Abstract

A new algorithm for three-dimensional image reconstruction in nuclear medicine in which scattered radiation rather than multiple projected images is used for determination of the source depth within the body is proposed. Images taken from numerous energy windows are combined for the reconstruction of the source distribution in the body. In the first paper of this series Gunter et al. [ IEEE Trans. Nucl. Sci. 37, 1300 ( 1990)] examined simple linear algorithms for recovering source depth information from scattered radiation. These linear algorithms were unsuccessful because the scattering process produces little signal in the low-energy images at high spatial frequencies. As a result, the reconstructed source distributions exhibited nodal patterns and blurring. The scattering kernel that was measured and reported in the first paper is now examined more carefully. The singular-value decomposition of the kernel matrices is used to break the reconstruction problem into distinct channels that relate energy spectra to source depth distributions. Based on this analysis, a new nonlinear reconstruction algorithm that avoids the earlier problems is proposed. The new algorithm does not degrade spatial resolution in the imaging plane and provides depth resolution with a standard deviation of 4 cm for point sources without requiring any camera motion. The algorithm also provides significant attenuation correction and, therefore, improved quantitation of the source distribution.

© 1992 Optical Society of America

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  1. D. L. Gunter, K. Hoffmann, R. N. Beck, “Three-dimensional imaging utilizing energy discrimination I,”IEEE Trans. Nucl. Sci. 37, 1300–1307 (1990).
  2. R. J. Jaszczak, C. E. Floyd, R. E. Coleman, “Scatter compensation techniques for SPECT,”IEEE Trans. Nucl. Sci. NS-32, 786–793 (1985).
  3. M. Singh, M. Horne, D. Maneval, J. Amartey, R. Brechner, “Non-uniform attenuation and scatter correction in SPECT,”IEEE Trans. Nucl. Sci. 35, 767–771 (1988).
  4. C. E. Floyd, R. J. Jaszczak, K. L. Greer, R. E. Coleman, “Inverse Monte Carlo: a unified reconstruction algorithm for SPECT,”IEEE Trans. Nucl. Sci. NS-32, 779–785 (1985).
  5. K. F. Koral, A. R. Johnston, “Estimation of organ depth by gamma ray spectral comparison,” Phys. Med. Biol. 22, 988–993 (1977).
  6. S. D. Egbert, R. S. May, “A spectral unfolding method to determine source depth distribution,” Phys. Med. Biol. 25, 453–461 (1980).
  7. D. Gagnon, A. Todd-Pokropek, A. Arsenualt, G. Dupras, “A method for the correction of scatter using energy information: holospectral imaging,”J. Nucl. Med. 29, 864 (1988); “Introduction to holospectral imaging in nuclear medicine for scatter subtraction,”IEEE Trans. Med. Imag. 8, 245–250 (1989); D. Gagnon, N. Pauliot, L. Laperriere, A. Arsenault, J. Gregoire, G. Dupras, “Post acquisition linearity correction for holospectral images in nuclear medicine,”IEEE Trans. Nucl. Sci. 37, 621–626 (1990).
  8. A. Siemens Gammasonics ZLC Digitrac 750 Camera was used for all our measurements. The HRP (high resolution parallel hole) collimator is a commercial collimator provided by Siemens Gammasonics with the camera. For a typical HRP collimator, the sensitivity is $ = 1.43 × 10−4, and the (collimator) resolution of a point source located 10 cm in front of the collimator is 0.63 cm FWHM. Details concerning HRP collimators are discussed in R. N. Beck, L. D. Redtung, “Collimator design using ray-tracing techniques,” IEEE Trans. Nucl. Sci. NS-32, 866–869 (1985).
  9. The distinction between Greek and lowercase Latin subscripted indices is significant. These subscripted indices indicate membership in a vector space. For Greek indices this vector space is the 16-dimensional space associated with source depth, while for lowercase Latin indices the vector space is the 45-dimensional space associated with the observed energy. This notation can be helpful in imaging theory, in which the object (source) space and the image space may have completely different structures. Any object in the source space must have a Greek index indicating source depth [e.g., the origin source (ρα) or a reconstruction (rα)], and any image must have a lowercase Latin index (e.g., Im) indicating a complete set of measurements in all the energy windows. Furthermore, the index structure of matrices, such as the kernel matrixK˜mα, indicates a mapping from object space to image space. The Einstein summation convention on Greek or lowercase Latin indices [which is introduced just before Eq. (7)] states that repeated indices are to be interpreted as an inner product in the appropriate vector space.The abstract indices are not intended to indicate specific components of the vectors but can be interpreted as specific components if desired. Occasionally, however, reference to a specific component of a vector or matrix is required. In those cases, I place parentheses around the index to indicate that a special component is being considered. Generally, this notation is needed when the SVD produces singular values that are, in turn, associated with special vectors. The set of singular values σ(i)does not constitute a vector space, but each singular value is associated with a source depth vector [D(i)α] and an energy spectrum [Wm(i)]. Because the indices in parentheses represent specific components of vectors, there is no summation convention implied by repetition of parenthetical indices.The harmonic indices, which are represented by uppercase Latin letters and are introduced in Eq. (43), label special spatial frequencies (QMN). Once again, because there is no vector space structure associated with the harmonic indices, no summation convention is implied by repeated indices.
  10. The specific energy windows are described by a central energy and a percentage window around that central energy. For the experiments described in this paper, the energy windows (in keV) are 44 (6%), 50 (6%), 56 (6%), 60 (4%), 66 (3%), 70 (3%), 74 (3%), 76 (3%), 80 (3%), 84 (2%), 86 (2%), 90 (2%), 94 (2%), 98 (2%), 100 (1%), 102 (1%), 104 (1%), 106 (2%), 108 (1%), 110 (1%), 112 (1%), 114 (1%), 116 (1%), 118 (1%), 120 (1%), 122 (1%), 124 (1%), 126 (1%), 128 (1%), 130 (1%), 132 (1%), 134 (1%), 136 (1%), 138 (1%), 140 (1%), 142 (1%), 144 (1%), 146 (1%), 148 (1%), 150 (1%), 152 (1%), 154 (1%), 156 (1%), 160 (1%), and 164 (21%).
  11. G. H. Golub, C. F. Van Loan, Matrix Computations, 2nd ed. (Johns Hopkins U. Press, Baltimore, Md., 1989).
  12. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986).
  13. To accumulate more counts, the energy window used in nuclear medicine is usually set much wider than in this study. The peak energy window [as defined in Eq. (39)] is centered at 140 keV and is only 1.4 keV (1%) wide. In normal clinical imaging, such a narrow energy window would produce substantial image noise owing to low counting statistics. Our data contain many more counts than a typical clinical image and, consequently, are unrealistically immune to this noise. For more realistic imaging, the peak energy channel ℘mmust be selected with a broader energy window around the photoemission peak. Considerable research has been done to optimize this energy window in traditional nuclear medicine [for example, see F. B. Aktins, R. N. Beck, P. B. Hoffer, D. Palmer, “Dependence of optimal baseline setting on scatter fraction and detector response function,” in Medical Radionuclide Imaging (International Atomic Energy Agency, Vienna, 1977), Vol. 1, pp. 101–118]. The 15% energy window that is used in the nuclear medicine clinic at the University of Chicago is a typical compromise between image noise and the loss of contrast owing to scattered radiation. In the nonlinear algorithm a similar compromise is necessary for the peak energy channel. According to Eq. (41), the peak image acts as a masking function that suppresses the nodal pattern and provides most of the high-spatial-frequency information in the reconstruction. Thus a sharp, high-resolution image ℘(x) is the crucial requirement for the peak energy channel. Indeed, the exclusion of scattered radiation may not be the best way to ensure a high-resolution peak image. In recent years the idea of energy-weighted acquisition (EWA) has replaced the simple energy window as the best method to achieve such high-resolution images. If the peak image ℘(x) is generated by using EWA, then the peak energy channel ℘mcorresponds to what is called the energy weighting function in EWA(which is not the same as the energy weighting functions in this paper). Two papers that discuss the selection of this energy weighting function in EWA are as follows: J. R. Halama, R. E. Henkin, L. E. Friend, “Gamma camera radionuclide images: improved contrast with energy-weighted acquisition,” Radiology 169, 533–538 (1988); R. P. DeVito, J. J. Hamill, “Determination of weighting functions for energy-weighted acquisition,” J. Nucl. Med. 32, 343–349 (1991). Whether a broader energy window or EWA is used to produce the peak image, the reconstructed results are not expected to differ significantly from those of the present study.

