How Random is Your Tomographic Noise? A Number Theoretic Transform (NTT) Approach

• Autorzy: Fiorini R. A.

Identyfikatory

DOI

10.3233/FI-2014-1116

• Języki publikacji: EN

Abstrakty

EN

Discrete Tomography (DT), differently from GT and CT, focuses on the case where only few specimen projections are known and the images contain a small number of different colours (e.g. black-and-white). A concise review on main contemporary physical and mathematical CT system problems is offered. Stochastic vs. Combinatorially Optimized Noise generation is compared and presented by two visual examples to emphasise a major double-bind problem at the core of contemporary most advanced instrumentation systems. Automatic tailoring denoising procedures to real dynamic system characteristics and performance can get closer to ideal self-registering and selflinearizing system to generate virtual uniform and robust probing field during its whole designed service life-cycle. The first attempt to develop basic principles for system background low-level noise source automatic characterization, profiling and identification by CICT, from discrete system parameter, is presented. As a matter of fact, CICT can supply us with cyclic numeric sequences perfectly tuned to their low-level multiplicative source generators, related to experimental high-level overall perturbation (according to high-level classic perturbation computational model under either additive or multiplicative perturbation hypothesis). Numeric examples are presented. Furthermore, a practical NTT example is given. Specifically, advanced CT system, HRO and Mission Critical Project (MCP) for very low Technological Risk (TR) and Crisis Management (CM) system will be highly benefitted mostly by CICT infocentric worldview. The presented framework, concepts and techniques can be used to boost the development of next generation algorithms and advanced applications quite conveniently.

Słowa kluczowe

EN

discrete tomography; computerized tomography; computational information conservation theory; denoising; modular arithmetic; number theoretic transform; biomedical cybernetics; biomedical engineering; HRO; public health; digital healthcare

• Wydawca: IOS Press
• Czasopismo: Fundamenta Informaticae
• Rocznik: 2014
• Tom: Vol. 135, nr 1/2
• Strony: 135--170
• Opis fizyczny: Bibliogr. 137 poz., rys.

Twórcy

autor

Fiorini R. A.

• Department of Electronics, Information and Bioengineering (DEIB), Politecnico di Milano, P.za Leonardo da Vinci 32, 20133 Milano, Italy

