DOI:
https://doi.org/10.14483/udistrital.jour.tecnura.2014.SE1.a01Publicado:
2014-12-01Número:
Vol. 18 (2014): Edición EspecialSección:
InvestigaciónProblema de tomografía local usando wavelets B-spline cúbicos
Local tomography problem using cubic B-spline wavelets
Descargas
Resumen (es)
En el presente artículo se describe e implementa una solución al problema de la tomografía local, equivalente a la inversión de la transformada de Radon, utilizando la transformada wavelet. Para ello se ejecuta un algoritmo basado en wavelets B-spline cúbicos de soporte compacto con suficientes momentos de desvanecimiento para que la función de escalado filtrada, la wavelet madre y su transformada de Hilbert tengan decaimiento rápido. Lo anterior favorece la localización de la transformada wavelet de la transformada de Radon y, por tanto, la inversión, es decir; la reconstrucción de una región central de interés del fantasma Shepp-Logan.Resumen (en)
This paper presents a formal description and subsequent implementation of a solution to the local tomography problem, equivalent to the so-called Radon Transform local Inversion. To do so, it is necessary to execute an algorithm based on compact-support B-spline cubic wavelets, including sufficient vanishing moments so that the filtered-scaling, the mother wavelet and its Hilbert Transform functions decay rapidly. This favors localization of Radon-Transform Wavelet Transform and, therefore, its corresponding inversion; that is, the reconstruction of a Shepp-Logan Phantom central interest region (ROI).
Referencias
Aldroubi, A. (1996). The wavelet transform: A surfing guide. Wavelets in Medicine and Biology. A. Audroubi, M. Unser (eds.), pp. 3-36. New York: CRC Press.
Antonini, M.; Barlaud, P.; Mathieu, P.; Daubechies, I. (1992). Image Coding Using Wavelet Transform. IEEE Trans. Image Proc., vol. 1(2), pp.205-220.
Berenstein, C.; Walnut, D. (1994). Local inversion of the Radon transform in even dimensions using Wavelets. S. Gindikin and P. Michor, eds., 75 years of Radon Transform, pp. 45-69. Cambridge, MA.: International Press Co., Ltd.
Berenstein, C.; Walnut, D. (1995.). Wavelets and local tomography. Wavelets in Medicine and Biology. CRC Press.
Brigham, E. (1988). The Fast Fourier Transform and its Applications. Ann Arbor, Michigan: Prentice Hall.
Boggess, A.; Narcowich, F. (2007). A First Course in Wavelets with Fourier analysis. Ann Arbor, Michigan: Prentice Hall.
Bonani, A.; Durand, S.; Weiss, G. Wavelets obtained by continuos deformations of the Haar wavelet. Revista Mat. Iberoamericana, vol. 12(1).
Chaudhury, K.; Unser, M. (April 2011). On the Hilbert Transform of Wavelets. IEEE, Transactions on Signal Processing, vol. 59(4).
Chui, C. (1992). An Introduction to Wavelets. New York: Academic Press.
Clarke, R. (1985). Transform Coding of Images. San Diego, CA: Academic Press.
Clarke, R. (1995). Digital Compression of Still Images and Video. San Diego, CA: Academic Press.
Daubechies, I. (1990). The wavelet transform, time-frequency localization and signal analysis. IEEE Transactions on Information. Theory, vol. 36(11), pp. 961-1005.
Daubechies, I. (1992). Ten Lectures on Wavelets. Philadelphia: CBMS Series 61, SIAM.
Daubechies, I. (Aug. 2006). Orthonormal bases of compactly supported wavelets. Communications on pure and Applied Mathematics, vol. 41(7), pp. 909-996.
Delaney, A.; Bresler, Y. (June 1995). Multiresolution tomographic reconstruction using wavelets. IEEE Transactions on Image Processing, vol. 4 (3).
Destefano, J.; Olson, T. (Aug. 1994). Wavelet Localization of the Radon Transform. IEEE Transactions on Signal Processing, vol. 42 (8), pp. 2055-2067.
