Abstract

Fourier-based transfer theory is extended into the temporal domain to describe both spatial and temporal noise processes in quantum-based medical imaging systems. Lag is represented as a temporal scatter in which the release of image quanta is delayed according to a probability density function. Expressions describing transfer of the spatiotemporal Wiener noise power spectrum through quantum gain and scatter processes are derived. Lag introduces noise correlations in the temporal domain in proportion to the correlated noise component only. The effect of lag is therefore dependent on both spatial and temporal physical processes. A simple model of a fluoroscopic system shows that image noise is reduced by a factor that is similar to Wagner’s information bandwidth integral, which depends on the temporal modulation transfer function.

© 2007 Optical Society of America

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References

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    [CrossRef]
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2007 (3)

Y. El-Mohri, L. E. Antonuk, Q. Zhao, Y. Wang, Y. Li, H. Du, and A. Sawant, "Performance of a high, fill factor indirect detection prototype flat-panel imager for mammography," Med. Phys. 34, 315-327 (2007).
[CrossRef] [PubMed]

D. C. Hunt, K. Tanioka, and J. A. Rowlands, "X-ray imaging using avalanche multiplication in amorphous selenium: Investigation of depth dependent avalanche noise," Med. Phys. 34, 976-986 (2007).
[CrossRef] [PubMed]

S. Richard and J. H. Siewerdsen, "Optimization of dual-energy imaging systems using generalized NEQ and imaging task," Med. Phys. 34, 127-139 (2007).
[CrossRef] [PubMed]

2006 (2)

S. Suryanarayanan, A. Karellas, S. Vedantham, and I. Sechopoulos, "Theoretical analysis of high-resolution digital mammography," Phys. Med. Biol. 51, 3041-3055 (2006).
[CrossRef] [PubMed]

G. Hajdok, J. Yao, J. J. Battista, and I. A. Cunningham, "Signal and noise transfer properties of photoelectric interactions in diagnostic x-ray imaging detectors," Med. Phys. 33, 3601-3620 (2006).
[CrossRef] [PubMed]

2005 (1)

M. Sattarivand and I. A. Cunningham, "Computational engine for development of complex cascade models of signal and noise in X-ray imaging systems," IEEE Trans. Med. Imaging 24, 211-222 (2005).
[CrossRef] [PubMed]

2003 (2)

P. R. Granfors, R. Aufrichtig, G. E. Possin, B. W. Giambattista, Z. S. Huang, J. Liu, and B. Ma, "Performance of a 41×41cm2 amorphous silicon flat panel x-ray detector designed for angiographic and R&F imaging applications," Med. Phys. 30, 2715-2726 (2003).
[CrossRef] [PubMed]

A. Ganguly, S. Rudin, D. R. Bednarek, and K. R. Hoffmann, "Micro-angiography for neuro-vascular imaging. II. Cascade model analysis," Med. Phys. 30, 3029-3039 (2003).
[CrossRef] [PubMed]

2002 (3)

J. H. Siewerdsen, I. A. Cunningham, and D. A. Jaffray, "A framework for noise-power spectrum analysis of multidimensional images," Med. Phys. 29, 2655-2671 (2002).
[CrossRef] [PubMed]

J. G. Mainprize, D. C. Hunt, and M. J. Yaffe, "Direct conversion detectors: the effect of incomplete charge collection on detective quantum efficiency," Med. Phys. 29, 976-990 (2002).
[CrossRef] [PubMed]

K. N. Jabri and D. L. Wilson, "Quantitative assessment of image quality enhancement due to unsharp-mask processing in x-ray fluoroscopy," J. Opt. Soc. Am. A 19, 1297-1307 (2002).
[CrossRef]

2001 (3)

J. Yao and I. A. Cunningham, "Parallel cascade: new ways to describe noise transfer in medical imaging systems," Med. Phys. 28, 2020-2038 (2001).
[CrossRef] [PubMed]

W. Zhao, W. G. Ji, and J. A. Rowlands, "Effects of characteristic x rays on the noise power spectra and detective quantum efficiency of photoconductive x-ray detectors," Med. Phys. 28, 2039-2049 (2001).
[CrossRef] [PubMed]

I. A. Cunningham, T. Moschandreou, and V. Subotic, "The detective quantum efficiency of fluoroscopic systems: The case for a spatialtemporal approach (or, Does the ideal observer have infinite patience?)," Proc. SPIE 4320, 479-488 (2001).
[CrossRef]

2000 (1)

P. R. Granfors and R. Aufrichtig, "DQE(f) of an amorphous silicon flat panel x-ray detector: detector parameter influences and measurement methodology," Proc. SPIE 3977, 2-13 (2000).
[CrossRef]

1999 (4)

1998 (4)

P. Xue and D. L. Wilson, "Detection of moving objects in pulsed-x-ray fluoroscopy," J. Opt. Soc. Am. A 15, 375-388 (1998).
[CrossRef]

T. Falco and B. G. Fallone, "Characteristics of metal-plate/film detectors at therapy energies. II. Detective quantum efficiency," Med. Phys. 25, 2463-2468 (1998).
[CrossRef]

