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Wavelet Modelling of Clinical Magnetic Resonance Tomography: An Ensemble Quantum Computing Approach

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Inverse Problems, Tomography, and Image Processing

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Abstract

Phase coherent wideband signal wavelets form a unified basis of the multichannel reconstructive analysis-synthesis filter bank of high resolution synthetic aperture radar (SAR) imaging and clinical MRI. The construction of unitary bank filters is performed by the Kepplerian quadrature detection strategy of physical astronomy which allows for the stroboscopic and synchronous cross-sectional filtering of phase histories in contiguous local frequency encoding subband multichannels relative to the rotating coordinate frame of quadrature reference. The Kepplerian quadrature detection strategy and the associated unitary filter bank construction take place in symplectic affine planes which are immersed into the three-dimensional compact super-encoding projective space. They are implemented in the quadrature format of a phase-splitting network by Fourier analysis of the Heisenberg nilpotent Lie group G. The action of G admits a matrix coordinatization by transvections. In terms of projective geometry, the longitudinal dilations jointly with the transvections of the G-action generate the group of homologies. The tomographic slices, frequency selected by the MRI scanner system, are identified with the projectively immersed, symplectic affine leaves O v (v ≠ 0) of the canonical coadjoint orbit foliation of G, on which the projective cohomologies linearly act.

Mensch, streck deine Vernunft hierher, diese Dinge zu begreiffen! — Johann Keppler (1571–1630)

There is nothing that nuclear spins will not do for you, as long as you treat them as human beings. — Erwin Louis Hahn (1949)

Die Philosophie ist Konstruktivismus, und der Konstruktivismus besitzt zwei komplementäre Aspekte, die sich wesensmäßig voneinander unterscheiden: Begriffe erschaffen und eine Ebene entwerfen. Die Begriffe sind gleichsam die mannigfachen Wellen, die sich heben und senken, die Immanenzebene aber ist die eine Welle, von der sie auf — und abgewickelt werden. Die Ebene ist es, die den Zusammenschluß der Begriffe mit stets wachsenden Verbindungen garantiert, und es sind die Begriffe, die die Besiedelung der Ebene in einer stets neuen, stets variablen Krümmung gewährleisten. Die Immanenzebene ist eine Blätterung. Sie ist gleichsam ein Schnitt durch das Chaos und wirkt wie ein Sieb. Das Chaos ist durch die Unmöglichkeit eines Bezugs zwischen zwei Bestimmungen gekennzeichnet. Indem sie einen Schnitt durch das Chaos legt, appelliert die Immanenzebene an eine Schöpfung von Begriffen. — Gilles Deleuze und Felix Guattari (1991)

The link between different energetic strata represented by symmetry groups is best sought in some appropriate Lie group transform. And since it is not possible to correlate events directly across energy gaps, the observer always being on one side of the gap, one may try to correlate the geometric patterns they form in each stratum as a result of the interactions they are subjected to and of the laws they obey, rather than in terms of the interactions themselves. —George L. Farre (1996)

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References

  1. C.M. Anderson, R. R. Edelman, and P. A. Turski, Clinical Magnetic Resonance Angiography. Raven Press, New York 1993.

    Google Scholar 

  2. S. W. Atlas, editor, Magnetic Resonance Imaging of the Brain and Spine. Second edition, Lippincott-Raven Publishers, Philadelphia, New York 1996.

    Google Scholar 

  3. A. J. Barkovich, Pediatric Neuroimaging, Second edition, Raven Press, New York 1995.

    Google Scholar 

  4. A. J. Barkovich, C. L. Truwit, Practical MRI Atlas of Neonatal Brain Development. Raven Press, New York 1990.

    Google Scholar 

  5. G. H. Barnett, C. P. Steiner, and J. Weisenberger, Intracranial meningioma resection using frameless stereotaxy. J. Image Guided Surgery 1, 46–52 (1955).

