001 |
| [50] | Bie, H. De, Xu, Y.: On the Clifford-Fourier transform. arXiv:1003.0689 (2010) · Zbl 1237.30018 |
| [51] | De Schepper, N., The generalized Clifford-Gegenbauer polynomials revisited, Adv. Appl. Clifford Algebras, 19, 253-268 (2009) · Zbl 1167.42009 |
| [52] | Delanghe, R., Clifford analysis: history and perspective, Comput. Methods Funct. Theory, 1, 1, 107-153 (2001) · Zbl 1011.30045 |
| [53] | Tunjung, N. Dian, Arifin, A. Zainal, Soelaiman, R.: Medical image segmentation using generalized gradient vector flow and clifford geometric algebra. International Conference on Biomedical Engineering, Surabaya, Indonesia, November 11, 5 pages (2008) |
| [54] | Di Claudio, Ed; Jacovitti, G.; Laurenti, A., On the inter-conversion between Hermite and Laguerre local image expansions, IEEE Trans. Image Process., 20, 12, 3553-3565 (2011) · Zbl 1374.94081 |
| [55] | Duits, R.; Felsberg, M.; Granlund, G.; Ter Romeny, Bh, Image analysis and reconstruction using a wavelet transform constructed from a reducible representation of the Euclidean motion group, Int. J. Comput. Vis., 72, 1, 79-102 (2006) · doi:10.1007/s11263-006-8894-5 |
| [56] | Escalante-Ramirez, B., The Hermite transform as an efficient model for local image analysis: an application to medical image fusion, Comput. Electr. Eng., 34, 99-110 (2008) · Zbl 1140.68516 |
| [57] | Estudillo-Romero, A., Escalante-Ramirez, B.: The Hermite transform: an alternative image representation model for Iris Recognition. pp. 86-93. In Progress in Pattern Recognition, Image Analysis and Applications, José Ruiz-Shulcloper Walter G. Kropatsch (Eds.), 13th Iberoamerican Congress on Pattern Recognition, CIARP 2008, Havana, Cuba, September 9-12, 2008 Proceedings. Lecture Notes in Computer Science 5197. Springer (2008) |
| [58] | Felsberg, M.; Duits, R.; Florack, L., The monogenic scale space on a rectangular domain and its features, Int. J. Comput. Vis., 64, 2-3, 187-201 (2005) · doi:10.1007/s11263-005-1843-x |
| [59] | Felsberg, M.; Sommer, G., The monogenic signal, IEEE Trans. Signal Process., 49, 12, 3136-3144 (2001) · Zbl 1369.94139 |
| [60] | Fischer, P., Multiresolution analysis for 2D turbulence Part 1: wavelets vs cosince packets, a comparative study, Discret. Contin. Dyn. Syst. B, 5, 659-686 (2005) · Zbl 1140.76350 |
| [61] | Fischer, P.; Baudoux, G.; Wouthers, J., Wavepred: a wavelet-based algorithm for the prediction of transmembrane proteins, Commun. Math. Sci., 1, 1, 44-56 (2003) · Zbl 1079.92033 |
| [62] | Fletcher, P., Discrete wavelets with Quaternion and Clifford coefficients, Adv. Appl. Clifford Algebras, 28, 59 (2018) · Zbl 1407.42031 · doi:10.1007/s00006-018-0876-5 |
| [63] | Gao, T., Liu, Z.-g., Gao, W.-ch., Zhang, J.: A robust technique for background subtraction in traffic video. In Advances in Neuro-Information Processing, Mario Köppen Nikola Kasabov George Coghill (Eds.), 15th International Conference, ICONIP 2008 Auckland, New Zealand, November 25-28, Revised Selected Papers, Part II 13, Springer, pp. 736-744 (2008) |
| [64] | Gao, T.; Liu, Z-G; Yue, S-H; Mei, J-Q; Zhang, J., Traffic video-based moving vehicle detection and tracking in the complex environment, Cybern. Syst. Int. J., 40, 7, 569-588 (2009) |
| [65] | Gu, Y. H.: Adaptive multiscaled polynomial transform for image compression based on local image property. (IPO rapport; Vol. 931). Eindhoven: Technische Universiteit Eindhoven, Institute for Perception Research, 56 pages (1993) |