J. H. Siewerdsen, L. E. Antonuk, Y. El-Mohri, J. Yorkston, W. Huang, and I. A. Cunningham, "Signal, noise power spectrum and detective quantum efficiency of indirect-detection flat-panel imagers for diagnostic radiology," Med. Phys. 25, 614-628 (1998).
[CrossRef] [PubMed]

P. Xue and D. L. Wilson, "Effects of motion blurring in x-ray fluoroscopy," Med. Phys. 25, 587-599 (1998).
[CrossRef] [PubMed]

1997 (3)

W. Zhao and J. A. Rowlands, "Digital radiology using active matrix readout of amorphous selenium: theoretical analysis of detective quantum efficiency," Med. Phys. 24, 1819-33 (1997).
[CrossRef]

H. H. Barrett, R. F. Wagner, and K. J. Myers, "Correlated point processes in radiological imaging," Proc. SPIE 3032, 110-125 (1997).
[CrossRef]

J. H. Siewerdsen, L. E. Antonuk, Y. el Mohri, J. Yorkston, W. Huang, J. M. Boudry, and I. A. Cunningham, "Empirical and theoretical investigation of the noise performance of indirect detection, active matrix flat-panel imagers (AMFPIs) for diagnostic radiology," Med. Phys. 24, 71-89 (1997).
[CrossRef] [PubMed]

1996 (2)

D. W. Mah, J. A. Rowlands, and J. A. Rawlinson, "Measurement of quantum noise in fluoroscopic systems for portal imaging," Med. Phys. 23, 231-238 (1996).
[CrossRef] [PubMed]

P. Xue, R. Aufrichtig, and D. L. Wilson, "Detectability of moving objects in fluoroscopy," Proc. SPIE 2712, 2-8 (1996).
[CrossRef]

1995 (1)

Y. Matsunaga, F. Hatori, T. Hiroyuki, and O. Yoshida, "Analysis of signal to noise ratio of photoconductive layered solid-state imaging device," IEEE Trans. Electron Devices 42, 38-42 (1995).
[CrossRef]

1994 (3)

R. Aufrichtig, P. Xue, C. W. Thomas, G. C. Gilmore, and D. L. Wilson, "Perceptual comparison of pulsed and continuous fluoroscopy," Med. Phys. 21, 245-256 (1994).
[CrossRef] [PubMed]

I. A. Cunningham, M. S. Westmore, and A. Fenster, "A spatial-frequency dependent quantum accounting diagram and detective quantum efficiency model of signal and noise propagation in cascaded imaging systems," Med. Phys. 21, 417-427 (1994).
[CrossRef] [PubMed]

R. Aufrichtig, C. W. Thomas, P. Xue, and D. L. Wilson, "Model for perception of pulsed fluoroscopy image sequences," J. Opt. Soc. Am. A 11, 3167-76 (1994).
[CrossRef]

1993 (1)

G. Murphy, W. Bitler, J. Coffin, and R. Langdon, "Lag vs. noise in fluoroscopic imaging," Proc. SPIE 1896, 174-179 (1993).
[CrossRef]

1990 (1)

R. M. Nishikawa and M. J. Yaffe, "Effect of various noise sources on the detective quantum efficiency of phosphor screens," Med. Phys. 17, 887-893 (1990).
[CrossRef] [PubMed]

1987 (2)

1985 (1)

M. J. Tapiovaara and R. F. Wagner, "SNR and DQE analysis of broad spectrum x-ray imaging," Phys. Med. Biol. 30, 519-529 (1985).
[CrossRef]

1984 (1)

M. J. Tapiovaara and R. F. Wagner, "A generalized detective quantum efficiency (DQE) approach to the analysis of x-ray imaging," Proc. SPIE 454, 540-549 (1984).

1979 (2)

R. F. Wagner and E. P. Muntz, "Detective quantum efficiency (DQE) analysis of electrostatic imaging and screen-film imaging in mammography," Proc. SPIE 173, 162-165 (1979).

R. F. Wagner, D. G. Brown, and M. S. Pastel, "Application of information theory to the assessment of computed tomography," Med. Phys. 6, 83-94 (1979).
[CrossRef] [PubMed]

1977 (1)

R. F. Wagner, "Toward a unified view of radiological imaging systems. Part II: Noisy images," Med. Phys. 4, 279-296 (1977).
[CrossRef] [PubMed]

1975 (1)

R. Shaw, "Some fundamental properties of xeroradiographic images," Proc. SPIE 70, 359-363 (1975).

1963 (1)

R. Shaw, "The equivalent quantum efficiency of the photographic process," J. Photogr. Sci. 11, 199-204 (1963).

IEEE Trans. Electron Devices (1)

Y. Matsunaga, F. Hatori, T. Hiroyuki, and O. Yoshida, "Analysis of signal to noise ratio of photoconductive layered solid-state imaging device," IEEE Trans. Electron Devices 42, 38-42 (1995).
[CrossRef]

IEEE Trans. Med. Imaging (1)

M. Sattarivand and I. A. Cunningham, "Computational engine for development of complex cascade models of signal and noise in X-ray imaging systems," IEEE Trans. Med. Imaging 24, 211-222 (2005).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (8)

J. Photogr. Sci. (1)

R. Shaw, "The equivalent quantum efficiency of the photographic process," J. Photogr. Sci. 11, 199-204 (1963).

Med. Phys. (22)