1990 (1)

D. L. Gunter, K. Hoffmann, R. N. Beck, “Three-dimensional imaging utilizing energy discrimination I,”IEEE Trans. Nucl. Sci. 37, 1300–1307 (1990).

1988 (2)

M. Singh, M. Horne, D. Maneval, J. Amartey, R. Brechner, “Non-uniform attenuation and scatter correction in SPECT,”IEEE Trans. Nucl. Sci. 35, 767–771 (1988).

D. Gagnon, A. Todd-Pokropek, A. Arsenualt, G. Dupras, “A method for the correction of scatter using energy information: holospectral imaging,”J. Nucl. Med. 29, 864 (1988); “Introduction to holospectral imaging in nuclear medicine for scatter subtraction,”IEEE Trans. Med. Imag. 8, 245–250 (1989); D. Gagnon, N. Pauliot, L. Laperriere, A. Arsenault, J. Gregoire, G. Dupras, “Post acquisition linearity correction for holospectral images in nuclear medicine,”IEEE Trans. Nucl. Sci. 37, 621–626 (1990).

1985 (3)

A. Siemens Gammasonics ZLC Digitrac 750 Camera was used for all our measurements. The HRP (high resolution parallel hole) collimator is a commercial collimator provided by Siemens Gammasonics with the camera. For a typical HRP collimator, the sensitivity is $ = 1.43 × 10−4, and the (collimator) resolution of a point source located 10 cm in front of the collimator is 0.63 cm FWHM. Details concerning HRP collimators are discussed in R. N. Beck, L. D. Redtung, “Collimator design using ray-tracing techniques,” IEEE Trans. Nucl. Sci. NS-32, 866–869 (1985).

C. E. Floyd, R. J. Jaszczak, K. L. Greer, R. E. Coleman, “Inverse Monte Carlo: a unified reconstruction algorithm for SPECT,”IEEE Trans. Nucl. Sci. NS-32, 779–785 (1985).

R. J. Jaszczak, C. E. Floyd, R. E. Coleman, “Scatter compensation techniques for SPECT,”IEEE Trans. Nucl. Sci. NS-32, 786–793 (1985).

1980 (1)

S. D. Egbert, R. S. May, “A spectral unfolding method to determine source depth distribution,” Phys. Med. Biol. 25, 453–461 (1980).

1977 (1)

K. F. Koral, A. R. Johnston, “Estimation of organ depth by gamma ray spectral comparison,” Phys. Med. Biol. 22, 988–993 (1977).

Aktins, F. B.

To accumulate more counts, the energy window used in nuclear medicine is usually set much wider than in this study. The peak energy window [as defined in Eq. (39)] is centered at 140 keV and is only 1.4 keV (1%) wide. In normal clinical imaging, such a narrow energy window would produce substantial image noise owing to low counting statistics. Our data contain many more counts than a typical clinical image and, consequently, are unrealistically immune to this noise. For more realistic imaging, the peak energy channel ℘mmust be selected with a broader energy window around the photoemission peak. Considerable research has been done to optimize this energy window in traditional nuclear medicine [for example, see F. B. Aktins, R. N. Beck, P. B. Hoffer, D. Palmer, “Dependence of optimal baseline setting on scatter fraction and detector response function,” in Medical Radionuclide Imaging (International Atomic Energy Agency, Vienna, 1977), Vol. 1, pp. 101–118]. The 15% energy window that is used in the nuclear medicine clinic at the University of Chicago is a typical compromise between image noise and the loss of contrast owing to scattered radiation. In the nonlinear algorithm a similar compromise is necessary for the peak energy channel. According to Eq. (41), the peak image acts as a masking function that suppresses the nodal pattern and provides most of the high-spatial-frequency information in the reconstruction. Thus a sharp, high-resolution image ℘(x) is the crucial requirement for the peak energy channel. Indeed, the exclusion of scattered radiation may not be the best way to ensure a high-resolution peak image. In recent years the idea of energy-weighted acquisition (EWA) has replaced the simple energy window as the best method to achieve such high-resolution images. If the peak image ℘(x) is generated by using EWA, then the peak energy channel ℘mcorresponds to what is called the energy weighting function in EWA(which is not the same as the energy weighting functions in this paper). Two papers that discuss the selection of this energy weighting function in EWA are as follows: J. R. Halama, R. E. Henkin, L. E. Friend, “Gamma camera radionuclide images: improved contrast with energy-weighted acquisition,” Radiology 169, 533–538 (1988); R. P. DeVito, J. J. Hamill, “Determination of weighting functions for energy-weighted acquisition,” J. Nucl. Med. 32, 343–349 (1991). Whether a broader energy window or EWA is used to produce the peak image, the reconstructed results are not expected to differ significantly from those of the present study.