    Article  Google Scholar 

  6. R. Bauer, E. van de Flierdt, K. Mörike, and C. Wagner-Manslau, MR Tomography of the Central Nervous System. Gustav Fischer Verlag, Stuttgart, Jena, New York 1993.

    Google Scholar 

  7. J. Beltran, editor, Current Review of MRI. First edition, Current Medicine, Philadelphia 1995.

    Google Scholar 

  8. J. H. Bisese, A.-M. Wang, Pediatric Cranial MRI: An Atlas of Normal Development. Springer-Verlag, New York, Berlin, Heidelberg 1994.

    Book  Google Scholar 

  9. S. Braitinger, J. Pahnke, Hrsg., editors, MR-Atlas der HNO-Anatomie, MR Atlas of ENT Anatomy. Bilingual Eurobook, F. K. Schattauer, Stuttgart, New York 1995.

    Google Scholar 

  10. J. J. Brown, F. J. Wippold II, Practical MRI: A Teaching File. Lippincott-Raven Publishers, Philadelphia, New York 1996.

    Google Scholar 

  11. D. R. Cahill, M. J. Orland, and G. M. Miller, Atlas of Human Cross-Sectional Anatomy, with CT and MR Images. Third edition, Wiley-Liss, New York, Chichester, Brisbane 1995.

    Google Scholar 

  12. W. G. Carrara, R. S. Goodman, and R. M. Majewski, Spotlight Synthetic Aperture Radar: Signal Processing Algorithms. Artech House, Boston, London 1995.

    MATH  Google Scholar 

  13. M. Castillo, S. K. Mukherji, Imaging of the Pediatric Head, Neck, and Spine. Lippincott-Raven Publishers, Philadelphia, New York 1996.

    Google Scholar 

  14. M. S. Cohen, Rapid MRI and functional applications. In: Brain Mappping — The Methods, A. W. Toga, J. C. Mazziotta, editors, pp. 223–255, Academic Press, San Diego, New York, Boston 1996.

    Google Scholar 

  15. L. J. Cutrona, E. M. Leith, L. J. Porcello, and W. E. Vivian, On the application of coherent optical processing techniques to synthetic-aperture radar. Proc. IEEE 54, 1026-1032 (1966).

    Google Scholar 

  16. H. Damasio, Human Brain Anatomy in Computerized Images. Oxford Univiversity Press, Oxford, New York 1995.

    Google Scholar 

  17. E. R. Davies, Electronics, Noise and Signal Recovery. Academic Press, London, San Diego, New York 1993.

    Google Scholar 

  18. P. L. Davis, editor, Breast Imaging. MRI Clinics of North America, Vol. 2, No. 4, November 1994, pp. 505–740, W. B. Saunders Company, Philadelphia, London, Toronto 1994.

    Google Scholar 

  19. G. Deleuze, F. Guattari, Q’est — ce que la philosophic? Les Editions de Minuit, Paris 1991.

    Google Scholar 

  20. P. De Potter, C. L. Shields, and J. A. Shields, MRI of the Eye and Orbit. Lippincott-Raven Publishers, Philadelphia, New York 1995.

    Google Scholar 

  21. S. J. Doran, P. Jakob, and M. Décorps, Rapid repetition of the “burst” sequence: The role of diffusion and consequences for imaging. Magn. Reson. Med.35, 547–553 (1996).

    Article  Google Scholar 

  22. R. R. Edelman, J. R. Hesselink, and M. B. Zlatkin, Clinical Magnetic Resonance Imaging. Two volumes, second edition, W. B. Saunders Company, Philadelphia, London, Toronto 1996.

    Google Scholar 

  23. G. Y. El-Khoury, R. A. Bergman, and W. J. Montgomery, Sectional Anatomy by MRI. Second edition, Churchill Livingstone, New York, Edinburgh, London 1995.

    Google Scholar 

  24. J. Fleckenstein, J. V. Crues III, and C. D. Reimers, Muscle Imgaging in Health and Disease. Springer-Verlag, Berlin, Heidelberg, New York 1996.