P. Xue and D. L. Wilson, "Effects of motion blurring in x-ray fluoroscopy," Med. Phys. 25, 587-599 (1998).
[CrossRef] [PubMed]

R. F. Wagner, "Toward a unified view of radiological imaging systems. Part II: Noisy images," Med. Phys. 4, 279-296 (1977).
[CrossRef] [PubMed]

R. F. Wagner, D. G. Brown, and M. S. Pastel, "Application of information theory to the assessment of computed tomography," Med. Phys. 6, 83-94 (1979).
[CrossRef] [PubMed]

W. Zhao and J. A. Rowlands, "Digital radiology using active matrix readout of amorphous selenium: theoretical analysis of detective quantum efficiency," Med. Phys. 24, 1819-33 (1997).
[CrossRef]

I. A. Cunningham, M. S. Westmore, and A. Fenster, "A spatial-frequency dependent quantum accounting diagram and detective quantum efficiency model of signal and noise propagation in cascaded imaging systems," Med. Phys. 21, 417-427 (1994).
[CrossRef] [PubMed]

J. Yao and I. A. Cunningham, "Parallel cascade: new ways to describe noise transfer in medical imaging systems," Med. Phys. 28, 2020-2038 (2001).
[CrossRef] [PubMed]

G. Hajdok, J. Yao, J. J. Battista, and I. A. Cunningham, "Signal and noise transfer properties of photoelectric interactions in diagnostic x-ray imaging detectors," Med. Phys. 33, 3601-3620 (2006).
[CrossRef] [PubMed]

P. R. Granfors, R. Aufrichtig, G. E. Possin, B. W. Giambattista, Z. S. Huang, J. Liu, and B. Ma, "Performance of a 41×41cm2 amorphous silicon flat panel x-ray detector designed for angiographic and R&F imaging applications," Med. Phys. 30, 2715-2726 (2003).
[CrossRef] [PubMed]

Y. El-Mohri, L. E. Antonuk, Q. Zhao, Y. Wang, Y. Li, H. Du, and A. Sawant, "Performance of a high, fill factor indirect detection prototype flat-panel imager for mammography," Med. Phys. 34, 315-327 (2007).
[CrossRef] [PubMed]

D. C. Hunt, K. Tanioka, and J. A. Rowlands, "X-ray imaging using avalanche multiplication in amorphous selenium: Investigation of depth dependent avalanche noise," Med. Phys. 34, 976-986 (2007).
[CrossRef] [PubMed]

S. Richard and J. H. Siewerdsen, "Optimization of dual-energy imaging systems using generalized NEQ and imaging task," Med. Phys. 34, 127-139 (2007).
[CrossRef] [PubMed]

J. H. Siewerdsen, I. A. Cunningham, and D. A. Jaffray, "A framework for noise-power spectrum analysis of multidimensional images," Med. Phys. 29, 2655-2671 (2002).
[CrossRef] [PubMed]

R. M. Nishikawa and M. J. Yaffe, "Effect of various noise sources on the detective quantum efficiency of phosphor screens," Med. Phys. 17, 887-893 (1990).
[CrossRef] [PubMed]

D. W. Mah, J. A. Rowlands, and J. A. Rawlinson, "Measurement of quantum noise in fluoroscopic systems for portal imaging," Med. Phys. 23, 231-238 (1996).
[CrossRef] [PubMed]

J. H. Siewerdsen, L. E. Antonuk, Y. el Mohri, J. Yorkston, W. Huang, J. M. Boudry, and I. A. Cunningham, "Empirical and theoretical investigation of the noise performance of indirect detection, active matrix flat-panel imagers (AMFPIs) for diagnostic radiology," Med. Phys. 24, 71-89 (1997).
[CrossRef] [PubMed]

T. Falco and B. G. Fallone, "Characteristics of metal-plate/film detectors at therapy energies. II. Detective quantum efficiency," Med. Phys. 25, 2463-2468 (1998).
[CrossRef]

J. H. Siewerdsen, L. E. Antonuk, Y. El-Mohri, J. Yorkston, W. Huang, and I. A. Cunningham, "Signal, noise power spectrum and detective quantum efficiency of indirect-detection flat-panel imagers for diagnostic radiology," Med. Phys. 25, 614-628 (1998).
[CrossRef] [PubMed]

J. H. Siewerdsen and D. A. Jaffray, "A ghost story: spatio-temporal response characteristics of an indirect- detection flat-panel imager," Med. Phys. 26, 1624-1641 (1999).
[CrossRef] [PubMed]

W. Zhao, W. G. Ji, and J. A. Rowlands, "Effects of characteristic x rays on the noise power spectra and detective quantum efficiency of photoconductive x-ray detectors," Med. Phys. 28, 2039-2049 (2001).
[CrossRef] [PubMed]

J. G. Mainprize, D. C. Hunt, and M. J. Yaffe, "Direct conversion detectors: the effect of incomplete charge collection on detective quantum efficiency," Med. Phys. 29, 976-990 (2002).
[CrossRef] [PubMed]

