Geometric Modeling of Medical Filtered Back-Projection Images
DOI:
https://doi.org/10.24203/ajas.v8i5.6327Keywords:
Filtered Back-Projection, Medical Imaging, Finsler Geometry, Symplectic Geometry, Minkowski Spaces, Holmes-Thomson VolumeAbstract
This paper deals with the geometric concepts of medical imaging. The main result is proving this novel achievement that the time-frequency projections from some angles in the filtered back-projection image can be calculated by the surface area of the convex body and symplectic Holmes-Thompson volume. For this purpose, the necessary notions are introduced by the surface area of a convex body of the corresponding projective spaces, Radon and Fourier transform in integral geometry, Holmes-Thompson measure in Finsler-Minkowsi-symplectic geometry. This result leads to the other correspondence between basic notions of medical imaging such as the projections from some angles which are equally spaced, the image-frequency of these projections, the low image correctness, and the corresponding notions of Finsler geometry.
References
Lipková, J., Angelikopoulos, P., Wu, S., Alberts, E., Wiestler, B., Diehl, C., Preibisch, C., Pyka, T., Combs, S.E., Hadjidoukas, P., Leemput Koumoutsakos, P., Lowengrub, J., Menze, B., “Personalized Radiotherapy Design for Glioblastoma: Integrating Mathematical Tumor Models”, Multimodal Scans, and Bayesian Inference, IEEE Transactions on Medical Imaging , vol. 38, no. 8, pp. 1875 – 1884, 2019.
Fioranelli, M., Sepehri, A., “A mathematical model for the virus medical imaging technique”, International Journal of Geometric Methods in Modern Physics, vol. 15, no. 7, 1850121, 2018.
Zayed, A.I., “A new perspective on the role of mathematics in medicine”, J Adv Res, vol. 17, pp. 49-54, 2019.
Farinelli, S., “Geometric arbitrage theory and market dynamics”, American Institute of Mathematical Sciences, vol. 7, no. 4, pp. 431-471, 2015.
Zabreiko, P.P., Lebedev, A.V., “Banach geometry of financial market models”, Doklady Mathematics, vol. 95, no. 2, pp. 164-167, 2015.
Chule, S., “Random Geometric Analysis in the Stochastic Volatility: Financial Markets States Degeneracy”, Analysis and Computations Journal, Forthcoming, SSRN: https://ssrn.com/abstract=2968295, 2017.
Piotrowski, E.W., Stadkowski, J., “Geometry of Financial Markets – Towards Information Theory Model of Markets”, Physica A: Statistical Mechanics and its Applications, vol. 382, no. 1, pp. 228-234, 2007.
Piotrowski, E.W., Stadkowski, J., “The merchandising mathematician model: profit intensities”, Physica A: Statistical Mechanics and its Applications, vol. 318, no. 3-4, pp. 496-504, 2003.
Holmes, R.D., Thompson, A.C., “N-dimensional area and content in Minkowski spaces”, Pacific Journal of Mathematics, vol. 85, no. 1, pp. 77-110, 1979.
Paiva, J.C.A., Fernande,s E., “Fourier transforms and Holmes-Thompson volume of Finsler manifolds”, International Mathematics Research Notices, vol. 19, pp. 1031-1042, 1999.
Paiva, J.C.A., Thompson, A.C., Volumes on normed and Finsler spaces, Chapter book of a Sampler of Riemann-Finsler Geometry, Cambridge University Press, pp. 1-48, 2004.
Paiva, J.C.A., Some problems on Finsler geometry, Handbook of Differential Geometry, vol. 2, pp. 1-33, 2006.
Koldobsky, A., Fourier Analysis in Convex Geometry, American Mathematical Society, vol. 116, 2006.
Benjancu, A.A., Finsler geometry and applications, Ellis Horwood, 1996.
Gelfand, I.M., Smirnov, M., “Lagrangians satisfying Crofton formulas, Radon transforms, and nonlocal differentials”, Advances in Mathematics, vol. 109, no. 2, pp. 188-227, 1994.
Gromov, M., “Filling Riemannian manifolds, Journal of Differential Geometry”, vol. 18, no. 1, pp. 1-147, 1998.
Rudin, W., Functional Analysis, Mc Graw-Hil. New York, 1973.
McDuff, D., Salamon, D., Introduction to symplectic topology, Oxford Mathematical Monographs, Clarendon Press, 1998.
Dym, H., McKean, H.P., Fourier series and Integrals, Academic Press, New York, 1997.
Schneider, R., “On integral geometry in projective Finsler spaces”, Izvestiya Natsional'noĭ Akademii Nauk Armenii. Matematika, vol. 37, pp. 34-51, 2002.
Schneider, R., Wieacker, J.A., “Integral geometry in Minkowski spaces”, Advances in Mathematics, vol. 129, no. 2, pp. 222-260, 1997.
Paiva, J.C.A, Fernandes, E., “Crofton formulas in Projective Finsler spaces”, Electronic Research Announcements of the American Mathematical Society, vol. 4, pp. 91-100, 1998.
Olson, T., De Stefano, J., “Wavelet localization of the radon transform”, IEEE Signal Process, vol. 42, no. 8, pp. 2055-2067, 1994.
Olson, T., “Optimal time-frequency projections for localized tomography”, Ann. Biomed, Eng., vol. 23, pp. 622-636, 1995.
Stanton, A., “Wilhelm Conrad Röntgen on a new kind of rays: translation of a paper read before the Würzburg Physical and Medical Society”, Nature, vol. 253, pp. 274–276, 1895.
Olson, T., Applied Fourier Analysis: From Signal Processing to Medical Imaging, Springer, 2017.
Downloads
Published
Issue
Section
License
Copyright (c) 2020 Atefeh Hasan-Zadeh
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
- Papers must be submitted on the understanding that they have not been published elsewhere (except in the form of an abstract or as part of a published lecture, review, or thesis) and are not currently under consideration by another journal published by any other publisher.
- It is also the authors responsibility to ensure that the articles emanating from a particular source are submitted with the necessary approval.
- The authors warrant that the paper is original and that he/she is the author of the paper, except for material that is clearly identified as to its original source, with permission notices from the copyright owners where required.
- The authors ensure that all the references carefully and they are accurate in the text as well as in the list of references (and vice versa).
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Attribution-NonCommercial 4.0 International that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
- The journal/publisher is not responsible for subsequent uses of the work. It is the author's responsibility to bring an infringement action if so desired by the author.