Bibliografia

• [1] Aert, S. V., Batenburg, K. J., Rossell, M. D., Erni, R., Tendeloo, G. V.: Three-dimensional atomic imaging of crystalline nanoparticles, Nature, (470), 2011, 374–377.
• [2] Agnew, G.: Random sources for cryptographic systems, in Advances in Cryptology - EUROCRYPT 87, A. Adamatzky, ed., 1987, 77–82, Springer, Berlin.
• [3] Arbuzov, E., Bukhgeim, A., Kazantsev, S.: Two-dimensional tomography problems and the theory of Aanalytic functions, Siberian Adv. Math., (8), 1998, 1–20.
• [4] Ashby, W. R.: An introduction to cybernetics, Chapman & Hall, 1956.
• [5] Bae, E.: Efficient Global Minimization Methods for Variational Problems in Imaging and Vision, Ph.D. Thesis, Department of Mathematics, University of Bergen, April 2011.
• [6] Batenburg, K., Sijbers, J.: Dart: A Fast Heuristic Algebraic Reconstruction Algorithm for Discrete Tomography, IEEE, 2007, IV–133–IV–136.
• [7] Batenburg, K. J., Bals, S., Sijbers, J., Kubel, C., Midgley, P. A., Hernandez, J. C., Kaiser, U., Encina, E. R., Coronado, E. A., Tendeloo, G. V.: 3D imaging of nanomaterials by discrete tomography, Ultramicroscopy, 109, 2009, 730–740.
• [8] Bergstrom, C. T., Lachmann, M.: Shannon information and biological fitness, In: IEEE information theory workshop 2004, 2004, 50–54.
• [9] Billingsley, P.: Probability and Measure, Wiley, New York, 1995.
• [10] Blank, A. A.: Axiomatics of Binocular Vision. The Foundations of Metric Geometry in Relation to Space Perception, Journal of the Optical Society of America, 48, 1958, 328–334.
• [11] Borel, E.: Les probabilités denombrables et leurs applications arithmétiques, Rendiconti del Circolo Matematico di Palermo (1884 - 1940), 27, 1909, 247–271.
• [12] Born, M.: Natural Philosophy of Cause and Chance, Clarendon Press, Oxford, 1949.
• [13] Bracewell, R., Riddle, A.: Inversion of fan-beam scans in radio astronomy, The Astrophysical Journal, 150, (1976), 427–434.
• [14] Bremermann, H.: Quantum noise and information, 5th Berkeley Symposium on Mathematical Statistics and Probability, Univ. of California Press, Berkeley, California, 1965.
• [15] Brunetti, S., Dulio, P., Peri, C.: Characterization of -1; 0; 1 valued functions in discrete tomography under sets of four directions, Discrete Geometry for Computer Imagery (DGCI) LCNS 6697, 2011, 394–405.
• [16] Brunetti, S., Dulio, P., Peri, C.: Discete Tomography determination of bounded lattice sets from four X-rays, Discrete Applied Mathematics, 2013, 2281–2293.
• [17] Brunetti, S., Dulio, P., Peri, C.: On the Non-Additive Sets of Uniqueness in a Finite Grid, Discrete Geometry for Computer Imagery (DGCI) LCNS 7749, 2013, 288–299, Sevilla 2013.
• [18] Calude, C.: Information and Randomness-An Algorithmic Perspective, Springer, 2nd edn., Springer, Berlin, 2002.
• [19] Calude, C., Svozil, K.: Quantum Randomness and Value Indefiniteness, Advanced Science Letters, (1), 2008, 165–168.
• [20] Cassirer, E.: An Essay on Man: Introduction to a Philosophy of Human Culture, Yale University Press, New Haven, 1944.
• [21] Chandler, B. M.: The history of combinatorial group theory: A case study in the history of ideas, studies in the history of mathematics and physical sciences (1st ed.), Springer, 1982, 387–407.
• [22] Chen, K.: Introduction to variational image-processing models and applications, International Journal of Computer Mathematics, 90(1), 2013, 1–8.
• [23] Chesler, D., Riederer, S.: Ripple suppression during reconstruction in transverse tomography, Phys Med Biol, 20(4), 1975, 632–636.
• [24] Chopra, D., Tanzi, R.: Super Brain, Three River Press, New York, 2001.
• [25] Courant, R., Hilbert, D.: Methods of Mathematical Physics, Vol. I, Interscience, New York, 1953.
• [26] Cruz, C. H. B., Fork, R. L., Knox, W. H., Shank, C. V.: Spectral Hole Burning in Large Molecules Probed with 10 Fs Optical Pulses, Chemical Physics Letters, (132), 1986, 341–344.
• [27] Cunningham, I. A., Judy, P. F.: Computed Tomography, CRC Press, January 2000.
• [28] Curtis, C. W.: Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer, History of Mathematics, American Mathematical Society, Providence, 2003.