A. Ganguly, S. Rudin, D. R. Bednarek, and K. R. Hoffmann, "Micro-angiography for neuro-vascular imaging. II. Cascade model analysis," Med. Phys. 30, 3029-3039 (2003).
[CrossRef] [PubMed]

R. Aufrichtig, P. Xue, C. W. Thomas, G. C. Gilmore, and D. L. Wilson, "Perceptual comparison of pulsed and continuous fluoroscopy," Med. Phys. 21, 245-256 (1994).
[CrossRef] [PubMed]

Phys. Med. Biol. (2)

S. Suryanarayanan, A. Karellas, S. Vedantham, and I. Sechopoulos, "Theoretical analysis of high-resolution digital mammography," Phys. Med. Biol. 51, 3041-3055 (2006).
[CrossRef] [PubMed]

M. J. Tapiovaara and R. F. Wagner, "SNR and DQE analysis of broad spectrum x-ray imaging," Phys. Med. Biol. 30, 519-529 (1985).
[CrossRef]

Proc. SPIE (8)

R. Shaw, "Some fundamental properties of xeroradiographic images," Proc. SPIE 70, 359-363 (1975).

R. F. Wagner and E. P. Muntz, "Detective quantum efficiency (DQE) analysis of electrostatic imaging and screen-film imaging in mammography," Proc. SPIE 173, 162-165 (1979).

M. J. Tapiovaara and R. F. Wagner, "A generalized detective quantum efficiency (DQE) approach to the analysis of x-ray imaging," Proc. SPIE 454, 540-549 (1984).

P. Xue, R. Aufrichtig, and D. L. Wilson, "Detectability of moving objects in fluoroscopy," Proc. SPIE 2712, 2-8 (1996).
[CrossRef]

H. H. Barrett, R. F. Wagner, and K. J. Myers, "Correlated point processes in radiological imaging," Proc. SPIE 3032, 110-125 (1997).
[CrossRef]

G. Murphy, W. Bitler, J. Coffin, and R. Langdon, "Lag vs. noise in fluoroscopic imaging," Proc. SPIE 1896, 174-179 (1993).
[CrossRef]

P. R. Granfors and R. Aufrichtig, "DQE(f) of an amorphous silicon flat panel x-ray detector: detector parameter influences and measurement methodology," Proc. SPIE 3977, 2-13 (2000).
[CrossRef]

I. A. Cunningham, T. Moschandreou, and V. Subotic, "The detective quantum efficiency of fluoroscopic systems: The case for a spatialtemporal approach (or, Does the ideal observer have infinite patience?)," Proc. SPIE 4320, 479-488 (2001).
[CrossRef]

Other (6)

H. H. Barrett and K. Myers, Image Science: Mathematical and Statistical Foundations (Wiley, 2004).

H. H. Barrett and W. Swindell, Radiological Imaging (Academic, 1981).

I. A. Cunningham, Handbook of Medical Imaging: Volume 1. Physics and Psychophysics (SPIE, 2000), Chap. 2.

J. C. Dainty and R. Shaw, Image Science (Academic, 1974).

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, 1968).

G. M. Jenkins and D. G. Watts, Spectral Analysis and Its Applications (Holden-Day, 1968).

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Figures (3)

Fig. 1
Fig. 1

Block diagram representing quantum scatter in the temporal domain followed by scatter in the spatial domain.

Fig. 2
Fig. 2

Schematic diagram of a simple cascaded model of noise propagation in an x-ray imaging system. The input q o ( r , t ) is a Poisson distribution of incident x-ray quanta in both space and time. The output d ( r , t ) is the detector presampling signal.

Fig. 3
Fig. 3

Presampling normalized NPS calculated with Eq. (20) for a phosphor-based detector having unity quantum efficiency ( α = 1 ) with high conversion gain ( μ g ̃ η = 200 , left); and modest conversion gain ( μ g ̃ η = 5 , right). As lag is increased (β decreases from unity), the correlated noise component is scaled by β. The uncorrelated component does not depend on β and is equal to the β = 0 line. The effects of noise aliasing are not included in this graph.

Tables (2)

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Table 1 Summary of Expressions for Mean Density of Image Quanta and Associated NPS under WSS Conditions for Quantum Gain and Quantum Scatter, in Units of mm 2 s 1

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Table 2 Summary of Mean Signal and Associated NPS Cascaded through the Fluoroscopic System Model Assuming a Continuous Exposure in Both Space and Time

Equations (65)