    Book  Google Scholar 

  25. E. Fukushima, editor, NMR in Biomedicine: The Physical Basis: Key Papers in Physics, Number 2, American Institute of Physics, New York 1989.

    Google Scholar 

  26. D. G. Gadian, NMR and its Applications to Living Systems. Second edition, Oxford University Press, Oxford, New York, Tokyo 1996.

    Google Scholar 

  27. A. E. George, Neurodegenerative diseases: Alzheimer’s disease and related disorders. Neuroimaging Clinics of North America, Vol. 5, No. 1, February 1995, pp. 1–159, W. B. Saunders Company, Philadelphia, London, Toronto 1995.

    Google Scholar 

  28. O. Gingerich, Kepler’s place in astronomy. In Kepler: Four Hundred Years. Proceedings of Conferences held in Honour of Johannes Kepler, A. Beer, and P. Beer, editors, Vistas in Astronomy, Vol. 18, pp. 261–278, Pergamon Press, Oxford, New York, Toronto 1975.

    Google Scholar 

  29. C. B. Grossman, Magnetic Resonance Imaging and Computed Tomography of the Head and Spine. Second edition, Williams & Wilkins, Baltimore, Philadelphia, London 1996.

    Google Scholar 

  30. R. I. Grossman, D. M. Yousem, Neuroradiology. Mosby-Year Book, St. Louis, Baltimore, Berlin 1994.

    Google Scholar 

  31. B. L. Hart, E. C. Benzel, and C. C. Ford, Fundamentals of Neuroimaging. W. B. Saunders Company, Philadelphia, London, Toronto 1997.

    Google Scholar 

  32. D. M. Healy, Jr., J. Lu, and J. B. Weaver, Two applications of wavelets and related techniques in medical imaging. Ann. Biomed. Eng. 23, 637–665 (1995).

    Article  Google Scholar 

  33. R. M. Henkelman, M. J. Bronskill, Artifacts in magnetic resonance imaging. Rev. Magn. Reson. Med. 2, 1–126 (1987).

    Google Scholar 

  34. A. Heuck, G. Luttke, and J. W Rohen, MR-Atlas der Extremitäten. F. K. Schattauer, Stuttgart, New York 1994.

    Google Scholar 

  35. L. Heuser, M. Oudkerk, editors, Advances in MRI. Blackwell Science, Oxford, London, Edinburgh 1996.

    Google Scholar 

  36. S. H. Heywang-Köbrunner, R. Beck, Contrast-Enhanced MRI of the Breast. Second edition, Springer-Verlag, Berlin, Heidelberg, New York 1996.

    Book  Google Scholar 

  37. S. H. Heywang-Köbrunner, I. Schreer, Bildgebende Mammadiagnostik: Untersuchungstechnik, Befundmuster und Differentialdiagnostik in Mammographie, Sonographie und Kernspintomographie. Georg Thieme Verlag, Stuttgart, New York 1996.

    Google Scholar 

  38. P. Horowitz, W. Hill, The Art of Electronics. Second edition, Cambridge University Press, Cambridge, New York, Port Chester 1990.

    Google Scholar 

  39. P. M. Jakob, F. Kober, and A. Haase, Radial BURST Imaging. Magn. Reson. Med. 36, 557–561 (1996).

    Article  Google Scholar 

  40. F. A. Jolesz, MRI-guided interventions. In: Progressi in RM, Note di tecnica, a cura di M. Cammisa, T. Scarabino, pp. 199-220, Guido Gnocchi, Editore, Napoli 1995.

    Google Scholar 

  41. W.A. Kaiser, MR Mammography (MRM). Springer-Verlag, Berlin, Heidelberg, New York 1993.

    Book  Google Scholar 

  42. M. King, Fourier optics and radar signal processing. In: Applications of Optical Fourier Transforms. H. Stark, editor, pp. 209–251, Academic Press, Orlando, San Diego, San Francisco 1982.

