Comparison of four natural pixel bases for SPECT imaging
In this paper the problem of restoring a two-dimensional tomographic medical image is considered. The emission function f ( P ) is defined on a plane circular domain Ω , and has to be restored from data collected by a SPECT machine. Through an approach known as natural pixel discretization, the solu...
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| Published in: | Journal of computational and applied mathematics Vol. 198; no. 2; pp. 361 - 377 |
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| Main Authors: | , , |
| Format: | Journal Article Conference Proceeding |
| Language: | English |
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Amsterdam
Elsevier B.V
15.01.2007
Elsevier |
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| ISSN: | 0377-0427, 1879-1778 |
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| Abstract | In this paper the problem of restoring a two-dimensional tomographic medical image is considered. The emission function
f
(
P
)
is defined on a plane circular domain
Ω
, and has to be restored from data collected by a SPECT machine. Through an approach known as
natural pixel discretization, the solution is expressed as a linear combination of functions belonging to a suitable basis. We consider here four different bases, all of them giving a highly structured coefficient matrix. The linear system obtained in this way can be solved efficiently by means of the fast Fourier transform. The computational cost and the performance of the bases are compared. When the data are contaminated by Poissonian noise, the numerical experimentation shows that all the bases are almost equivalent from the point of view of the restoration efficiency. Hence the choice of a basis should rely on other considerations, as for instance the computational cost. |
|---|---|
| AbstractList | In this paper the problem of restoring a two-dimensional tomographic medical image is considered. The emission function f(P) is defined on a plane circular domain Omega, and has to be restored from data collected by a SPECT machine. Through an approach known as natural pixel discretization, the solution is expressed as a linear combination of functions belonging to a suitable basis. We consider here four different bases, all of them giving a highly structured coefficient matrix. The linear system obtained in this way can be solved efficiently by means of the fast Fourier transform. The computational cost and the performance of the bases are compared. When the data are contaminated by Poissonian noise, the numerical experimentation shows that all the bases are almost equivalent from the point of view of the restoration efficiency. Hence the choice of a basis should rely on other considerations, as for instance the computational cost. In this paper the problem of restoring a two-dimensional tomographic medical image is considered. The emission function f ( P ) is defined on a plane circular domain Ω , and has to be restored from data collected by a SPECT machine. Through an approach known as natural pixel discretization, the solution is expressed as a linear combination of functions belonging to a suitable basis. We consider here four different bases, all of them giving a highly structured coefficient matrix. The linear system obtained in this way can be solved efficiently by means of the fast Fourier transform. The computational cost and the performance of the bases are compared. When the data are contaminated by Poissonian noise, the numerical experimentation shows that all the bases are almost equivalent from the point of view of the restoration efficiency. Hence the choice of a basis should rely on other considerations, as for instance the computational cost. |
| Author | Bozzo, E. Menchi, O. Bevilacqua, R. |
| Author_xml | – sequence: 1 givenname: R. surname: Bevilacqua fullname: Bevilacqua, R. email: bevilacq@di.unipi.it – sequence: 2 givenname: E. surname: Bozzo fullname: Bozzo, E. email: bozzo@di.unipi.it – sequence: 3 givenname: O. surname: Menchi fullname: Menchi, O. email: menchi@di.unipi.it |
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| Keywords | Image processing Restoration Regularization Ill-posedness Fast Fourier transformation Numerical linear algebra Image restoration Discretization method Direct method Numerical analysis Linear system Matrix inversion Applied mathematics Imaging Two-dimensional calculations Ill posed problem |