Amartey, J.

M. Singh, M. Horne, D. Maneval, J. Amartey, R. Brechner, “Non-uniform attenuation and scatter correction in SPECT,”IEEE Trans. Nucl. Sci. 35, 767–771 (1988).

Arsenualt, A.

D. Gagnon, A. Todd-Pokropek, A. Arsenualt, G. Dupras, “A method for the correction of scatter using energy information: holospectral imaging,”J. Nucl. Med. 29, 864 (1988); “Introduction to holospectral imaging in nuclear medicine for scatter subtraction,”IEEE Trans. Med. Imag. 8, 245–250 (1989); D. Gagnon, N. Pauliot, L. Laperriere, A. Arsenault, J. Gregoire, G. Dupras, “Post acquisition linearity correction for holospectral images in nuclear medicine,”IEEE Trans. Nucl. Sci. 37, 621–626 (1990).

Beck, R. N.

D. L. Gunter, K. Hoffmann, R. N. Beck, “Three-dimensional imaging utilizing energy discrimination I,”IEEE Trans. Nucl. Sci. 37, 1300–1307 (1990).

A. Siemens Gammasonics ZLC Digitrac 750 Camera was used for all our measurements. The HRP (high resolution parallel hole) collimator is a commercial collimator provided by Siemens Gammasonics with the camera. For a typical HRP collimator, the sensitivity is $ = 1.43 × 10−4, and the (collimator) resolution of a point source located 10 cm in front of the collimator is 0.63 cm FWHM. Details concerning HRP collimators are discussed in R. N. Beck, L. D. Redtung, “Collimator design using ray-tracing techniques,” IEEE Trans. Nucl. Sci. NS-32, 866–869 (1985).

To accumulate more counts, the energy window used in nuclear medicine is usually set much wider than in this study. The peak energy window [as defined in Eq. (39)] is centered at 140 keV and is only 1.4 keV (1%) wide. In normal clinical imaging, such a narrow energy window would produce substantial image noise owing to low counting statistics. Our data contain many more counts than a typical clinical image and, consequently, are unrealistically immune to this noise. For more realistic imaging, the peak energy channel ℘mmust be selected with a broader energy window around the photoemission peak. Considerable research has been done to optimize this energy window in traditional nuclear medicine [for example, see F. B. Aktins, R. N. Beck, P. B. Hoffer, D. Palmer, “Dependence of optimal baseline setting on scatter fraction and detector response function,” in Medical Radionuclide Imaging (International Atomic Energy Agency, Vienna, 1977), Vol. 1, pp. 101–118]. The 15% energy window that is used in the nuclear medicine clinic at the University of Chicago is a typical compromise between image noise and the loss of contrast owing to scattered radiation. In the nonlinear algorithm a similar compromise is necessary for the peak energy channel. According to Eq. (41), the peak image acts as a masking function that suppresses the nodal pattern and provides most of the high-spatial-frequency information in the reconstruction. Thus a sharp, high-resolution image ℘(x) is the crucial requirement for the peak energy channel. Indeed, the exclusion of scattered radiation may not be the best way to ensure a high-resolution peak image. In recent years the idea of energy-weighted acquisition (EWA) has replaced the simple energy window as the best method to achieve such high-resolution images. If the peak image ℘(x) is generated by using EWA, then the peak energy channel ℘mcorresponds to what is called the energy weighting function in EWA(which is not the same as the energy weighting functions in this paper). Two papers that discuss the selection of this energy weighting function in EWA are as follows: J. R. Halama, R. E. Henkin, L. E. Friend, “Gamma camera radionuclide images: improved contrast with energy-weighted acquisition,” Radiology 169, 533–538 (1988); R. P. DeVito, J. J. Hamill, “Determination of weighting functions for energy-weighted acquisition,” J. Nucl. Med. 32, 343–349 (1991). Whether a broader energy window or EWA is used to produce the peak image, the reconstructed results are not expected to differ significantly from those of the present study.

Brechner, R.

M. Singh, M. Horne, D. Maneval, J. Amartey, R. Brechner, “Non-uniform attenuation and scatter correction in SPECT,”IEEE Trans. Nucl. Sci. 35, 767–771 (1988).

Coleman, R. E.

C. E. Floyd, R. J. Jaszczak, K. L. Greer, R. E. Coleman, “Inverse Monte Carlo: a unified reconstruction algorithm for SPECT,”IEEE Trans. Nucl. Sci. NS-32, 779–785 (1985).

R. J. Jaszczak, C. E. Floyd, R. E. Coleman, “Scatter compensation techniques for SPECT,”IEEE Trans. Nucl. Sci. NS-32, 786–793 (1985).

Dupras, G.

D. Gagnon, A. Todd-Pokropek, A. Arsenualt, G. Dupras, “A method for the correction of scatter using energy information: holospectral imaging,”J. Nucl. Med. 29, 864 (1988); “Introduction to holospectral imaging in nuclear medicine for scatter subtraction,”IEEE Trans. Med. Imag. 8, 245–250 (1989); D. Gagnon, N. Pauliot, L. Laperriere, A. Arsenault, J. Gregoire, G. Dupras, “Post acquisition linearity correction for holospectral images in nuclear medicine,”IEEE Trans. Nucl. Sci. 37, 621–626 (1990).

Egbert, S. D.

S. D. Egbert, R. S. May, “A spectral unfolding method to determine source depth distribution,” Phys. Med. Biol. 25, 453–461 (1980).

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986).

Floyd, C. E.

R. J. Jaszczak, C. E. Floyd, R. E. Coleman, “Scatter compensation techniques for SPECT,”IEEE Trans. Nucl. Sci. NS-32, 786–793 (1985).

C. E. Floyd, R. J. Jaszczak, K. L. Greer, R. E. Coleman, “Inverse Monte Carlo: a unified reconstruction algorithm for SPECT,”IEEE Trans. Nucl. Sci. NS-32, 779–785 (1985).

Gagnon, D.