Donoho, D. (1993). Nonlinear Wavelet Methods for Recovery of Signals, Densities and Spectra from Indirect and Noyse Data. Different Perspectives on Wavelets. Proceedings of Symposia in Pure Math., AMS., I. Daubechies, Edt., 47, pp. 173-205.
Faridani, A.; Keinert, F.; Natterer, F.; Ritman, E.; Smith, K. (1990). Local and global tomography. New York: Springer-Verlag.
Faridani, A.; Ritman, E.; Smith, K. (1993). Local Tomography. SIAM Journal on Applied Mathematics, vol. 52(2), pp. 459-484.
Faridani, A.; Finch, D.; Ritman, E.; Smith, K. (Aug. 1997). Local Tomography II. SIAM Journal on Applied Mathematics, vol. 57( 4), pp. 1095-1127.
Folland, G. (1999). Real Analysis. New York: John Wiley & Sons, Inc, 2nd edition.
Guédon, J.; Unser, M. (1992). Least squares and spline filtered back-projection. NCRR Rep. 52/92, Nat. Inst. Health.
Goswami, J.; Chan, A. (1999). Fundamentals of Wavelets Theory, Algorithms, and Applications. John Wiley & sons, Inc.
Helgason, S. (1999). The Radon Transform. Boston: Birkhäuser. Second Edition.
Hernández, E.; Weiss, G. (1996). A First Course on Wavelets. Boca Raton FL: CRC Press.
Hong, D.; Wang, J.; Gardner, R. (2005). Real Analysis with an Introduction to Wavelets and Applications. Academic Press, Elsevier.
Irino, T.; Kawahara, H. (1993). Signal reconstruction from modifed auditory wavelet transform. IEEE Trans. Signal Process, vol. 41(32), pp. 3549-3553.
Kincaid, D.; Cheney, W. (1994). Análisis Numérico. Edit. E.U.A. pp. 323-363.
Lebon, F.; Rodríguez-Ramos, R.; López-Realpozo, J.C.; Bravo-Castillero, J.; Guinovart-Díaz, R.; Mesejo, A. (January 2006). Effective Properties of Nonlinear Laminated Composites With Perfect Adhesion. Transactions of the ASME, vol. 73, pp. 174-178.
López, R.; De Armas, R. (2010). Algoritmo para detectar eventos epilépticos a partir de wavelets analizando la función de energía. Tesis de maestría en matemáticas aplicadas. Universidad EAFIT, Medellín.
Maass, P. (agosto 1992). The interior Radon Transform. SIAM Journal on Applied Mathematics, vol. 52(3), pp. 710-724.
Mallat, S. (1989). A theory for multiresolution signal decomposition: The wavelet representation. IEEE Trans. Pattern Anal. Mach. Intell, vol. 11(33), pp. 674-693.
Mallat, S. (1989). Multiresolution approximations and wavelet orthonormal bases for L2.Trans. of Amer. Math. Soc. 315, pp. 69-87.
Mallat, S.; Hwang, W. (1992). Singularity detection and processing with wavelets. IEEE Trans. Inform. Theory (Special Issue on Wavelet Transforms and Multiresolution Signal Analysis), 38(2), pp. 617-643.
Mallat, S. (2008). Wavelet Tour of signal Processing. Boston: American Press.
Mertins, A. (1996). Signal Analysis Wavelets, Filter Banks, Time-Frequency Transforms and Applications. New York: John Wiley & sons.
Meyer, Y. (1990). Ondelettes et operateurs, I: Ondelettes. Paris: Herman.
Natterer, F. (2001). The Mathematics of Computarized Tomography. SIAM.
Natterer, F.; Wübbeling, F. (2001). Mathematical Methods in Image Reconstruction. SIAM.
Pinsky, M. (2001). Introduction to Fourier Analysis and wavelet. Brooks/Cole, NJ.
Quak, E.; Weyrich, N. (1994). Decomposition and reconstruction algorithms for spline wavelet on a bounded interval. Appl. and Comp. Harmonic Anal. (ACHA), vol. 1, pp. 217-231.