• [29] Dacquino, G., Fiorini, R. A.: The morphometric approach for effective automatic data classification, Editrice Esculapio, July 1994, 99–124.
• [30] Dacquino, G., Fiorini, R. A., Cattaneo, B., Fabiani, A.: Numerical fovea: an efficient solution to discrete inverse problems, Proc. 40th SPIE Annual Symposium, July 10-11, San Diego, California, USA, USA, (1995), 319–331.
• [31] DeGiacomo, P.: Mente e Creativita’, Franco Angeli, Milano, Italy, 1999.
• [32] Donaldson-Matasci, M. C., Bergstrom, C. T., Lachmann, M.: Shannon information and biological fitness, Oikos, (119), 2010, 219–230.
• [33] Dulio, P., Gardner, R., Peri, C.: Discrete point X-rays, SIAM J. Discrete Math, 20(1), 2006, 171–188.
• [34] Eco, U.: A Theory of Semiotics, Macmillan: London, 1976.
• [35] Eskin, G.: On non-abelian Radon transform, Russ. J. Math. Phys, (11), 2004, 391–408.
• [36] Filloux, J.-C.: The Problem of Space with Particular Emphasis on Specific Sensory Spaces, La Nature, 85, 1957, 403–407,438–443,490–493.
• [37] Finch, D., Uhlmann, G.: The X-ray transform for a non-abelian connection in two dimensions, Inverse Problems, (17), 2001, 695–701.
• [38] Fiorini, R. A.: Strumentazione Biomedica: Sistemi di Supporto Attivo, CUSL, Collana Scientifica, Milano. Italy, (pp.180), 1994.
• [39] Fiorini, R. A.: Sanitá 5.0, La visione evolutiva, Parte A: visione multiscala, CUSL, Collana Scientifica, Milano. Italy, (pp.438), 2010.
• [40] Fiorini, R. A., Condorelli, A., Laguteta, G.: Discrete Tomography Data Footprint Reduction via Natural Compression, Fundamenta Informaticae, (125), 2013, 273–284.
• [41] Fiorini, R. A., Laguteta, G.: Discrete Tomography Data Footprint Reduction by Information Conservation, Fundamenta Informaticae, (125), 2013, 261–272.
• [42] Fiorini, R. A., Santacroce, G.: Economic Competitivity in Healthcare Safety Management by Biomedical Cybernetics ALS, Proc. International Symposium, The Economic Crisis: Time For A Paradigm Shift - Towards a Systems Approach, Universitat de València, January 24-25, 2013.
• [43] Fiorini, R. A., Santacroce, G.: Systems Science and Biomedical Cybernetics for Healthcare Safety Management, Proc. International Symposium, The Economic Crisis: Time For A Paradigm Shift - Towards a Systems Approach, Universitat de València, January 24-25, 2013.
• [44] Fletcher, J.: Integrating Noise Reduction Technology into your Practice to Reduce Patient Dose without Sacrificing Image Quality, CT Dose Summit 2011, CT Clinical Innovation Center, Dept. of Radiology, Mayo Clinic, Rochester, MN.).
• [45] Ford, L., Fulkerson, D.: Maximal flow through a network, Canadian Journal of Mathematics, (8), 1956, 399–404.
• [46] Furstenberg, H.: Ergodic behavior of diagonal measures and a theorem of Szemeredi on arithmetic progressions, J. Analyse Math., (31), 1977, 204–256.
• [47] Furstenberg, H.: Recurrence in Ergodic theory and Combinatorial Number Theory, Princeton Univ. Press, Princeton NJ, 1981.
• [48] Gale, D.: A theorem on flows in networks, Pacific Journal of Mathematics, 7(2), 1957, 1073–1082.
• [49] Gardner, R.: Geometric Tomography, Cambridge University Press, Cambridge, UK, 2nd ed., 2006.
• [50] Gardner, R., Gritzmann, P., Prangenberg, D.: On the computational complexity of reconstructing lattice sets from their x-rays, Discrete Mathematics, 202(13), 1999, 45–71.
• [51] Gardner., R., Kiderlen, M.: A solution to Hammers X-ray reconstruction problem, Adv. Math., (214), 2007, 323–343.
• [52] Gardner, R., McMullen, P.: On Hammer’s X-ray Problem, J. London Math. Soc., 2(21), 1980, 171–175.
• [53] Gibson, J. J.: Perception as a Function of Stimulation, McGraw-Hill, New York, 1959, 456–501.
• [54] Good, I. J.: Why Physical Space Has Three Dimensions, British Journal for the Philosophy of Science, 9, 1959, 317–319.
• [55] Green, B., Tao, T.: The primes contain arbitrarily long arithmetic progressions, Annals of Math., (167), 2008, 481–547.
• [56] Gregor, J., Benson, T.: Computational analysis and improvement of SIRT, IEEE Transactions on Medical Imaging, 27(7), 2008, 918–924.