D. Gagnon, A. Todd-Pokropek, A. Arsenualt, G. Dupras, “A method for the correction of scatter using energy information: holospectral imaging,”J. Nucl. Med. 29, 864 (1988); “Introduction to holospectral imaging in nuclear medicine for scatter subtraction,”IEEE Trans. Med. Imag. 8, 245–250 (1989); D. Gagnon, N. Pauliot, L. Laperriere, A. Arsenault, J. Gregoire, G. Dupras, “Post acquisition linearity correction for holospectral images in nuclear medicine,”IEEE Trans. Nucl. Sci. 37, 621–626 (1990).

Golub, G. H.

G. H. Golub, C. F. Van Loan, Matrix Computations, 2nd ed. (Johns Hopkins U. Press, Baltimore, Md., 1989).

Greer, K. L.

C. E. Floyd, R. J. Jaszczak, K. L. Greer, R. E. Coleman, “Inverse Monte Carlo: a unified reconstruction algorithm for SPECT,”IEEE Trans. Nucl. Sci. NS-32, 779–785 (1985).

Gunter, D. L.

D. L. Gunter, K. Hoffmann, R. N. Beck, “Three-dimensional imaging utilizing energy discrimination I,”IEEE Trans. Nucl. Sci. 37, 1300–1307 (1990).

Hoffer, P. B.

To accumulate more counts, the energy window used in nuclear medicine is usually set much wider than in this study. The peak energy window [as defined in Eq. (39)] is centered at 140 keV and is only 1.4 keV (1%) wide. In normal clinical imaging, such a narrow energy window would produce substantial image noise owing to low counting statistics. Our data contain many more counts than a typical clinical image and, consequently, are unrealistically immune to this noise. For more realistic imaging, the peak energy channel ℘mmust be selected with a broader energy window around the photoemission peak. Considerable research has been done to optimize this energy window in traditional nuclear medicine [for example, see F. B. Aktins, R. N. Beck, P. B. Hoffer, D. Palmer, “Dependence of optimal baseline setting on scatter fraction and detector response function,” in Medical Radionuclide Imaging (International Atomic Energy Agency, Vienna, 1977), Vol. 1, pp. 101–118]. The 15% energy window that is used in the nuclear medicine clinic at the University of Chicago is a typical compromise between image noise and the loss of contrast owing to scattered radiation. In the nonlinear algorithm a similar compromise is necessary for the peak energy channel. According to Eq. (41), the peak image acts as a masking function that suppresses the nodal pattern and provides most of the high-spatial-frequency information in the reconstruction. Thus a sharp, high-resolution image ℘(x) is the crucial requirement for the peak energy channel. Indeed, the exclusion of scattered radiation may not be the best way to ensure a high-resolution peak image. In recent years the idea of energy-weighted acquisition (EWA) has replaced the simple energy window as the best method to achieve such high-resolution images. If the peak image ℘(x) is generated by using EWA, then the peak energy channel ℘mcorresponds to what is called the energy weighting function in EWA(which is not the same as the energy weighting functions in this paper). Two papers that discuss the selection of this energy weighting function in EWA are as follows: J. R. Halama, R. E. Henkin, L. E. Friend, “Gamma camera radionuclide images: improved contrast with energy-weighted acquisition,” Radiology 169, 533–538 (1988); R. P. DeVito, J. J. Hamill, “Determination of weighting functions for energy-weighted acquisition,” J. Nucl. Med. 32, 343–349 (1991). Whether a broader energy window or EWA is used to produce the peak image, the reconstructed results are not expected to differ significantly from those of the present study.

Hoffmann, K.

D. L. Gunter, K. Hoffmann, R. N. Beck, “Three-dimensional imaging utilizing energy discrimination I,”IEEE Trans. Nucl. Sci. 37, 1300–1307 (1990).

Horne, M.

M. Singh, M. Horne, D. Maneval, J. Amartey, R. Brechner, “Non-uniform attenuation and scatter correction in SPECT,”IEEE Trans. Nucl. Sci. 35, 767–771 (1988).

Jaszczak, R. J.

R. J. Jaszczak, C. E. Floyd, R. E. Coleman, “Scatter compensation techniques for SPECT,”IEEE Trans. Nucl. Sci. NS-32, 786–793 (1985).

C. E. Floyd, R. J. Jaszczak, K. L. Greer, R. E. Coleman, “Inverse Monte Carlo: a unified reconstruction algorithm for SPECT,”IEEE Trans. Nucl. Sci. NS-32, 779–785 (1985).

Johnston, A. R.

K. F. Koral, A. R. Johnston, “Estimation of organ depth by gamma ray spectral comparison,” Phys. Med. Biol. 22, 988–993 (1977).

Koral, K. F.

K. F. Koral, A. R. Johnston, “Estimation of organ depth by gamma ray spectral comparison,” Phys. Med. Biol. 22, 988–993 (1977).

Maneval, D.

M. Singh, M. Horne, D. Maneval, J. Amartey, R. Brechner, “Non-uniform attenuation and scatter correction in SPECT,”IEEE Trans. Nucl. Sci. 35, 767–771 (1988).

May, R. S.

S. D. Egbert, R. S. May, “A spectral unfolding method to determine source depth distribution,” Phys. Med. Biol. 25, 453–461 (1980).

Palmer, D.

To accumulate more counts, the energy window used in nuclear medicine is usually set much wider than in this study. The peak energy window [as defined in Eq. (39)] is centered at 140 keV and is only 1.4 keV (1%) wide. In normal clinical imaging, such a narrow energy window would produce substantial image noise owing to low counting statistics. Our data contain many more counts than a typical clinical image and, consequently, are unrealistically immune to this noise. For more realistic imaging, the peak energy channel ℘mmust be selected with a broader energy window around the photoemission peak. Considerable research has been done to optimize this energy window in traditional nuclear medicine [for example, see F. B. Aktins, R. N. Beck, P. B. Hoffer, D. Palmer, “Dependence of optimal baseline setting on scatter fraction and detector response function,” in Medical Radionuclide Imaging (International Atomic Energy Agency, Vienna, 1977), Vol. 1, pp. 101–118]. The 15% energy window that is used in the nuclear medicine clinic at the University of Chicago is a typical compromise between image noise and the loss of contrast owing to scattered radiation. In the nonlinear algorithm a similar compromise is necessary for the peak energy channel. According to Eq. (41), the peak image acts as a masking function that suppresses the nodal pattern and provides most of the high-spatial-frequency information in the reconstruction. Thus a sharp, high-resolution image ℘(x) is the crucial requirement for the peak energy channel. Indeed, the exclusion of scattered radiation may not be the best way to ensure a high-resolution peak image. In recent years the idea of energy-weighted acquisition (EWA) has replaced the simple energy window as the best method to achieve such high-resolution images. If the peak image ℘(x) is generated by using EWA, then the peak energy channel ℘mcorresponds to what is called the energy weighting function in EWA(which is not the same as the energy weighting functions in this paper). Two papers that discuss the selection of this energy weighting function in EWA are as follows: J. R. Halama, R. E. Henkin, L. E. Friend, “Gamma camera radionuclide images: improved contrast with energy-weighted acquisition,” Radiology 169, 533–538 (1988); R. P. DeVito, J. J. Hamill, “Determination of weighting functions for energy-weighted acquisition,” J. Nucl. Med. 32, 343–349 (1991). Whether a broader energy window or EWA is used to produce the peak image, the reconstructed results are not expected to differ significantly from those of the present study.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986).