| [66] | Hitzer, E.: New developments in Clifford Fourier transforms. In: Mastorakis, N.E., Pardalos, P.M., Agarwal, R.P., Kocinac, L. (eds.) Advances in Applied and Pure Mathematics, Proceedings of the 2014 International Conference on Pure Mathematics, Applied Mathematics, Computational Methods (PMAMCM 2014), Santorini Island, Greece, July 17-21, 2014, Mathematics and Computers in Science and Engineering Series, Vol. 29, pp. 19-25 (2014) |
| [67] | Hitzer, E.: Clifford (Geometric) Algebra Wavelet Transform. In: Skala V., Hildenbrand, D. (eds.) Proc. of GraVisMa 2009, 02-04 Sep. 2009, Plzen, Czech Republic, pp. 94-101 (2009) |
| [68] | Heydari, M.H., Hooshmandasl, M.R., Cattani, C., Li, M.: Legendre Wavelets Method for Solving Fractional Population Growth Model in a Closed System, Hindawi Publishing Corporation Mathematical Problems in Engineering, Volume 2013, Article ID 161030, 8 pages · Zbl 1296.65108 |
| [69] | Heydari, Mh; Hooshmandasl, Mr; Shakiba, A.; Cattani, C., Legendre wavelets Galerkin method for solving nonlinear stochastic integral equations, Nonlinear Dyn., 85, 1185-1202 (2016) · Zbl 1355.65011 |
| [70] | Huang, Zj; Huang, Gh; Cheng, L. L., Medical image segmentation of blood vessels based on Clifford algebra and Voronoi diagram, J. Softw., 13, 6, 361-373 (2018) |
| [71] | Mahmoud, Mm; Ibrahim, Mabrouk; Ben, A.; Hashim, Mha, Wavelet multifractal models for transmembrane proteins’ series, Int. J. Wavelets Multires Inf. Process., 14, 6, 36 (2016) · Zbl 1353.42034 |
| [72] | Ikeda, M.; Arai, M.; Okuno, T.; Shimizu, T., TMPDB: a database of experimentally-characterized transmembrane topologies, Nucleic Acids Res., 31, 406-409 (2003) |
| [73] | Janssen, Mhj; Janssen, Ajem; Bekkers, Ej; Bescos, Jo; Duits, R., Design and processing of invertible orientation scores of 3D images, J. Math. Imaging Vis., 60, 1427-1458 (2018) · Zbl 1433.94012 |
| [74] | Kindratenko, Vv, On using functions to describe the shape, J. Math. Imaging Vis., 18, 225-245 (2003) · Zbl 1020.68081 |
| [75] | Khobragade, Ka; Kale, Kv, Multi-wavelet based feature extraction Algorithm for Iris recognition, Int. J. Eng. Innov. Technol. (IJEIT), 3, 12, 368-373 (2014) |
| [76] | Kyte, J.; Doolittle, Rf, A simple method for displaying the hydrophathic character of a protein, J. Mol. Biol., 157, 105-132 (1982) |
| [77] | Lahar, S.: Clifford Algebra: a visual introduction. A topnotch WordPress.com site, March 18 (2014) |
| [78] | Leibon, G.; Rockmore, Dn; Park, W.; Taintor, R.; Chirikjian, Gs, A fast Hermite transform, Theor. Comput. Sci., 409, 2, 211-228 (2008) · Zbl 1156.65104 |
| [79] | Levin, E.; Lubinsky, D., Orthogonal polynomials for exponential weights \(x^{2\rho }e^{-2{\cal{Q}}(x)}\) on \([0, d)]\), J. Approx. Theory, 134, 199-256 (2005) · Zbl 1079.42017 |
| [80] | Li, M., He, Y., Long, Y.: Analogue implementation of wavelet transform using discrete time switched-current filters. In: M. Zhu (Ed.): Electrical Engineering and Control, LNEE 98, pp. 677-682 |
| [81] | Li, L-W; Kang, X-K; Leong, M-S, Spheroidal Wave Functions in Electromagnetic Theory (2002), Hoboken: Wiley-Interscience Publication, Hoboken |
| [82] | Li, L.; Tang, Yy, Wavelet-Hough transform with applications in edge and target detections, Int. J. Wavelets Multiresolut Inf. Process., 4, 567-587 (2006) · Zbl 1141.68592 |