• [57] Guédon, J.-P., Bizais, Y.: Bandlimited and Haar filtered back-projection reconstruction, IEEE Trans Medical Imaging, 13(3), 1994, 430–440.
• [58] Hajdu, L.: Unique reconstruction of bounded sets in discrete tomography, Electron. Notes Discrete Math., (20), 2005, 15–25.
• [59] Hajdu, L., Tijdeman, R.: Algebraic aspects of emission tomography with absorption, Theoretical Computer Science, (290), 2003, 2169–2181.
• [60] Hajdu, L., Tijdeman, R.: Algebraic discrete tomography, Ch.4 of Advances in Discrete Tomography and its Applications, ed. By G.T. Herman and A. Kuba, Birkhauser, Boston, 2007, 55–81.
• [61] Hamilton, W. R.: Memorandum respecting a new System of Roots of Unity, Philosophical Magazine, 12, 1856, 446.
• [62] Hammer, P. C.: Problem 2, American Mathematical Society in: V.L. Klee (Ed.), Proc. Symp. in Pure Mathematics, vol. VII: Convexity, 1963, 498–499.
• [63] Hansen, P., Saxild-Hansen, M.: AIR tools, a MATLAB package of algebraic iterative construction methods, Journal of Computational and Applied Mathematics, 236(8), 2012, 2167–2178.
• [64] Hanson, K. M., Mosby, C. V.: Noise and computed contrast discrimination in tomography, chap.113 in Radiology of the Skull and Brain, Vol. 5: Technical Aspects of Computed Tomography, T. H. Newton and D. G. Potts, (eds.), 1981, St. Louis.
• [65] Herman, G., Rowland, S.: Three methods for reconstruction objects for x-rays, a comparative study, Comp. Graph. Imag. Process, (2), 1973, 151–178.
• [66] Hertz, R.: Death and the Right Hand, Free Press, Glencoe (IL), 1960.
• [67] Hjelmslev, L.: Prolegomena to a Theory of Language, (trans. Francis J. Whitfield), Univ. of Wisconsin: Madison, 1961.
• [68] Hockett, C.: A Course in Modern Linguistics, Macmillan: New York, 1958.
• [69] Hockett, C.: The Origin of Speech, Scientific American, (203), 1960, 89–96.
• [70] Hsieh, J., Nett, B., Yu, Z., Sauer, K., Thibault, J.-B., Bouman, C. A.: Recent Advances in CT Image Reconstruction, Curr. Radiol. Rep., Advances in CT Imaging, Springer Science + Business Media, New York, online, 2013,1-13).
• [71] Jiang, M., Wang, G.: Convergence of the simultaneous algebraic reconstruction technique (SART), IEEE Transactions on Image Processing, 12(8), August 2003, 957–961.
• [72] Jinschek, J. R., Batenburg, K. J., Calderon, H. A., Kilaas, R., Radmilovic, V., Kisielowski, C.: 3-D reconstruction of the atomic positions in a simulated gold nanocrystal based on discrete tomography, Ultramicroscopy, 108, 2008, 589–604.
• [73] Judd, D., Wyszecki, G.: Color in Business, Science and Industry, Wiley Series in Pure and Applied Optics (third ed.), Berkeley, California, 1975.
• [74] Kac, M.: Can one hear the shape of a drum?, American Mathematical Monthly, 73(4, part 2), 1966, 1–23.
• [75] Kak, A., M.Slaney: Principles of Computerized Tomographic Imaging, IEEE Press, New York, 1987.
• [76] Knox, W. H., Fork, R. L., Downer, M. C., Miller, D. A. B., Chemla, D. S., Shank, C. V., Gossard, A. C., Wiegmann,W.: Femtosecond Dynamics of Resonantly Excited Excitons in Room-Temperature Gaas Quantum Wells, Physical Review Letters, (54), 1985, 1306–1309.
• [77] Knox,W. H., Hirlimann, C., Miller, D. A. B., Shah, J., Chemla, D. S., Shank, C. V.: Femtosecond Excitation of Nonthermal Carrier Populations in GaAs Quantum-Wells, Physical Review Letters, 56, 1986, 1191–1193.
• [78] Kolmogorov, A.: On Tables of Random Numbers, Sankhya Ser. A, (25), 1963, 369–375.
• [79] Kolmogorov, A.: Logical basis for information theory and probability theory, IEEE Transactions on Information Theory, 14(5), 1968, 662–664.
• [80] Kolmogorov, A.: Logical basis for information theory and probability theory, Theoretical Computer Science, 207(2), 1998, 387–395.
• [81] Komlós, J., Sárközy, G., Szemerédi, E.: Blow-up Lemma, Combinatorica, 17(1), 1997, 109–123.
• [82] Komlos, J., Sárközy, G., Szemerédi, E.: An algorithmic version of the Blow-up Lemma, Random Structures and Algorithms, (12), 1998, 297–312.
• [83] Kroupa, P., Famaey, B., de Boer, K., Dabringhausen, J., Pawlowski, M., Boily, C., Jerjen, H., Forbes, D., Hensler, G., Metz, M.: Local-Group tests of dark-matter concordance cosmology. Towards a new paradigm for structure formation, Astronomy and Astrophysics, 523(id.A32), 2010, 22pp.