Redtung, L. D.

A. Siemens Gammasonics ZLC Digitrac 750 Camera was used for all our measurements. The HRP (high resolution parallel hole) collimator is a commercial collimator provided by Siemens Gammasonics with the camera. For a typical HRP collimator, the sensitivity is $ = 1.43 × 10−4, and the (collimator) resolution of a point source located 10 cm in front of the collimator is 0.63 cm FWHM. Details concerning HRP collimators are discussed in R. N. Beck, L. D. Redtung, “Collimator design using ray-tracing techniques,” IEEE Trans. Nucl. Sci. NS-32, 866–869 (1985).

Singh, M.

M. Singh, M. Horne, D. Maneval, J. Amartey, R. Brechner, “Non-uniform attenuation and scatter correction in SPECT,”IEEE Trans. Nucl. Sci. 35, 767–771 (1988).

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986).

Todd-Pokropek, A.

D. Gagnon, A. Todd-Pokropek, A. Arsenualt, G. Dupras, “A method for the correction of scatter using energy information: holospectral imaging,”J. Nucl. Med. 29, 864 (1988); “Introduction to holospectral imaging in nuclear medicine for scatter subtraction,”IEEE Trans. Med. Imag. 8, 245–250 (1989); D. Gagnon, N. Pauliot, L. Laperriere, A. Arsenault, J. Gregoire, G. Dupras, “Post acquisition linearity correction for holospectral images in nuclear medicine,”IEEE Trans. Nucl. Sci. 37, 621–626 (1990).

Van Loan, C. F.

G. H. Golub, C. F. Van Loan, Matrix Computations, 2nd ed. (Johns Hopkins U. Press, Baltimore, Md., 1989).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986).

IEEE Trans. Nucl. Sci. (5)

D. L. Gunter, K. Hoffmann, R. N. Beck, “Three-dimensional imaging utilizing energy discrimination I,”IEEE Trans. Nucl. Sci. 37, 1300–1307 (1990).

R. J. Jaszczak, C. E. Floyd, R. E. Coleman, “Scatter compensation techniques for SPECT,”IEEE Trans. Nucl. Sci. NS-32, 786–793 (1985).

M. Singh, M. Horne, D. Maneval, J. Amartey, R. Brechner, “Non-uniform attenuation and scatter correction in SPECT,”IEEE Trans. Nucl. Sci. 35, 767–771 (1988).

C. E. Floyd, R. J. Jaszczak, K. L. Greer, R. E. Coleman, “Inverse Monte Carlo: a unified reconstruction algorithm for SPECT,”IEEE Trans. Nucl. Sci. NS-32, 779–785 (1985).

A. Siemens Gammasonics ZLC Digitrac 750 Camera was used for all our measurements. The HRP (high resolution parallel hole) collimator is a commercial collimator provided by Siemens Gammasonics with the camera. For a typical HRP collimator, the sensitivity is $ = 1.43 × 10−4, and the (collimator) resolution of a point source located 10 cm in front of the collimator is 0.63 cm FWHM. Details concerning HRP collimators are discussed in R. N. Beck, L. D. Redtung, “Collimator design using ray-tracing techniques,” IEEE Trans. Nucl. Sci. NS-32, 866–869 (1985).

J. Nucl. Med. (1)

D. Gagnon, A. Todd-Pokropek, A. Arsenualt, G. Dupras, “A method for the correction of scatter using energy information: holospectral imaging,”J. Nucl. Med. 29, 864 (1988); “Introduction to holospectral imaging in nuclear medicine for scatter subtraction,”IEEE Trans. Med. Imag. 8, 245–250 (1989); D. Gagnon, N. Pauliot, L. Laperriere, A. Arsenault, J. Gregoire, G. Dupras, “Post acquisition linearity correction for holospectral images in nuclear medicine,”IEEE Trans. Nucl. Sci. 37, 621–626 (1990).

Phys. Med. Biol. (2)

K. F. Koral, A. R. Johnston, “Estimation of organ depth by gamma ray spectral comparison,” Phys. Med. Biol. 22, 988–993 (1977).

S. D. Egbert, R. S. May, “A spectral unfolding method to determine source depth distribution,” Phys. Med. Biol. 25, 453–461 (1980).

Other (5)

The distinction between Greek and lowercase Latin subscripted indices is significant. These subscripted indices indicate membership in a vector space. For Greek indices this vector space is the 16-dimensional space associated with source depth, while for lowercase Latin indices the vector space is the 45-dimensional space associated with the observed energy. This notation can be helpful in imaging theory, in which the object (source) space and the image space may have completely different structures. Any object in the source space must have a Greek index indicating source depth [e.g., the origin source (ρα) or a reconstruction (rα)], and any image must have a lowercase Latin index (e.g., Im) indicating a complete set of measurements in all the energy windows. Furthermore, the index structure of matrices, such as the kernel matrixK˜mα, indicates a mapping from object space to image space. The Einstein summation convention on Greek or lowercase Latin indices [which is introduced just before Eq. (7)] states that repeated indices are to be interpreted as an inner product in the appropriate vector space.The abstract indices are not intended to indicate specific components of the vectors but can be interpreted as specific components if desired. Occasionally, however, reference to a specific component of a vector or matrix is required. In those cases, I place parentheses around the index to indicate that a special component is being considered. Generally, this notation is needed when the SVD produces singular values that are, in turn, associated with special vectors. The set of singular values σ(i)does not constitute a vector space, but each singular value is associated with a source depth vector [D(i)α] and an energy spectrum [Wm(i)]. Because the indices in parentheses represent specific components of vectors, there is no summation convention implied by repetition of parenthetical indices.The harmonic indices, which are represented by uppercase Latin letters and are introduced in Eq. (43), label special spatial frequencies (QMN). Once again, because there is no vector space structure associated with the harmonic indices, no summation convention is implied by repeated indices.