| [83] | Lina, J-M, Image processing with complex Daubechies wavelets, J. Math. Imaging Vis., 7, 211-223 (1997) · Zbl 1490.94021 |
| [84] | Rahman, S. M Mahbubur; Ahmad, Mo; Swamy, Mns, A new statistical detector for DWT-based additive image watermarking using the Gauss-Hermite expansion, IEEE Trans. Image Process., 18, 8, 1782-1796 (2009) · Zbl 1371.94669 |
| [85] | Malonek, H.R., Falcão, M.I.: 3D-Mappings Using Monogenic Functions. ICNAAM 3-7 (2006) |
| [86] | Martens, J.-B.: The Hermite transform: a survey. EURASIP J. Appl. Signal Process., Article ID 26145, pp. 1-20 (2006) · Zbl 1122.94016 |
| [87] | Mawardi, B., Construction of quaternion-valued wavelets, Matematika, 26, 1, 107-114 (2010) |
| [88] | Michel, V., Lectures on constructive approximation (2013), Basel: Birkhäuser, Basel · Zbl 1295.65003 |
| [89] | Morais, J.; Kou, Ki; Sprössig, W., Generalized holomorphic Szegö kernel in 3D spheroids, Comput. Math. Appl., 65, 4, 576-588 (2013) · Zbl 1319.33001 |
| [90] | Moussa, M.-M.: Calcul efficace et direct des représentations de maillages 3D utilisant les harmoniques sphériques, Thèse de Doctorat de l’université Claude Bernard, Lyon 1, France (2007) |
| [91] | Osipov, A.; Rokhlin, V.; Xiao, H., Prolate Spheroidal Wave Functions of Order Zero. Mathematical Tools for Bandlimited Approximation, Applied Mathematical Sciences (2013), Berlin: Springer, Berlin · Zbl 1287.65015 |
| [92] | Paula, Ic; Medeiros, Fns; Bezerra, Fn; Ushizima, Dm, Multiscale corner detection in planar shapes, J. Math. Imaging Vis., 45, 251-263 (2013) |
| [93] | Park, W.; Leibon, G.; Rockmore, Dn; Chirikjian, Gs, Accurate image rotation using Hermite expansions, IEEE Trans. Image Process., 18, 9, 1988-2003 (2009) · Zbl 1371.94290 |
| [94] | Pena, D. P.: Cauchy-Kowalevski extensions, Fueter’s theorems and boundary values of special systems in Clifford analysis. A Ph.D. thesis in Mathematics, Ghent University (2008) |
| [95] | Rizo-Rodríguez, D.; Mendez-Vazquez, H.; Garcia-Reyes, E., Illumination invariant face recognition using quaternion-based correlation filters, J. Math. Imaging Vis., 45, 164-175 (2013) · Zbl 1276.68138 |
| [96] | Saillard, J., Bunel, G.: Apport des fonctions sphéroidales pour l’estimation des paramètres d’une cible radar. 12ème Colloque Gretsi-Juan-Les-Pins, 12-19 Juin 4 pages (1989) |
| [97] | Sau, Kartik; Basak, Ratan Kumar; Chanda, Amitabha, Image Compression based on Block Truncation Coding Using Clifford Algebra, Procedia Technology, 10, 699-706 (2013) |
| [98] | Shi, Yg, Orthogonal polynomials for Jacobi-Exponential weights, Acta Math. Hungar., 140, 4, 363-376 (2013) · Zbl 1299.42089 |
| [99] | Skibbe, H.; Reisert, M., Spherical tensor algebra: a toolkit for 3D image processing, J. Math. Imaging Vis., 58, 3, 349-381 (2017) · Zbl 1387.68289 |
| [100] | Soulard, R.; Carré, P., Characterization of color images with multiscale monogenic maxima, IEEE Trans. Pattern Anal. Mach. Intell., 40, 10, 2289-2302 (2018) |
| [101] | Stanković, Srdjan; Stanković, Ljubiša; Orović, Irena, Compressive Sensing Approach in the Hermite Transform Domain, Mathematical Problems in Engineering, 2015, 1-9 (2015) · Zbl 1395.94163 |
| [102] | Stratton, Ja, Spheroidal functions, Physics, 21, 51-56 (1935) · Zbl 0010.35801 |
| [103] | Stratton, Ja; Morse, Pm; Chu, Lj; Little, Jdc; Corbato, Fj, Spheroidal Wave Functions (1956), New York: Wiley, New York |