• [84] Lachmann, M., Newman, M., Moore, C.: The physical limits of communication or Why any sufficiently advanced technology is indistinguishable from noise?, American Journal of Physics, 72(10), 2004, 1290–1293.
• [85] Lam, T. Y., Leung, K.: On vanishing sums of roots of unity, J. Algebra, (224), 2000, 91–109.
• [86] Lehmer, D. H.: Review of Walther von Dyck (1856-1934). Mathematik, Technik und Wissenschaftsorganisation an der TH Mnchen, Annals of Mathematics, 31(3), 1930, 419–448.
• [87] Liao, H., Herman, G.: A coordinate ascent approach to tomographic reconstruction of label images from a few projection, Discrete Applied Mathematics, 151(1-3), 2005, 184–197.
• [88] Lowenthal, D.: Geography, Experience, and Imagination: Towards a Geographical Epistemology, Annals Association of American Geographers, 51(3), 1961, 241–260.
• [89] Ltzen, J.: Joseph Liouville 1809-1882, Master of Pure and Applied Mathematics, Springer-Verlag New-York, Inc., New York, 1990.
• [90] Lukic, T.: Regularized Problems in Image Processing, Ph.D. Thesis, 2011 Doctoral Thesis, Faculty of Technical Sciences, University of Novi Sad, Serbia.
• [91] Luneburg, R. K.: Mathematical Analysis of Binocular Vision, Princeton Univ. Press, Princeton NJ, 1947.
• [92] MacKay, D.: Information Theory, Inference, and Learning Algorithms, Cambridge University Press, Cambridge, 2003.
• [93] Manovich, L.: The Langage of New Media, MIT Press, Cambridge, 2001.
• [94] Manovich, L.: Software Takes Command, Bloomsbury Academic, New York, 2013.
• [95] Martin, N., England, J.: Mathematical Theory of Entropy, Cambridge University Press, Cambridge, 2011.
• [96] McCullough, E.: Specifying and evaluating the performance of computed tomographic (CT) scanners, Med Phys, (7), 1979, 291–296.
• [97] Mitchell, S., Michels, N.-L., Kunze, K., Pérez-Ramrez, J.: Visualization of hierarchically structured zeolite bodies from macro to nano length scales, Nature Chemistry, 4, (2012), 825–831.
• [98] Moore, S., Judy, P.: The effect of misregistration of the projections on spatial resolution of CT scanners, Med Phys, (10), 1983, 169–175.
• [99] Mredhula, L., Dorairangasamy, M.: An Extensive Review of Significant Researches on Medical Image Denoising Techniques, International Journal of Computer Applications, 64(14), February 2013, 1–12.
• [100] Natterer, F.: The Mathematics of Computerized Tomography, SIAM, Philadelphia, 2001.
• [101] Novikov, R. G.: Une formule dinversion pour la transformation dun rayonnement X atténué, C. R. Acad. Sci. Paris Ser. I Math., 332(12), 2001, 1059–1063.
• [102] Novikov, R. G.: An inversion formula for the attenuated X-ray transformation, Ark. Mat., (40), 2002, 145–167.
• [103] Park, J., Johnson, J. T., Majurec, N., Niamsuwan, N., Piepmeier, J. R., Mohammed, P. N., Ruf, C. S., Misra, S., Yueh, S. H., Dinardo, S. J.: Airborne L-Band Radio Frequency Interference Observations From the SMAPVEX08 Campaign and Associated Flights, IEEE Trans. on Geoscience and Remote Sensing, 49(12), 2011, 3359–3370.
• [104] Pineda, A. R., Tward, D. J., Gonzalez, A., Siewerdsen, J. H.: Beyond noise power in 3D computed tomography: The local NPS and off-diagonal elements of the Fourier domain covariance matrix, Med. Phys., 39(6), June 2012, 3240–3252.
• [105] Ramsey, F. P.: On a problem of formal logic, Proc. London Math. Soc., s2-, 30(1), 1930, 264–286.
• [106] Revesz, G.: The Problem of Space with Particular Emphasis on Specific Sensory Spaces, American Journal of Psychology, 50, 1937, 434–436.
• [107] Rowe, D. E.: Review of Walther von Dyck (1856-1934). Mathematik, Technik und Wissenschaftsorganisation an der TH Mnchen, Historia Mathematica, 35(4), 2008, 333–334.
• [108] Russell, B.: Human Knowledge: Its Scope and Limits, Simon and Schuster, New York, 1948.
• [109] Ryser, H.: Combinatorial properties of matrices of zeros and ones, Canadian Journal of Mathematics, (9), 1957, 371–377.
• [110] Salo, M., Paternain, G., Uhlmann, G.: The attenuated ray transform for connections and Higgs fields, Geom. Funct. Anal., 22(5), 2012, 1460–1489.