The specific energy windows are described by a central energy and a percentage window around that central energy. For the experiments described in this paper, the energy windows (in keV) are 44 (6%), 50 (6%), 56 (6%), 60 (4%), 66 (3%), 70 (3%), 74 (3%), 76 (3%), 80 (3%), 84 (2%), 86 (2%), 90 (2%), 94 (2%), 98 (2%), 100 (1%), 102 (1%), 104 (1%), 106 (2%), 108 (1%), 110 (1%), 112 (1%), 114 (1%), 116 (1%), 118 (1%), 120 (1%), 122 (1%), 124 (1%), 126 (1%), 128 (1%), 130 (1%), 132 (1%), 134 (1%), 136 (1%), 138 (1%), 140 (1%), 142 (1%), 144 (1%), 146 (1%), 148 (1%), 150 (1%), 152 (1%), 154 (1%), 156 (1%), 160 (1%), and 164 (21%).

G. H. Golub, C. F. Van Loan, Matrix Computations, 2nd ed. (Johns Hopkins U. Press, Baltimore, Md., 1989).

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986).

To accumulate more counts, the energy window used in nuclear medicine is usually set much wider than in this study. The peak energy window [as defined in Eq. (39)] is centered at 140 keV and is only 1.4 keV (1%) wide. In normal clinical imaging, such a narrow energy window would produce substantial image noise owing to low counting statistics. Our data contain many more counts than a typical clinical image and, consequently, are unrealistically immune to this noise. For more realistic imaging, the peak energy channel ℘mmust be selected with a broader energy window around the photoemission peak. Considerable research has been done to optimize this energy window in traditional nuclear medicine [for example, see F. B. Aktins, R. N. Beck, P. B. Hoffer, D. Palmer, “Dependence of optimal baseline setting on scatter fraction and detector response function,” in Medical Radionuclide Imaging (International Atomic Energy Agency, Vienna, 1977), Vol. 1, pp. 101–118]. The 15% energy window that is used in the nuclear medicine clinic at the University of Chicago is a typical compromise between image noise and the loss of contrast owing to scattered radiation. In the nonlinear algorithm a similar compromise is necessary for the peak energy channel. According to Eq. (41), the peak image acts as a masking function that suppresses the nodal pattern and provides most of the high-spatial-frequency information in the reconstruction. Thus a sharp, high-resolution image ℘(x) is the crucial requirement for the peak energy channel. Indeed, the exclusion of scattered radiation may not be the best way to ensure a high-resolution peak image. In recent years the idea of energy-weighted acquisition (EWA) has replaced the simple energy window as the best method to achieve such high-resolution images. If the peak image ℘(x) is generated by using EWA, then the peak energy channel ℘mcorresponds to what is called the energy weighting function in EWA(which is not the same as the energy weighting functions in this paper). Two papers that discuss the selection of this energy weighting function in EWA are as follows: J. R. Halama, R. E. Henkin, L. E. Friend, “Gamma camera radionuclide images: improved contrast with energy-weighted acquisition,” Radiology 169, 533–538 (1988); R. P. DeVito, J. J. Hamill, “Determination of weighting functions for energy-weighted acquisition,” J. Nucl. Med. 32, 343–349 (1991). Whether a broader energy window or EWA is used to produce the peak image, the reconstructed results are not expected to differ significantly from those of the present study.

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Figures (16)

Fig. 1
Fig. 1

The four largest singular values, σi, of the imaging kernel, K ˜ m α(|ν|), as a function of the spatial frequency up to the Nyquist frequency of the camera.

Fig. 2
Fig. 2

The four largest singular values, σi, of the imaging kernel, K ˜ m α(|ν|), as a function of the spatial frequency up to 0.75 cm−1.

Fig. 3
Fig. 3

(a) Depth distribution functions associated with channel 1 for spatial frequencies ν = 0.0, 0.023, 0.046, and 0.069 cm−1. (b) Energy weighting functions associated with channel 1 for spatial frequencies ν = 0.0, 0.023, 0.046, and 0.069 cm−1.

Fig. 4
Fig. 4

(a) Depth distribution functions associated with channel 2 for spatial frequencies ν = 0.0, 0.023, 0.046, and 0.069 cm−1. (b) Energy weighting functions associated with channel 2 for spatial frequencies ν = 0.0, 0.023, 0.046, and 0.069 cm−1.

Fig. 5
Fig. 5

(a) Depth distribution functions associated with channel 3 for spatial frequencies ν = 0.0, 0.023, 0.046, and 0.069 cm−1. (b) Energy weighting functions associated with channel 3 for spatial frequencies ν = 0.0, 0.023, 0.046, and 0.069 cm−1.

Fig. 6
Fig. 6

(a) Depth distribution functions associated with channel 4 for spatial frequencies ν = 0.0, 0.023, 0.046, and 0.069 cm−1. (b) Energy weighting functions associated with channel 4 for spatial frequencies ν = 0.0, 0.023, 0.046, and 0.069 cm−1.

Fig. 7
Fig. 7

Number of channels N(ν) having singular values larger than the Tikhonov parameter as a function of spatial frequency for σTik = 0.02, 0.035, and 0.05 σmax.

Fig. 8
Fig. 8

Number of projected channels n(ν) [as defined by Eq. (30)] as a function of spatial frequency for σTik = 0.02, 0.035, and 0.05 σMax.

Fig. 9
Fig. 9

Projected fractions of the depth distributions, f(i) [as defined by Eq. (32)], for σTik = 0.035 σMax as a function of spatial frequency for channels i 1 through 4.

Fig. 10
Fig. 10

Projected fractions of the energy weighting functions, e(i) [as defined by Eq. (38)], for σTik = 0.035 σMax as a function of spatial frequency for channels i 1 through 4.

Fig. 11
Fig. 11

Schematic drawing of the simulated four-point source phantom. The point sources are indicated by stars (*), and the cylindrical ROI’s that are used to estimate the depth resolution of the algorithm are displayed by dashed lines.