| [104] | Stein, Em; Weiss, G., Introduction to Fourier Analysis On Euclidien Spaces (1971), Princeton, New Jersey: Princeton University Press, Princeton, New Jersey · Zbl 0232.42007 |
| [105] | Suk, T.: 2D and 3D Image Analysis by Moments. Ph.D. Thesis in Informatics and Cybernetics. Institute of Information Theory and Automation, Czech Republic Academy of Sciences (2016) |
| [106] | Unser, M.; Sage, D.; Van De Ville, D., Multiresolution monogenic signal analysis using the Riesz-Laplace wavelet transform, IEEE Trans. Image Process., 18, 11, 2402-2418 (2009) · Zbl 1371.94541 |
| [107] | Van De Ville, D., Sage, D., Balac, K., Unser, M.: The Marr wavelet pyramid and multiscale directional image analysis. 16th European Signal Processing Conference (EUSIPCO 2008), Lausanne, Switzerland, August 25-29, 5 pages (2008) |
| [108] | Van De Ville, D.; Unser, M., Complex wavelet bases, steerability, and the Marr-Like pyramid, IEEE Trans. Image Process., 17, 11, 2063-2080 (2008) · Zbl 1371.42046 |
| [109] | Venkatesh, Yv, Hermite polynomials for signal reconstruction from zero-crossings, 1. One-dimensional signals, IEEE Proceed. I Commun. Speech Vis., 139, 6, 587-596 (1992) |
| [110] | Venkatesh, Yv; Ramani, K.; Nandini, R., Wavelet array decomposition of images using a Hermite sieve, Sadhana, 18, 2, 301-324 (1993) |
| [111] | Ville, J., Theorie et applications de la notion de signal analytique (theory and applications of the notion of analytic signal), Cables et transmission, 2, 1, 61-74 (1948) |
| [112] | Wang, R., Introduction to Orthogonal Transforms With Applications in Data Processing and Analysis (2012), Cambridge: Cambridge University Press, Cambridge · Zbl 1272.94001 |
| [113] | Wang, Lifeng; Hu, Ruifeng; Zhang, Jun; Ma, Yunpeng, On the Vortex Detection Method Using Continuous Wavelet Transform with Application to Propeller Wake Analysis, Mathematical Problems in Engineering, 2015, 1-9 (2015) · Zbl 1394.76031 |
| [114] | Wang, X-Y; Zhao, L.; Niu, P-P; Fu, Z-K, Image denoising using Gaussian scale mixtures with Gaussian-Hermite PDF in steerable pyramid domain, J. Math. Imaging Vis., 39, 245-258 (2011) · Zbl 1255.68271 |
| [115] | Wietzke, L.; Sommer, G., The signal multi-vector, J. Math. Imaging Vis., 37, 132-150 (2010) · Zbl 1490.94031 |
| [116] | Yang, B.; Flusser, J.; Kautsky, J., Rotation of 2D orthogonal polynomials, Pattern Recognit. Lett., 102, 44-49 (2018) |
| [117] | Yang, B.; Flusser, J.; Suk, Jt, Steerability of Hermite kernel, Int. J. Pattern Recognit. Artif. Intell., 27, 4, 25 (2013) |
| [118] | Yang, B., Suk, T., Dai, M., Flusser, J.: Chapter \(7, 2D\) and \(3D\) Image Analysis by Gaussian-Hermite Moments. In Moments and Moment Invariants—Theory and Applications. G.A. Papakostas (ed.), GCSR Vol. 1, Science Gate Publishing, pp. 143-173 (2014) |
| [119] | Yu, M., Zhu, Q.: Tilt correction method of text image based on wavelet pyramid. AIP Conference Proceedings 1834, 040028 7 pages (2017) |
| [120] | Zhang, B.; Zhang, L.; Zhang, L.; Karray, F., Retinal vessel extraction by matched filter with first-order derivative of Gaussian, Comput. Biol. Med., 40, 438-445 (2010) |
| [121] | Zou, C., Kou, K. I.: Hypercomplex signal energy concentration in the spatial and quaternionic linear canonical frequency domains. arXiv:1609.00890 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