| Language | English |
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| References | Hsieh, Gullberg, Zeng, Huesman (bib14) 1996; 43 Groch, Erwin (bib10) 2000; 28 Hoffman, Cutler, Digby, Mazziotta (bib13) 1990; 37 Press, Teukolsky, Vetterling, Flannery (bib15) 1990; 37 Brianzi, Di Benedetto (bib7) 2001 Bertero, Boccacci (bib1) 1998 Bex, Makous (bib4) 2002; 19 Bindi, Brianzi, Di Benedetto (bib5) 2002; 12 R. Bevilacqua, E. Bozzo, O. Menchi, SPECT imaging reconstruction through discretization on a polar grid, Technical Report TR-02-17, Dipartimento di Informatica, Università di Pisa, 2002. Gullberg, Zeng (bib12) 1994; 41 P. Boccacci, P. Bonetto, P. Brianzi, P. Calvini, F. Di Benedetto, A.R. Formiconi, A. Passeri, The 2D+1 method for an efficient modelling of collimator blur in 3D SPECT imaging, Ann. Univ. Ferrara Sez. VII Sci. Mat. XIV (Suppl.) (2000). R. Bevilacqua, E. Bozzo, O. Menchi, SPECT imaging reconstruction: proposal of an algorithm which exploits the matrix structure, Technical Report TR-01-12, Dipartimento di Informatica, Università di Pisa, 2001. Gullberg, Hsieh, Zeng (bib11) 1996; 43 Golub, Van Loan (bib9) 1996 Buonocore, Brody, Macovsky (bib8) 1981; 28 Golub (10.1016/j.cam.2005.07.027_bib9) 1996 Press (10.1016/j.cam.2005.07.027_bib15) 1990; 37 Buonocore (10.1016/j.cam.2005.07.027_bib8) 1981; 28 Gullberg (10.1016/j.cam.2005.07.027_bib11) 1996; 43 Bindi (10.1016/j.cam.2005.07.027_bib5) 2002; 12 10.1016/j.cam.2005.07.027_bib3 10.1016/j.cam.2005.07.027_bib2 Bex (10.1016/j.cam.2005.07.027_bib4) 2002; 19 Gullberg (10.1016/j.cam.2005.07.027_bib12) 1994; 41 10.1016/j.cam.2005.07.027_bib6 Bertero (10.1016/j.cam.2005.07.027_bib1) 1998 Groch (10.1016/j.cam.2005.07.027_bib10) 2000; 28 Hoffman (10.1016/j.cam.2005.07.027_bib13) 1990; 37 Brianzi (10.1016/j.cam.2005.07.027_bib7) 2001 Hsieh (10.1016/j.cam.2005.07.027_bib14) 1996; 43 |
| References_xml | – start-page: 165 year: 2001 end-page: 175 ident: bib7 article-title: Exploiting matrix structures in SPECT models publication-title: Structured Matrices: Recent Developments in Theory and Computation – volume: 37 start-page: 616 year: 1990 end-page: 620 ident: bib13 article-title: 3-D phantom to simulate cerebral blood flow and metabolic images for PET publication-title: IEEE Trans. Nucl. Sci. – volume: 43 start-page: 2306 year: 1996 end-page: 2319 ident: bib14 article-title: Image reconstruction using a generalized natural pixel basis publication-title: IEEE Trans. Nucl. Sci. – reference: R. Bevilacqua, E. Bozzo, O. Menchi, SPECT imaging reconstruction: proposal of an algorithm which exploits the matrix structure, Technical Report TR-01-12, Dipartimento di Informatica, Università di Pisa, 2001. – volume: 12 start-page: 1 year: 2002 end-page: 8 ident: bib5 article-title: A Fourier approach to the natural pixel discretization of brain single-photon emission computed tomography publication-title: Int. J. Imaging Syst. Technol. – reference: R. Bevilacqua, E. Bozzo, O. Menchi, SPECT imaging reconstruction through discretization on a polar grid, Technical Report TR-02-17, Dipartimento di Informatica, Università di Pisa, 2002. – volume: 37 start-page: 616 year: 1990 end-page: 620 ident: bib15 article-title: 3-D phantom to simulate cerebral blood flow and metabolic images for PET publication-title: IEEE Trans. Nucl. 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| Snippet | In this paper the problem of restoring a two-dimensional tomographic medical image is considered. The emission function
f
(
P
)
is defined on a plane circular... In this paper the problem of restoring a two-dimensional tomographic medical image is considered. The emission function f(P) is defined on a plane circular... |
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| SubjectTerms | Applied sciences Artificial intelligence Computer science; control theory; systems Exact sciences and technology Fourier analysis Ill-posedness Image processing Information, signal and communications theory Mathematics Numerical analysis Numerical analysis. Scientific computation Numerical linear algebra Pattern recognition. Digital image processing. Computational geometry Regularization Restoration Sciences and techniques of general use Signal processing Telecommunications and information theory |
| Title | Comparison of four natural pixel bases for SPECT imaging |
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