Fig. 12
Fig. 12

Reconstruction of the four-point-source phantom (see Fig. 11) for the nonlinear algorithm with parameters σTik = 0.035 σMax, HMax = 4, MMat = 8, and μTik = 0.01 μMax. The reconstructed source density is proportional to the local density of points in the figure. Negative source densities were suppressed as nonphysical.

Fig. 13
Fig. 13

(a) Peak image (x) produced by a simulated four-point-source phantom as a function of coordinates (x, y) in the imaging plane. (b) Function C1(x) associated with channel 1 [as defined by Eq. (64)] as a function of coordinates (x, y) in the imaging plane. The function C1(x) was evaluated by using the nonlinear algorithm with parameters σTik = 0.035 σMax, HMax = 4, MMat = 8, and μTik = 0.01 μMax. (c) Function C2(x) associated with channel 2 [as defined by Eq. (64)] as a function of coordinates (x, y) in the imaging plane. The function C2(x) was evaluated by using the nonlinear algorithm with parameters σTik = 0.0350 σMax, HMax = 4, MMat = 8, and μTik = 0.01 μMax. (d) Function C3(x) associated with channel 3 [as defined by Eq. (64)] as a function of coordinates (x, y) in the imaging plane. The function C3(x) was evaluated by using the nonlinear algorithm with parameters σTik = 0.035 σMax, HMax = 4, MMat = 8, and μTik = 0.01 μMax. (e) Function C1(x) associated with channel 1 [as defined by Eq. (64)] as a function of coordinates (x, y) in the imaging plane. The function C1(x) was evaluated by using the nonlinear algorithm with parameters σTik = 0.035 σMax, HMax = 3, MMat = 8, and μTik = 0.005 μMax. The oscillatory nature of this function indicates amplification of noise in the reconstruction process because the parameter μTik was too low.

Fig. 14
Fig. 14

Relative source density as a function of depth in the four ROI’s. The source density was evaluated by using the nonlinear algorithm with parameters σTik = 0.035 σMax, HMax = 4, MMat = 8, and μTik = 0.01 μMax.

Fig. 15
Fig. 15

(a) Depth resolution of point sources for the reconstruction algorithms with parameters HMax = 4, MMat = 8, μTik = 0.01 μMax, and σTik = 0.035 σMax (three channels) and 0.050 σMax (two channels). (b) Depth resolution of point sources for the reconstruction algorithms with parameters HMax = 4, MMat = 8, μTik = 0.01 μMax, and σTik = 0.020 σMax (four channels) and 0.050 σMax (two channels).

Fig. 16
Fig. 16

Attenuation corrections measured in each of the ROI’s of Fig. 11 for the reconstruction algorithm with parameters HMax = 4, MMat = 8, μTik = 0.01 μMax, and σTik = 0.035 σMax.

Tables (2)

Tables Icon

Table 1 Number of Independent Coefficients of C ( i ) M N.

Tables Icon

Table 2 Number of Equations Satisfied by C ( i ) M N, Based on the Assumption that σTik = 0.05 σMax

Equations (64)