• [111] Salo, M., Uhlmann, G.: The attenuated ray transform on simple surfaces, J. Diff. Geom., 88(1), 2011, 161–187.
• [112] Schle, T., Schnörr, C.,Weber, S., Hornegger, J.: Discrete tomography by convex-concave regularization and d.c. programming, Discrete Applied Mathematics, 151(1-3), 2005, 229–243.
• [113] Secchi, P., Gupta, A.: Toward A Reformed Empiricism, Ph.D. Thesis, Guido Nardi Honor Ph.D. in ”Scientific and Technological Research Epistemology”, June 12, 2013, Politecnico di Milano, Milano, Italy.
• [114] Segal, D.: Polycyclic groups, Cambridge University Press, Cambridge, 1983.
• [115] Shannon, C. E.: A Mathematical Theory of Communication, Bell System Technical Journal, 27(3), July/October 1948, 379–423.
• [116] Sharafutdinov, V.: On an inverse problem of determining a connection on a vector bundle, J. Inverse and Ill-Posed Problems, (8), 2000, 51–88.
• [117] Shepp, L., Logan, B.: Optimal Reconstruction of a Function from its Projections, Duke Math. J., (42), 1975, 645–659.
• [118] Sivek, G.: On Vanishing Sums of Distinct Roots of Unity, Integers, (10), 2010, 365–368.
• [119] van der Sluis, A., van der Vorst, H. A.: Numerical solution of large, sparse linear algebraic systems arising from tomographic problems, In G. Nolet, editor, Seismic Tomography With Applications in Global Seismology and Exploration Geophysics, 1987, 49–83, D. Reidel Publishing Company, Dordrecht.
• [120] Stam, R.: Film Theory, Blackwell: Oxford, 2000.
• [121] Stillwell, J.: Mathematics and its history, Springer, New York, 2002.
• [122] Svozil, K.: The quantum coin toss-Testing microphysical undecidability, Physics Letters A, (143), 1990, 433–437.
• [123] Symul, T., Assad, S., Lam, P.: Real time demonstration of high bitrate quantum random number generation with coherent laser light, Appl. Phys. Lett. 98, 231103, 2011, 1–3.
• [124] Szemerédi, E.: On sets of integers containing no k elements in arithmetic progression, Acta Arith., (27), 1975, 299–345.
• [125] Szemerédi, E.: Regular partitions of graphs, Problémes combinatoires et théorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976), Colloq. Internat. CNRS 260, Paris: CNRS, 1978, 399–401.
• [126] Tabak, F.: Robust Algorithms for Discrete Tomography, Master Thesis, Delft University of Technology, June 2012, Delft, the Netherlands.
• [127] Taubes, C. H.: On the Yang-Mills-Higgs equations, Bulletin of the American Mathematical Society, New Series, 10(2), 1984, 295–297.
• [128] Velayudham, A., Kanthavel, R.: A Survey on Medical Image De-noising Techniques, International journal of Advanced research in Electronics and Communication Engineering (IJARECE), 2(3), March 2013.
• [129] Vincent, C.: The generation of truly random binary numbers, Journal of Physics E: Scientific Instruments, (3), 1970, 594–598.
• [130] Viot, P.: Physics of complex systems and Modeling, Statistics and Algorithms for out-of-equilibrium systems, Numerical Simulation in Statistical Physics, Lecture in Master 2, 2012, Laboratoire de Physique Thorique de la Matire Condense, Paris.
• [131] Volčič, A.: A three-point solution to Hammer’s X-ray problem, J.London Math.Soc, (34), 1986, 349–359.
• [132] Wallace, C.: Physically random generator, Computer Systems Science & Engineering, (5), 1990, 82–88.
• [133] Website:, C. E. U.: European Laboratory for Particle Physics, Conseil Européen pour la Recherche Nuclaire, 2013, http://home.web.cern.ch/about/updates/2013/11/atlas-sees-higgs-boson-decay-fermions.
• [134] Whitrow, G. J.: Why Physical Space Has Three Dimensions, British Journal for the Philosophy of Science, 6, 1955, 13–31.
• [135] Wildberger, N. J.: Universal Hyperbolic Geometry I: trigonometry, Geom. Dedicata, 163, 2013, 215–274.
• [136] Yu, D., Fessler, J.: Introduction to variational image-processing models and applications, IEEE Trans. Med. Imag., 21(2), February 2002, 159–173.
• [137] Zhou, Z.: Relation between the solitons of Yang-Mills-Higgs fields in 2+1 dimensional Minkowski spacetime and anti-de Sitter space-time, J. Math. Phys, (42), 2001, 4938.