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NEQ ( k ) = q ¯ 2 G ¯ 2 MTF 2 ( k ) NPS ( k ) ,
DQE ( k ) = NEQ ( k ) q ¯ = q ¯ G ¯ 2 MTF 2 ( k ) NPS ( k ) ,
q ̃ ( r , t ) = n = 1 N ̃ δ ( r r ̃ n ) δ ( t t ̃ n ) ,
q ̃ out ( r , t ) = n = 1 N ̃ m = 1 g ̃ n δ ( r r ̃ n ) δ ( t t ̃ n ) ,
q ¯ out = μ g ̃ N TA = μ g ̃ q ¯ in
K q out ( τ r , τ t ) = μ g ̃ 2 K q in ( τ r , τ t ) + σ g ̃ 2 q ¯ in δ ( τ r ) δ ( τ t ) .
NPS q out ( k , ν ) = μ g ̃ 2 NPS q in ( k , ν ) + σ g ̃ 2 q ¯ in
q ̃ out ( r , t ) = n = 1 N ̃ δ ( r r ̃ n Δ r ̃ n ) δ ( t t ̃ n Δ t ̃ n ) .
q ¯ out = q ¯ in T A p ( r , t r , t ) d 2 r d t = q ¯ in ,
K q out ( r , r , t , t ) = q ¯ in δ ( r r ) δ ( t t ) T A p ( r ρ , t λ ) d 2 ρ d λ + T T A A p ( r ρ , t λ ) p ( r ρ , t λ ) R q in ( ρ ρ , λ λ ) d 2 ρ d 2 ρ d λ d λ q ¯ in T A p ( r ρ , t λ ) p ( r ρ , t λ ) d 2 ρ d λ q ¯ in T A p ( r ρ , t λ ) d 2 ρ d λ q ¯ in T A p ( r ρ , t λ ) d 2 ρ d λ .
K q out ( r r , t t ) = q ¯ in δ ( r r ) δ ( t t ) + T T A A p ( ξ , ζ ) p ( ξ , ζ ) K q in ( ( r r ) ( ξ ξ ) , ( t t ) ( ζ ζ ) ) d 2 ξ d 2 ξ d ζ d ζ q ¯ in T A p ( ξ , ζ ) p ( ( r r ) + ξ , ( t t ) + ζ ) d 2 ξ d ζ = q ¯ in δ ( τ r ) δ ( τ t ) + T T A A p ( ξ , ζ ) p ( ξ , ζ ) K q in ( τ r ( ξ ξ ) , τ t ( ζ ζ ) ) d 2 ξ d 2 ξ d ζ d ζ q ¯ in T A p ( ξ , ζ ) p ( τ r + ξ , τ t + ζ ) d 2 ξ d ζ .
K q out ( τ r , τ t ) = [ K q in ( τ r , τ t ) q ¯ in ] * p ( τ r , τ t ) p ( τ r , τ t ) + q ¯ in δ ( τ r ) δ ( τ t ) .
NPS q out ( k , ν ) = [ NPS q in ( k , ν ) q ¯ in ] T ( k , ν ) 2 + q ¯ in
Π ( x a x , y a y , t a t + 1 2 ) = { 1 if ( x a x 2 ) ( y a y 2 ) ( a t t 0 ) 0 otherwise .
T ( k , ν ) = a x sinc ( π a x u ) a y sinc ( π a y v ) a t sinc ( π a t ν ) e i π ν a t ,
psNPS F ( k ) = [ ( μ g ̃ 2 + σ g ̃ 2 μ g ̃ ) κ 2 η 2 α q ¯ o T r ( k ) 2 a t 2 T t ( ν ) 2 sinc 2 ( π a t ν ) d ν + κ η μ g ̃ α a t q ¯ o ] a x 2 sinc 2 ( π a x u ) a y 2 sinc 2 ( π a y v ) ,
β = a t T t ( ν ) 2 sinc 2 ( π a t ν ) d ν = a t a L ,
psNPS F ( k ) = [ ( μ g ̃ 2 + σ g ̃ 2 μ g ̃ ) κ 2 η 2 α a t q ¯ o T r ( k ) 2 β + κ 2 η μ g ̃ α a t q ¯ o ] a x 2 sinc 2 ( π a x u ) a y 2 sinc 2 ( π a y v ) = β psNPS C ( k ) T r ( k ) 2 + psNPS U ( k ) ,
psNPS R ( k ) = psNPS C ( k ) T r ( k ) 2 + psNPS U ( k ) .
psNPS F ( k ) psNPS R ( k ) = β psNPS C ( k ) T r ( k ) 2 + psNPS U ( k ) psNPS C ( k ) T r ( k ) 2 + psNPS U ( k ) .
psNPS F N ( k ) = β psNPS C ( k ) T r ( k ) 2 + psNPS U ( k ) ( μ g ̃ 2 + σ g ̃ 2 μ g ̃ ) κ 2 η 2 α 2 a t q ¯ o a x 2 sinc 2 ( π a x u ) a y 2 sinc 2 ( π a y v ) 1 α [ β T r ( k ) 2 + 1 μ g ̃ η ] ,
q ̃ out ( r , t ) = T T A A q out ( r , t ) p { j } ( { r j , t j } ) { d 2 r j , 1 j N } { d t j , 1 j N } ,
q ̃ out ( r , t ) = n = 1 N T T A A g n δ ( r r n ) δ ( t t n ) p { j } ( { r j , t j } ) { d 2 r j , 1 j N } { d t j , 1 j N } .
q ̃ out ( r , t ) = n = 1 N T A g n δ ( r r n ) δ ( t t n ) ( T T A A p { j } ( { r j , t j } ) { d 2 r j , 1 j N , j n } { d t j , 1 j N , j n } ) d 2 r n d t n .
q ̃ out ( r , t ) = n = 1 N T A g n δ ( r r n ) δ ( t t n ) p n ( r n , t n ) d 2 r n d t n = n = 1 N g n p n ( r , t ) .