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I ( x , E ) = 0 d z - d 2 y K ( x - y , E , z ) ρ ( y , z ) + n ( x , E ) ,
K ( x , E , z ) = K ( x , E , z ) .
I ˜ ( ν , E ) = - d 2 x I ( x , E ) exp ( - 2 π i ν · x ) , ρ ˜ ( ν , z ) = = d 2 x ρ ( x , z ) exp ( - 2 π i ν · x ) , K ˜ ( ν , E , z ) = - d 2 x K ( x , E , z ) exp ( - 2 π i ν · x ) , n ˜ ( ν , E ) = - d 2 x n ( x , E ) exp ( - 2 π i ν · x ) .
I ˜ ( ν , E ) = - d z K ˜ ( v , E , z ) ρ ˜ ( ν , z ) + n ˜ ( ν , E ) ,
I ˜ m ( ν ) = α = 1 16 K ˜ m α ( ν ) ρ ˜ α ( ν ) + n ˜ m ( ν ) ,
r ˜ α ( ν ) = m = 1 N E R ˜ α m ( ν ) I ˜ m ( ν ) ,
r ˜ α ( ν ) = Δ ˜ α β ( ν ) ρ ˜ β ( ν ) + R ˜ α m ( ν ) n ˜ m ( ν ) ,
Δ ˜ α β ( ν ) R ˜ α m ( ν ) K ˜ m β ( ν ) .
χ 2 0 d z - d 2 x r ( x , z ) = ρ ( x , z ) 2 = - d 2 x α = 1 16 r α ( x ) - ρ α ( x ) 2 = - d 2 ν α = 1 16 r ˜ α ( ν ) - ρ ˜ α ( ν ) 2 ,
χ 2 ( ν ) α = 1 16 r ˜ α ( ν ) - ρ ˜ α ( ν ) 2 = [ r ˜ α ( ν ) - ρ ˜ α ( ν ) ] ¯ [ r ˜ α ( ν ) - ρ ˜ α ( ν ) ] = [ Δ ˜ α β ( ν ) - δ α β ] ρ ˜ β ( ν ) 2 + R ˜ α m ( ν ) n ˜ m ( ν ) 2 = [ Δ ˜ α β ( ν ) - δ α β ] [ Δ ˜ α γ ( ν ) - δ α γ ] ρ ˜ β ( ν ) ¯ ρ ˜ γ ( ν ) + R ˜ α m ( ν ) R ˜ α n ( ν ) n ˜ m ( ν ) ¯ n ˜ n ( ν ) ,
n ˜ m ( ν ) = 0.
ρ ˜ β ( ν ) ¯ ρ ˜ γ ( ν ) = S 2 δ β γ .
n ˜ m ( ν ) ¯ n ˜ n ( ν ) = N 2 δ m n .
χ 2 ( ν ) = S 2 [ Δ ˜ α β ( ν ) - δ α β ] [ Δ ˜ α β ( ν ) - δ α β ] + N 2 R ˜ α m ( ν ) R ˜ α m ( ν ) .
R ˜ α m [ S 2 K ˜ m β K ˜ n β + N 2 δ m n ] = S 2 K ˜ n α ,
K ˜ m α = W m n Σ n β D β α = W Σ D ,
Σ = diag { σ ( i ) , ( i ) = 1 , , 16 } ,
R ˜ = D T Λ W T
Λ = diag { λ ( i ) , ( i ) = 1 , , 16 } ,
λ ( i ) = σ ( i ) σ ( i ) 2 + σ Tik 2
σ Tik N S .
Π ^ α β ( ν ) = diag { 1 if σ ( α ) ( ν ) σ Tik 0 if σ ( α ) ( ν ) < σ Tik } .
Π ^ 2 ( ν ) = Π ^ ( ν ) .
Σ m α Π ^ α β Σ m β .
Π α β ( ν ) D υ α ( ν ) Π ^ υ ω ( ν ) D ω β ( ν ) = D T Π ^ D .
K ˜ = W Σ D W Σ Π ^ D ( W Σ D ) ( D T Π ^ D ) K ˜ Π ,
Π ^ α β ( ν ) diag { σ ( α ) 2 ( ν ) σ ( α ) 2 ( ν ) + σ Tik 2 } = Λ α m ( ν ) Σ m β ( ν ) ,
r ˜ α ( ν ) Π α β ( ν ) ρ ˜ β ( ν ) + R ˜ α m ( ν ) n ˜ m ( ν ) .
N ( ν ) Π α α ( ν ) = Π ^ α α ( ν ) = [ number of singular values , σ α ( ν ) , such that σ α ( ν ) σ Tik ] .
n ( ν ) Π α β ( 0 ) Π α β ( ν ) .
Π ( 0 ) Π ( ν ) Π ( ν ) , Π ( ν ) Π ( 0 ) Π ( ν ) ,
f ( i ) ( ν ) [ Π α β D ( i ) α ( ν ) D ( i ) β ( ν ) ] 1 / 2 .
P ^ m n ( ν ) diag { 1 if σ ( m ) ( ν ) σ Tik 0 if σ ( m ) ( ν ) < σ Tik 0 if ( m ) > 16 } ,
P m n ( ν ) W m s ( ν ) P ^ s t ( ν ) W n t ( ν ) = W P ^ W T .
K ˜ P K ˜ = K ˜ Π ,
P ^ Σ Λ ,
P K R
e ( i ) ( ν ) [ P m n ( 0 ) W m ( i ) ( ν ) W n ( i ) ( ν ) ] 1 / 2 ,
P ( x ) P m I m ( x ) .
P ˜ ( ν ) = - d 2 x P ( x ) exp ( - 2 π i ν · x ) .
r ^ α ( x ) = P ( x ) c α ( x ) ,
c ˜ α ( ν ) = - d 2 x c α ( x ) exp ( - 2 π i ν · x ) .
Q M N 1 L ( M e ^ 1 + N e ^ 2 ) ,
c α ( x ) = ( i ) = 1 N ( 0 ) D ( i ) α ( 0 ) M N C ( i ) M N exp ( 2 π i Q M N · x ) ,
Q M N L = ( M 2 + N 2 ) 1 / 2 < H Max + 1 / 2 ν c L ,
c ˜ α ( ν ) = ( i ) = 1 N ( 0 ) D ( i ) α ( 0 ) M N C ( i ) M N δ ( ν - Q M N ) .
r ^ ˜ α ( ν ) = ( i ) = 1 N ( 0 ) D ( i ) α ( 0 ) M N C ( i ) M N P ˜ ( ν - Q M N ) .
r ^ α ( x ) = P ( x ) ( i ) = 1 N ( 0 ) D ( i ) α ( 0 ) M N C ( i ) M N exp ( 2 π i Q M N · x ) .
Π α β ( ν ) r ˜ β ( ν ) Π α β ( ν ) ρ ˜ β ( ν ) .
Π α β ( ν ) r ^ ˜ β ( ν ) = Π α β ( ν ) r ˜ β ( ν ) .
( J 2 + K 2 ) 1 / 2 < M Mat + 1 / 2 ,
Π α β ( ν ) r ^ ˜ β ( ν ) = Π α β ( ν ) R ˜ β m ( ν ) I ˜ m ( ν ) .
K ˜ n α ( ν ) Π α β ( ν ) r ^ ˜ β ( ν ) = K ˜ n α ( ν ) Π α β ( ν ) R ˜ β m ( ν ) I ˜ m ( ν ) .
P n j ( ν ) K ˜ j β ( ν ) r ^ ˜ β ( ν ) = P n j ( ν ) K ˜ j β ( ν ) R ˜ β m ( ν ) I ˜ m ( ν ) P n j ( ν ) P j m ( ν ) I ˜ m ( ν ) = P n m ( ν ) I ˜ m ( ν ) .
P K ˜ r ^ ˜ = ( W P ^ W T ) ( W Σ D ) r ^ ˜ = ( W P ^ W T ) I ˜ ,
P ^ n j ( ν ) Σ j α ( ν ) D α β ( ν ) r ^ ˜ β ( ν ) = P ^ n j ( ν ) W m j ( ν ) I ˜ m ( ν ) .
σ ( j ) ( ν ) D ( j ) α ( ν ) ( i ) = 1 N ( 0 ) D ( i ) α ( 0 ) M N C ( i ) M N P ˜ ( ν - Q M N ) = W m ( j ) ( ν ) I ˜ m ( ν ) ,
( i ) = 1 N ( 0 ) M N σ ( j ) ( Q J K ) D ( j ) α ( Q J K ) D ( i ) α ( 0 ) P ˜ ( Q J K - Q M N ) C ( i ) M N = W m ( j ) ( Q j k ) I ˜ m ( Q J K ) .
I ^ ( j ) J K W m ( j ) ( Q J K ) I ˜ m ( Q J K ) ,
M ^ ( j ) ( i ) J K M N σ ( j ) ( Q J K ) D ( j ) α ( Q J K ) D ( i ) α ( 0 ) P ˜ ( Q J K - Q M N ) ,
( i ) M N M ^ ( j ) ( i ) J K M N C ( i ) M N = I ^ ( j ) J K .
S 1 = ( 11.05 , 10.71 , 0.5 ) , S 2 = ( 10.71 , 32.47 , 4.5 ) , S 3 = ( 32.81 , 11.90 , 8.5 ) , S 4 = ( 32.47 , 32.13 , 12.5 )
( fractional error ) = ( M ^ C - I ^ 2 I ^ 2 ) 1 / 2
C i ( x ) D ( i ) α ( 0 ) c α ( x ) = M N C ( i ) M N exp ( 2 π i Q M N · x ) .

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