Abstract

We investigated the detectability of moving, low-contrast objects in white-noise image sequences. The computer-generated, cylindrical phantoms mimicked arteries, catheters, and guide wires in medical, x-ray fluoroscopy image sequences at 16 acquisitions/s (pulsed-16) or 32 acquisitions/s (pulsed-32). We measured detectability by using a reference–test, adaptive forced-choice method whereby reference and test presentations were alternated during an experimental session to minimize effects of subject attention and accuracy criteria. In the case of the largest cylinder (diameter 0.48 deg), the highest speed (5.86 deg/s) increased absolute detectability by 42% compared with that in the stationary case. With the smallest cylinder (diameter 0.023 deg), this motion decreased detectability by 51%. The dose savings of pulsed-16 was 18% of that for pulsed-32, with relatively little effect of velocity or object size. In general, subjects took slightly longer to respond in the case of low-acquisition fluoroscopy. Detectability data were modeled with a nonprewhitening matched filter that included a physiological, spatiotemporal contrast sensitivity function and a suboptimal, spatiotemporal signal template with time-limited memory.

© 1998 Optical Society of America

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1997

M. J. Tapiovaara, “Efficiency of low-contrast detail detectability in fluoroscopic imaging,” Med. Phys. 24, 655–664 (1997).
[CrossRef] [PubMed]

1996

D. L. Wilson, K. N. Jabri, P. Xue, R. Aufrichtig, “Perceived noise versus display noise in temporally filtered image sequences,” J. Electron. Imaging 5, 490–495 (1996).
[CrossRef]

P. Xue, D. L. Wilson, “Pulsed fluoroscopy detectability from interspersed adaptive forced choice measurements,” Med. Phys. 23, 1833–1843 (1996).
[CrossRef] [PubMed]

M. P. Eckstein, J. S. Whiting, J. P. Thomas, “Role of knowledge in human visual temporal integration in spatiotemporal noise,” J. Opt. Soc. Am. A 13, 1960–1968 (1996).
[CrossRef]

1995

R. Aufrichtig, D. L. Wilson, “X-ray fluoroscopy spatio-temporal filtering with object detection,” IEEE Trans. Med. Imaging 14, 733–746 (1995).
[CrossRef] [PubMed]

T. B. Shope, “Radiation-induced skin injuries from fluoroscopy,” Radiology 197(P), 209 (1995) (abstract).

T. Carney, D. A. Silverstein, S. A. Klein, “Vernier acuity during image rotation and translation: visual performance limits,” Vision Res. 35, 1951–1964 (1995).
[CrossRef] [PubMed]

A. E. Burgess, “Comparison of receiver operating characteristic and forced choice observer performance measurement methods,” Med. Phys. 22, 643–655 (1995).
[CrossRef] [PubMed]

1994

D. L. Wilson, P. Xue, R. Aufrichtig, “Perception of fluoroscopy last-image-hold,” Med. Phys. 21, 1875–1883 (1994).
[CrossRef] [PubMed]

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

N. L. Eigler, M. P. Eckstein, K. N. Mahrer, J. S. Whiting, “Improving detection of coronary morphological features from digital angiograms: effect of stenosis stabilization display,” Circulation 89, 2700–2709 (1994).
[CrossRef] [PubMed]

A. Abdel-Malek, F. Yassa, J. Bloomer, “An adaptive gating approach for x-ray dose reduction during interventional procedures,” IEEE Trans. Med. Imaging 13, 2–12 (1994).
[CrossRef]

A. E. Burgess, “Statistically defined backgrounds: performance of a modified nonprewhitening observer model,” J. Opt. Soc. Am. A 11, 1237–1242 (1994).
[CrossRef]

R. Aufrichtig, C. Thomas, P. Xue, D. L. Wilson, “A model for perception of pulsed fluoroscopy image sequences,” J. Opt. Soc. Am. A 11, 3167–3176 (1994).
[CrossRef]

1991

L. B. Stelmach, P. J. Hearty, “Requirements for static and dynamic spatial resolution in advanced television systems: a psychophysical evaluation,” J. Soc. Motion Pict. TV Eng. 100, 5–9 (1991).

1989

M. J. Morgan, S. Benton, “Motion-deblurring in human vision,” Nature (London) 340, 385–386 (1989).
[CrossRef]

G. M. Long, D. A. Rourke, “Training effects on the resolution of moving targets—dynamic visual acuity,” Human Factors 31, 443–451 (1989).

1988

S. L. Fritz, S. E. Mirvis, S. O. Pais, S. Roys, “Phantom evaluation of angiographer performance using low frame rate acquisition fluoroscopy,” Med. Phys. 15, 600–603 (1988).
[CrossRef] [PubMed]

M. Livingstone, D. Hubel, “Segregation of form, color, movement, and depth: anatomy, physiology, and perception,” Science 240, 740–749 (1988).
[CrossRef] [PubMed]

1986

1985

A. E. Burgess, H. Ghandeharian, “Visual signal detection. III. On Bayesian use of prior knowledge and cross-correlation,” J. Opt. Soc. Am. A 2, 1498–1507 (1985).
[CrossRef] [PubMed]

L. D. Loo, K. Doi, C. E. Metz, “Investigation of basic imaging properties in digital radiography. 4. Effect of unsharp masking on the detectability of simple patterns,” Med. Phys. 12, 209–214 (1985).
[CrossRef] [PubMed]