q ̃ in ( r , t ) = n = 1 N p n ( r , t ) .
q ̃ out ( r , t ) = n = 1 N g n p ( r , t ) = N μ g ̃ p ( r , t ) = μ g ̃ q ̃ in ( r , t ) .
R q out ( r , r , t , t ) = q ̃ out ( r , t ) q ̃ out ( r , t ) ( A , T ) .
R q out ( r , r , t , t ) = T T A A ( n = 1 N g n δ ( r r n ) δ ( t t n ) ) ( n = 1 N g n δ ( r r n ) δ ( t t n ) ) p { j , j } ( { r j , r j , t j , t j } ) { d 2 r j , 1 j N } { d 2 r j , 1 j N } { d t j , 1 j N } { d t j , 1 j N } .
R q out ( r , r , t , t ) = n = 1 N n = 1 N T T A A g n g n δ ( r r n ) δ ( r r n ) δ ( t t n ) δ ( t t n ) ( T T A A p { j , j } ( { r j , r j , t j , t j } ) { d 2 r j , 1 j N , j n } { d 2 r j , 1 j N , j n } { d t j , 1 j N , j n } { d t j , 1 j N , j n } ) d 2 r n d 2 r n d t n d t n .
R q out ( r , r , t , t ) = n = 1 N n = 1 N T T A A g n g n δ ( r r n ) δ ( r r n ) δ ( t t n ) δ ( t t n ) p ( n , n ) ( r n , r n , t n , t n ) d 2 r n d 2 r n d t n d t n .
R q out ( r , r , t , t ) n = n ̂ = n = 1 N T A g n 2 δ ( r r n ) δ ( r r n ) δ ( t t n ) δ ( t t n ) ( T A p ( n , n ) ( r n , r n , t n , t n ) d 2 r n d t n ) d 2 r n d t n .
R q out ( r , r , t , t ) n = n = n = 1 N T A g n 2 δ ( r r ) δ ( r ρ ) δ ( t t ) δ ( t λ ) p n ( ρ , λ ) d 2 ρ d λ = n = 1 N g n 2 δ ( r r ) δ ( t t ) p n ( r , t ) .
R q out ( r , r , t , t ) n n = n = 1 N n = 1 n n N T A g n g n δ ( r r n ) δ ( t t n ) ( T A δ ( r ρ ) δ ( t λ ) p ( n , n ) ( r n , ρ , t n , λ ) d 2 ρ d λ ) d 2 r n d t n = n = 1 N n = 1 n n N T A g n g n δ ( r ρ ) δ ( t λ ) p ( n , n ) ( ρ , r , λ , t ) d 2 ρ d λ = n = 1 N n = 1 n n N g n g n p ( n , n ) ( r , r , t , t ) .
R q out ( r , r , t , t ) = n = 1 N g n 2 δ ( r r ) δ ( t t ) p n ( r , t ) + n = 1 N n = 1 n n N g n g n p ( n , n ) ( r , r , t , t ) .
R q in ( r , r , t , t ) n = n = n = 1 N δ ( r r ) δ ( t t ) p n ( r , t ) and R q in ( r , r , t , t ) n n = n = 1 N n = 1 n n N p ( n , n ) ( r , r , t , t ) .
R q out ( r , r , t , t ) = N ( σ g ̃ 2 + μ g ̃ 2 ) p ( r , t ) δ ( r r ) δ ( t t ) + ( N 2 μ g ̃ 2 N ( σ g ̃ 2 + μ g ̃ 2 ) ) p ( r , r , t , t ) = μ g ̃ 2 ( N p ( r , t ) δ ( r r ) δ ( t t ) + N 2 p ( r , r , t , t ) ) + N σ g ̃ 2 p ( r , t ) δ ( r r ) δ ( t t ) .
R q in ( r , r , t , t ) = R q in ( r , r , t , t ) n = n + R q in ( r , r , t , t ) n n = N p ( r , t ) δ ( r r ) δ ( t t ) + N 2 p ( r , r , t , t ) .
R q out ( r , r , t , t ) = μ g ̃ 2 R q in ( r , r , t , t ) + N σ g ̃ 2 p ( r , t ) δ ( r r ) δ ( t t ) .
K q out ( r , r , t , t ) = R q out ( r , r , t , t ) q ̃ out ( r , t ) q ̃ out ( r , t ) = μ g ̃ 2 R q in ( r , r , t , t ) + N σ g ̃ 2 p ( r , t ) δ ( r r ) δ ( t t ) N 2 μ g ̃ 2 p ( r , t ) p ( r , t ) = μ g ̃ 2 K q in ( r , r , t , t ) + N σ g ̃ 2 p ( r , t ) δ ( r r ) δ ( t t ) .
q ̃ out ( r , t ) = E { r ̃ j , t ̃ j } { E { Δ r ̃ j , Δ t ̃ j } { q ̃ out ( r , t ) } } ,
q ¯ A ( r , t ) = E { Δ r ̃ j , Δ t ̃ j } { q out ( r , t ) } = T T A A ( i = n N δ ( r r n Δ r n ) δ ( t t n Δ t n ) ) p { j } ( { r j + Δ r j , t j + Δ t j r j , t j } ) { d 2 Δ r j , 1 j N } { d Δ t j , 1 j N } .