    Chapter  Google Scholar 

  43. H.-J. Kretschmann, W. Weinrich, Dreidimensionale Computergraphik neurofunktioneller Systeme: Grundlagen für die neurologisch-topische Diagnostik und die kranielle Bilddiagnostik (Magnetresonanztomographie und Computertomographie). Georg Thieme Verlag, Stuttgart, New York 1996.

    Google Scholar 

  44. J. Kucharczyk, M. E. Moseley, T. Roberts, and W. W. Orrison, Jr., editors, Functional neuroimaging. Neuroimaging Clinics of North America, Vol. 5, No. 2, May 1995, pp. 161-308, W. B. Saunders Company, Philadelphia, London, Toronto 1995.

    Google Scholar 

  45. A. Kumar, D. Welti, and R. R. Ernst, NMR Fourier zeugmatography. J. Magn. Reson. 18, 69–83 (1975).

    Google Scholar 

  46. R. I. Kuzniecky, G. D. Jackson, Magnetic Resonance in Epilepsy. Raven Press, New York 1995.

    Google Scholar 

  47. K. Kwong, Functional magnetic resonance imaging with echo planar imaging. Magn. Reson. Quart.11, 1–20 (1995).

    Google Scholar 

  48. E. N. Leith, Synthetic aperture radar. In: Optical Data Processing, D. Casasent, editor, pp. 89–117, Topics in Applied Physics, Vol. 23, Springer-Verlag, Berlin, Heidelberg, New York 1978.

    Chapter  Google Scholar 

  49. E. N. Leith, Optical processing of synthetic aperture radar data. In: Photonic Aspects of Modern Radar, H. Zmuda, E. N. Toughlian, editors, pp. 381-401, Artech House, Boston, London 1994.

    Google Scholar 

  50. P. J. Marcer, W. Schempp, A mathematically specified template for DNA and the genetic code in terms of the physically realisable processes of quantum holography. In: Proc. Symp. Living Computers: The Nature of Turing and Non-Turing Reproducible Order in Living Organisms, A. M. Fedorec, P. J. Marcer, editors, pp. 45-62, The British Computer Society, Greenwich University Press, London 1996.

    Google Scholar 

  51. J. Mattson, M. Simon, The Pioneers of NMR and Magnetic Resonance in Medicine: The Story of MRI. Bar-Ilan University Press, Ramat Gan 1996.

    Google Scholar 

  52. C. Mead, L. Conway, Introduction to VLSI Systems. Addison-Wesley Publishing Company, Reading, Menlo Park, London 1980.

    Google Scholar 

  53. T. B. Möller, E. Reif, MRI Atlas of the Musculoskeletal System. Blackwell Scientific Publications, Boston, Oxford, London 1993.

    Google Scholar 

  54. C. T. W. Moonen, P. van Gelderen, N. Ramsey, G. Liu, J. H. Duyn, J. Frank, and D. R. Weinberger, PRESTO, a rapid 3D approach for functional MRI of human brain. In: Syllabus Functional MRI, P. Pavone and P. Rossi, editors, pp. 105–110, Springer-Verlag, Berlin, Heidelberg, New York 1996.

    Chapter  Google Scholar 

  55. P. L. Munk, C. A. Helms, MRI of the Knee. Second edition, Lippincott-Raven Publishers, Philadelphia, New York 1996.

    Google Scholar 

  56. M. Nagele, G. Adam, Moderne Kniegelenkdiagnostik: Bildgebende Verfahren und klinische Aspekte. Springer-Verlag, Berlin, Heidelberg, New York 1995.

    Book  Google Scholar 

  57. W. Noack, editor, X-SAR Picture Book. Springer-Verlag, Berlin, Heidelberg, New York 1997.

    Google Scholar 

  58. E. J. Potchen, E. M. Haacke, J. E. Siebert, and A. Gottschalk, Magnetic Resonance Angiography: Concepts and Applications. Mosby-Year Book, St. Louis, Baltimore, Boston 1993.