Ramm, A.; Zaslavsky, A. (1993). Reconstructing singularities of a function from its Radon transform. Math. and Comput. Modelling, vol. 18(1), pp. 109-138.
Ramm, A.; Zaslavsky, A. (1993). Singularities of the Radon transform. Bull. AMS, vol. 25(1), pp. 109-115.
Rashid-Farrokhi, F.; Liu, K.; Berenstein, C.; Walnut, D. (1995). Localized wavelet based computerized tomography. Washington: Proceedings ICIP-95.
Rashid-Farrokhi, F., Liu, R., K.J.; Berenstein, C.; Walnut, D. (Oct. 1997). Wavelet-Based Multiresolution Local Tomography. IEEE. Trans. On image Process, vol. 6(10), pp. 1412-1429.
Rioul, O.; Duhamel, P. (1992). Fast algorithms for discrete and continuous wavelet transform. IEEE Trans. Inform. Theory, vol. 38, pp. 569-586.
Shapiro, J. (1993). Embedded image coding using zerotrees of wavelet coeficients. IEEE Trans. Signal Proc., vol. 41, pp. 3445-3462.
Shen, X.; Galerkin, A. (2000). Wavelet method for a singular convolution equation on the real line. J. Int. Equa. Appl., vol. 12, pp. 157-176.
Stark, H. (2005). Wavelets and Signal Processing An Application-Based Introduction. Springer.
Tang, Y.; Yang, L. (2000). Wavelet Theory and Its Application to Pattern Recognition. World Scientific.
Tao, Q.; Mang, V.; Yuesheng, X. (2000). Wavelet Analysis and Applications. Berlin: Birkhäuser Verlag Basel.
Unser, M.; Aldroubi, A.; Murray, E. (1993). A family of polynomial spline wavelet transforms. Signal Processing, vol. 30, pp. 141-162.
Unser, M.; Aldroubi, A. (1993). B-spline processing I: Theory and II: Eficient design and application. IEEE Trans. Signal Process, vol. 41, pp. 821-848.
Unser, M. (1996). A practical guide to implementation of the wavelet transforms. Wavelets in Medicine and Biology. New York: CRC Press, pp. 37-76.
Walker, J. (2008). Local inversion of the Radon Transform in the plane using wavelets. A Primer on wavelets and Their Scientific Applications. Second edition. Chapman & Hall/CRC. Taylor & Francis Group.
Walnut, D. (1992). Application of Gabor and Wavelet expansions to the Radon Transform, Probabilistic and Stochastic Methods in Analysis, with applications. J. Byrnes et al. eds., Kluwer Academic Publishers, Inc., pp. 187-205.
Walnut, D. (1993). Local inversion of the Radon Transform in the plane using wavelets. San Diego: Proceedings of SPIE's. International Symposium on Optics, Imaging, and Instrumentation.
Walnut, D. (2002). An Introduction to Wavelet Analysis. Birkhäuser.
Walter, G.; Shen, X. (1999). Deconvolution using Meyer wavelets. J. Integral Equations Appl., vol. 11, pp. 515-534.
Wojtaszczyk, P. (1997). A Mathematical Introduction to Wavelets. Cambridge University Press.
Yin, X.; Ferguson, B.; Abbott, D. (2009). Wavelet based local tomographic image using terahertz techniques. Digital Signal Processing, vol. 19, pp.750–763. Elsevier.
Cómo citar
APA
ACM
ACS
ABNT
Chicago
Harvard
IEEE
MLA
Turabian
Vancouver
Descargar cita
Licencia
Esta licencia permite a otros remezclar, adaptar y desarrollar su trabajo incluso con fines comerciales, siempre que le den crédito y concedan licencias para sus nuevas creaciones bajo los mismos términos. Esta licencia a menudo se compara con las licencias de software libre y de código abierto “copyleft”. Todos los trabajos nuevos basados en el tuyo tendrán la misma licencia, por lo que cualquier derivado también permitirá el uso comercial. Esta es la licencia utilizada por Wikipedia y se recomienda para materiales que se beneficiarían al incorporar contenido de Wikipedia y proyectos con licencias similares.