q ¯ A ( r , t ) = n = 1 N T A δ ( r r n Δ r n ) δ ( t t n Δ t n ) ( T T A A p { j } ( { r j + Δ r j , t j + Δ t j r j , t j } ) { d 2 Δ r j , 1 j N , j n } { d Δ t j , 1 j N , j n } ) d 2 Δ r n d Δ t n .
q ¯ A ( r , t ) = n = 1 N T A δ ( r ρ Δ ) δ ( t λ Δ ) p n ( ρ Δ , λ Δ r n , t n ) d 2 ρ Δ d λ Δ = n = 1 N p n ( r , t r n , t n ) .
q ̃ out ( r , t ) = E { r ̃ j , t ̃ j } { q ¯ A ( r , t ) } = n = 1 N T T A A p n ( r , t r n , t n ) p { j } ( { r j , t j } ) { d 2 r j , 1 j N } { d t j , 1 j N } .
q ̃ out ( r , t ) = n = 1 N T A p n ( r , t r n , t n ) ( T T A A p { j } ( { r j , t j } ) { d 2 r j , 1 j N , j n } { d t j , 1 j N , j n } ) d 2 r n d t n .
q ̃ out ( r , t ) = n = 1 N T A p n ( r , t r , t ) p n ( r , t ) d 2 r d t .
q ̃ out ( r , t ) = T A p ( r , t r , t ) q ̃ in ( r , t ) d 2 r d t .
R q ( r , r , t , t ) = E { r ̃ j , r ̃ j , t ̃ j t ̃ j } { E { Δ r ̃ j , Δ r ̃ j , Δ t ̃ j , Δ t ̃ j } { q ̃ out ( r , t ) q ̃ out ( r , t ) } } .
R q A ( r , r , t , t ) = E { Δ r ̃ j , Δ r ̃ j , Δ t ̃ j , Δ t ̃ j } { q ̃ out ( r , t ) q ̃ out ( r , t ) } = T T A A ( n = 1 N δ ( r r n Δ r n ) δ ( t t n Δ t n ) ) ( n = 1 N δ ( r r n Δ r n ) δ ( t t n Δ t n ) ) p { j , j } ( { r j + Δ r j , r j + Δ r j , t j + Δ t j , t j + Δ t j r j , r j , t j , t j } ) { d 2 Δ r j , 1 j N } { d 2 Δ r j , 1 j N } { d Δ t j , 1 j N } { d Δ t j , 1 j N } .
R q A ( r , r , t , t ) = n = 1 N n = 1 N T T A A δ ( r r n Δ r n ) δ ( t t n Δ t n ) δ ( r r n Δ r n ) δ ( t t n Δ t n ) ( T T A A p { j , j } ( { r j + Δ r j , r j + Δ r j , t j + Δ t j , t j + Δ t j r j , r j , t j , t j } ) { d 2 Δ r j , 1 j N , j n } { d 2 Δ r j , 1 j N , j n } { d Δ t j , 1 j N , j n } { d Δ t j , 1 j N j n } ) d 2 Δ r n d 2 Δ r n d Δ t n d Δ t n .
R q A ( r , r , t , t ) = n = 1 N n = 1 N T T A A δ ( r r n Δ r n ) δ ( t t n Δ t n ) δ ( r r n Δ r n ) δ ( t t n Δ t n ) p ( n , n ) ( r n + Δ r n , r n + Δ r n , t n + Δ t n , t n + Δ t n r n , r n , t n , t n ) d 2 Δ r n d 2 Δ r n d Δ t n d Δ t n .
R q A ( r , r , t , t ) n = n = n = 1 N T A δ ( r r n Δ r n ) δ ( t t n Δ t n ) δ ( r r n Δ r n ) δ ( t t n Δ t n ) p n ( r n + Δ r n , t n + Δ t n r n , t n ) d 2 Δ r n d Δ t n = n = 1 N δ ( r r ) δ ( t t ) p n ( r , t r n , t n ) .
R q A ( r , r , t , t ) n n = n = 1 N n = 1 n n N T A δ ( r r n Δ r n ) δ ( t t n Δ t n ) ( T A δ ( r r n Δ r n ) δ ( t t n Δ t n ) p ( n , n ) ( r n + Δ r n , r n + Δ r n , t n + Δ t n , t n + Δ t n r n , r n , t n , t n ) d 2 Δ r n d Δ t n ) d 2 Δ r n d Δ t n .
R q A ( r , r , t , t ) n n = n = 1 N n = 1 n n N T A δ ( r r n Δ r n ) δ ( t t n Δ t n ) ( T A δ ( r ρ Δ ) δ ( t λ Δ ) p ( n , n ) ( r n + Δ r n , ρ Δ , t n + Δ t n , λ Δ r n , r n , t n , t n ) d 2 ρ Δ d λ Δ ) d 2 Δ r n d Δ t n = n = 1 N n = 1 n n N T A δ ( r r n Δ r n ) δ ( t t n Δ t n ) p ( n , n ) ( r n + Δ r n , r , t n + Δ t n , t r n , r n , t n , t n ) d 2 Δ r n d Δ t n = n = 1 N n = 1 n n N T A δ ( r ρ Δ ) δ ( t λ Δ ) p ( n , n ) ( ρ Δ , r , λ Δ , t r n , r n , t n , t n ) d 2 ρ Δ d λ Δ = n = 1 N n = 1 n n N p ( n , n ) ( r , r , t , t r n , r n , t n , t n ) .