    Google Scholar 

  59. V. Rasche, R. W. de Boer, D. Holz, and R. Proksa, Continuous radial data acquisition for dynamic MRI. Magn. Reson. Med. 34, 754–761 (1995).

    Article  Google Scholar 

  60. V. Rasche, D. Holz, and W. Schepper, Radial turbo spin echo imaging. Magn. Reson. Med. 32, 629–38 (1994).

    Article  Google Scholar 

  61. A. W. Rihaczek, Principles of High-Resolution Radar. Artech House, Boston, London 1996.

    MATH  Google Scholar 

  62. P. B. Roemer, W. A. Edelstein, C. E. Hayes, S. P. Souza, and O. M. Mueller, The NMR phased array. Magn. Reson. Med. 16, 192–252 (1990).

    Article  Google Scholar 

  63. V. M. Runge, editor, Magnetic Resonance Imaging: Clinical Principles. J. B. Lippincott Company, Philadelphia, New York, London 1992.

    Google Scholar 

  64. V. M. Runge, Magnetic Resonance Imaging of the Brain. J. B. Lippincott Company, Philadelphia 1994.

    Google Scholar 

  65. J. A. Sanders, Functional magnetic resonance imaging. In: Functional Brain Imaging, W. W. Orrison, Jr., J. D. Lewine, J. A. Sanders, and M. F. Hartshorne, editors, pp. 239–326, Mosby-Year Book, St. Louis, Baltimore, Berlin 1995.

    Google Scholar 

  66. K. Sartor, MR Imaging of the Skull and Brain. Springer-Verlag, Berlin, Heidelberg, New York 1995.

    Google Scholar 

  67. W. Schempp, Harmonic Analysis on the Heisenberg Nilpotent Lie Group, with Applications to Signal Theory. Pitman Research Notes in Mathematics Series, Vol. 147, Longman Scientific and Technical, London 1986.

    Google Scholar 

  68. W. Schempp, Geometric analysis: The double-slit interference experiment and magnetic resonance imaging. Cybernetics and Systems’ 96, Vol. 1, pp. 179–183, Austrian Society for Cybernetic Studies, Vienna 1996.

    Google Scholar 

  69. W. Schempp, Geometric analysis and symbol calculus: Fourier transform magnetic resonance imaging and wavelets. In: Synergie, Syntropie, Nichtlineare Systeme, Wiener-Symposium, Heft 4, W. Eisenberg, U. Renner, S. Trimper, M. Kunz und K. Vogelsang, Hrsg., pp. 133-186, Verlag im Wissenschaftszentrum Leipzig, Leipzig 1996.

    Google Scholar 

  70. W. Schempp, Non-commutative affine geometry and symbolic calculus: Fourier transform magnetic resonance imaging and wavelets. In: Signal and Image Representation in Combined Spaces, J. Zeevi and R. R. Coifman, editors, pp. 1–47, Academic Press, London, San Diego, New York 1997.

    Google Scholar 

  71. W. Schempp, Wavelets in high-resolution radar imaging and clinical resonance tomography. Cybernetics and Systems: An International Journal 28, 1–23 (1997).

    Article  MATH  Google Scholar 

  72. W. Schempp, Wavelets in high resolution radar imaging and clinical magnetic resonance imaging. Proc. IWISP’ 96: Third International Workshop on Image and Signal Processing on the Theme of Advance in Computational Intelligence, Manchester, United Kingdom, B. G. Mertzios, P. Liatsis, editors, pp. 73-80, Elsevier, Amsterdam, Lausanne, New York 1996.

    Google Scholar 

  73. W. Schempp, Magnetic Resonance Imaging: Mathematical Foundations and Applications. John Wiley & Sons, New York, Chichester, Brisbane (in print).

    Google Scholar 

  74. A. G. Sorensen, MR imaging in hyperacute stroke. In: Syllabus Functional MRI, P. Pavone and P. Rossi, editors, pp. 99–102, Springer-Verlag, Berlin, Heidelberg, New York 1996.