R q A ( r , r , t , t ) = n = 1 N δ ( r r ) δ ( t t ) p n ( r , t r n , t n ) + n = 1 N n = 1 n n N p ( n , n ) ( r , r , t , t r n , r n , t n , t n ) .
R q out ( r , r , t , t ) = E ( { r ̃ j , r ̃ j , t ̃ j , t ̃ j } ) ( R q A ( r , r , t , t ) ) = T T A A R q A ( r , r , t , t ) p { j , j } ( { r j , r j , t j , t j } ) { d 2 r j , 1 j N } { d 2 r j , 1 j N } { d t j , 1 j N } { d t j , 1 j N } .
R q out ( r , r , t , t ) = T T A A ( i = 1 N δ ( r r ) δ ( t t ) p n ( r , t r n , t n ) + n = 1 N n = 1 n n N p ( n , n ) ( r , r , t , t r n , r n , t n , t n ) ) p { j , j } ( { r j , r j , t j , t j } ) { d 2 r j , 1 j N } { d 2 r j , 1 j N } { d t j , 1 j N } { d t j , 1 j N } ,
R q out ( r , r , t , t ) = n = 1 N T A δ ( r r ) δ ( t t ) p n ( r , t r n , t n ) ( T T A A p { j , j } ( { r j , r j , t j , t j } ) { d 2 r j , 1 j N , j n } { d 2 r j , 1 j N } { d t j , 1 j N , j n } { d t j , 1 j N } ) d 2 r n d t n + n = 1 N n = 1 n n N T T A A p ( n , n ) ( r , r , t , t r n , r n , t n , t n ) ( T T A A p { j , j } ( { r j , r j , t j , t j } ) { d 2 r j , 1 j N , j n } { d 2 r j , 1 j N , j n } { d t j , 1 j N , j n } { d t j , 1 j N , j n } ) d 2 r n d 2 r n d t n d t n .
R q out ( r , r , t , t ) = n = 1 N T A δ ( r r ) δ ( t t ) p n ( r , t r n , t n ) p n ( r n , t n ) d 2 r n d t n + n = 1 N n = 1 n n N T T A A p ( n , n ) ( r , r , t , t r n , r n , t n , t n ) p n ( r n , r n , t n , t n ) d 2 r n d 2 r n d t n d t n .
R q out ( r , r , t , t ) = n = 1 N T A δ ( r r ) δ ( t t ) p n ( r , t ρ , λ ) p n ( ρ , λ ) d 2 ρ d λ + n = 1 N n = 1 n n N T T A A p ( n , n ) ( r , r , t , t ρ , ρ , λ , λ ) p ( n , n ) ( ρ , ρ , λ , λ ) d 2 ρ d 2 ρ d λ d λ .
R q out ( r , r , t , t ) = T A δ ( r r ) δ ( t t ) p ( r , t ρ , λ ) ( n = 1 N p n ( ρ , λ ) ) d 2 ρ d λ + T T A A p ( r , r , t , t ρ , ρ , λ , λ ) ( n = 1 N n = 1 n n N p ( n , n ) ( ρ , ρ , λ , λ ) ) d 2 ρ d 2 ρ d λ d λ .
R q out ( r , r , t , t ) = T A δ ( r r ) δ ( t t ) p ( r , t ρ , λ ) q ̃ in ( ρ , λ ) d 2 ρ d λ + T T A A p ( r , r , t , t ρ , ρ , λ , λ ) ( R q in ( ρ , ρ , λ , λ ) R q in ( ρ , ρ , λ , λ ) n = n ) d 2 ρ d 2 ρ d λ d λ ,
R q out ( r , r , t , t ) = T A δ ( r r ) δ ( t t ) p ( r , t ρ , λ ) q ̃ in ( ρ , λ ) d 2 ρ d λ + T T A A p ( r , r , t , t ρ , ρ , λ , λ ) R q in ( ρ , ρ , λ , λ ) d 2 ρ d 2 ρ d λ d λ T A p ( r , r , t , t ρ , ρ , λ , λ ) q ̃ in ( ρ , λ ) d 2 ρ d λ .
K q out ( r , r , t , t ) = R q out ( r , r , t , t ) q ̃ out ( r , t ) q ̃ out ( r , t ) = T A δ ( r r ) δ ( t t ) p ( r , t ρ , λ ) q ̃ in ( ρ , λ ) d 2 ρ d λ + T T A A p ( r , r , t , t ρ , ρ , λ , λ ) R q in ( ρ , ρ , λ , λ ) d 2 ρ d 2 ρ d λ d λ T A p ( r , r , t , t ρ , ρ , λ , λ ) q ̃ in ( ρ , λ ) d 2 ρ d λ ( T A p ( r , t ρ , λ ) q ̃ in ( ρ , λ ) d 2 ρ d λ ) ( T A p ( r , t ρ , λ ) q ̃ in ( ρ , λ ) d 2 ρ d λ ) .

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