    Chapter  Google Scholar 

  75. H. Stefan, Epilepsien: Diagnose und Behandlung. Chapman & Hall, London, Glasgow, Weinheim 1995.

    Google Scholar 

  76. B. Stephenson, Kepler’s Physical Astronomy. Princeton University Press, Princeton, NJ 1994.

    MATH  Google Scholar 

  77. P. Stoeter, P. Gutjahr, und K. Brühl, Tumoren bei Kindern: Moderne Bildgebung mit MRT und CT, Band 1: ZNS. Georg Thieme Verlag, Stuttgart, New York 1996.

    Google Scholar 

  78. D. W. Stoller, editor, Magnetic Resonance Imaging in Orthopaedics & Sports Medicine. Second edition, Lippincott-Raven Publishers, Philadelphia, New York 1997.

    Google Scholar 

  79. H. Strunk, P. Gutjahr, Tumoren bei Kindern: Moderne Bildgebung mit MRT und CT, Band 2: Körperstamm und Extremitäten. Georg Thieme Verlag, Stuttgart, New York 1996.

    Google Scholar 

  80. K. R. Thulborn, J. Voyvodic, B. McCurtain, J. Gillen, S. Chang, M. Just, P. Carpenter, and J. A. Sweeney, High field functional MRI in humans: Applications to cognitive functions. In: Syllabus Functional MRI, P. Pavone and P. Rossi, editors, pp. 91–96, Springer-Verlag, Berlin, Heidelberg, New York 1996.

    Chapter  Google Scholar 

  81. C. L. Truwit, T. E. Lempert, High Resolution Atlas of Cranial Neuroanatomy. Williams & Wilkins, Baltimore, Philadelphia, Hong Kong 1994.

    Google Scholar 

  82. D. Uhlenbrock, MRT und MRA des Kopfes: Indikationsstellung, Wahl der Untersuchungsparameter, Befundinterpretation. Georg Thieme Verlag, Stuttgart, New York 1996.

    Google Scholar 

  83. M. S. van der Knaap, J. Valk, Magnetic Resonance of Myelin, Myelination, and Myelin Disorders. Second edition, Springer-Verlag, Berlin, Heidelberg, New York 1995.

    Book  Google Scholar 

  84. T. J. Vogl, MR-Angiographie und MR-Tomographie des Gefäßsystems: Klinische Diagnostik. Springer-Verlag, Berlin, Heidelberg, New York 1995.

    Book  Google Scholar 

  85. D. R. Wehner, High Resolution Radar. Artech House, Norwood, MA 1987.

    Google Scholar 

  86. A. Weil, Sur certains groupes d’opérateurs unitaires. Acta Math. 111, 143–211 (1964). In: Œuvres Scientifiques, Collected Papers, Vol. III (1964-1978), pp. 1-69, Springer-Verlag, New York, Heidelberg, Berlin 1980.

    Article  MathSciNet  MATH  Google Scholar 

  87. M. V. Wickerhauser, Costum wavelet packet image compression design. Proc. IWISP’ 96: Third International Workshop on Image and Signal Processing on the Theme of Advance in Computational Intelligence, Manchester, United Kingdom, B. G. Mertzios, P. Liatsis, editors, pp. 47–52, Elsevier, Amsterdam, Lausanne, New York 1996.

    Google Scholar 

  88. D. H. Yock, Jr., Magnetic Resonance Imaging of CNS Disease: A Teaching File. Mosby-Year Book, St. Louis, Baltimore, Berlin 1995.

    Google Scholar 

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  1. Lehrstuhl für Mathematik I, University of Siegen, Germany

    Walter Schempp

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  1. Kansas State University, Manhattan, Kansas, USA

    Alexander G. Ramm

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Schempp, W. (1998). Wavelet Modelling of Clinical Magnetic Resonance Tomography: An Ensemble Quantum Computing Approach. In: Ramm, A.G. (eds) Inverse Problems, Tomography, and Image Processing. Springer, Boston, MA. https://doi.org/10.1007/978-1-4020-7975-7_10

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