R. F. Wagner, D. G. Brown, “Unified SNR analysis of medical imaging systems,” Phys. Med. Biol. 30, 489–518 (1985).
[CrossRef]

H. Fujita, K. Doi, M. L. Giger, “Investigation of basic imaging properties in digital radiography. 6. MTFs of II-TV digital imaging systems,” Med. Phys. 12, 713–719 (1985).
[CrossRef] [PubMed]

1984

1981

S. M. Kay, J. S. L. Marple, “Spectrum analysis—a modern perspective,” Proc. IEEE 69, 1380–1419 (1981).
[CrossRef]

A. E. Burgess, R. F. Wagner, R. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93–94 (1981).
[CrossRef] [PubMed]

M. Fahle, T. Poggio, “Visual hyperacuity: spatiotemporal interpolation in human vision,” Proc. R. Soc. London 213, 451–477 (1981).
[CrossRef]

1979

1967

1966

1962

J. W. Miller, E. J. Ludvigh, “The effect of relative motion on visual acuity,” Surv. Ophthalmol. 7, 83–116 (1962).
[PubMed]

Abdel-Malek, A.

A. Abdel-Malek, F. Yassa, J. Bloomer, “An adaptive gating approach for x-ray dose reduction during interventional procedures,” IEEE Trans. Med. Imaging 13, 2–12 (1994).
[CrossRef]

Ahumada, J.

Albert, J.

Ammann, E.

E. Ammann, G. Wiede, “Generators and tubes in interventional radiology,” in Syllabus: A Categorial Course in Physics, Physical and Technical Aspects of Angiography and Interventional Angiography, S. Balter, T. B. Shope, eds., (RSNA, Oak Brook, Ill., 1995), pp. 59–74.

Aufrichtig, R.

D. L. Wilson, K. N. Jabri, P. Xue, R. Aufrichtig, “Perceived noise versus display noise in temporally filtered image sequences,” J. Electron. Imaging 5, 490–495 (1996).
[CrossRef]

R. Aufrichtig, D. L. Wilson, “X-ray fluoroscopy spatio-temporal filtering with object detection,” IEEE Trans. Med. Imaging 14, 733–746 (1995).
[CrossRef] [PubMed]

R. Aufrichtig, C. Thomas, P. Xue, D. L. Wilson, “A model for perception of pulsed fluoroscopy image sequences,” J. Opt. Soc. Am. A 11, 3167–3176 (1994).
[CrossRef]

D. L. Wilson, P. Xue, R. Aufrichtig, “Perception of fluoroscopy last-image-hold,” Med. Phys. 21, 1875–1883 (1994).
[CrossRef] [PubMed]

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

D. L. Wilson, K. N. Jabri, R. Aufrichtig “Perception of temporally filtered x-ray fluoroscopy images,” submitted to Med. Phys.

Barlow, H. B.

A. E. Burgess, R. F. Wagner, R. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93–94 (1981).
[CrossRef] [PubMed]

Benton, S.

M. J. Morgan, S. Benton, “Motion-deblurring in human vision,” Nature (London) 340, 385–386 (1989).
[CrossRef]

Bloomer, J.

A. Abdel-Malek, F. Yassa, J. Bloomer, “An adaptive gating approach for x-ray dose reduction during interventional procedures,” IEEE Trans. Med. Imaging 13, 2–12 (1994).
[CrossRef]

Bouman, M. A.

Brown, D. G.

R. F. Wagner, D. G. Brown, “Unified SNR analysis of medical imaging systems,” Phys. Med. Biol. 30, 489–518 (1985).
[CrossRef]

Burgess, A. E.

A. E. Burgess, “Comparison of receiver operating characteristic and forced choice observer performance measurement methods,” Med. Phys. 22, 643–655 (1995).
[CrossRef] [PubMed]

A. E. Burgess, “Statistically defined backgrounds: performance of a modified nonprewhitening observer model,” J. Opt. Soc. Am. A 11, 1237–1242 (1994).
[CrossRef]

A. E. Burgess, H. Ghandeharian, “Visual signal detection. III. On Bayesian use of prior knowledge and cross-correlation,” J. Opt. Soc. Am. A 2, 1498–1507 (1985).
[CrossRef] [PubMed]

A. E. Burgess, R. F. Wagner, R. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93–94 (1981).
[CrossRef] [PubMed]

Burr, D.

D. Burr, J. Ross, M. C. Marrone, “Seeing objects in motion,” Proc. R. Soc. London 227, 249–265 (1986).
[CrossRef]

Carney, T.

T. Carney, D. A. Silverstein, S. A. Klein, “Vernier acuity during image rotation and translation: visual performance limits,” Vision Res. 35, 1951–1964 (1995).
[CrossRef] [PubMed]

Carterette, E.

J. Whiting, D. Honig, E. Carterette, N. Eigler, “Observer performance in dynamic displays: effect of frame rate on visual signal detection in noisy images,” in Human Vision, Visual Processing, and Digital Display II, B. E. Rogowitz, M. H. Brill, J. P. Allebach, eds., Proc. SPIE1453, 165–176 (1991).
[CrossRef]

Doi, K.

L. D. Loo, K. Doi, C. E. Metz, “Investigation of basic imaging properties in digital radiography. 4. Effect of unsharp masking on the detectability of simple patterns,” Med. Phys. 12, 209–214 (1985).
[CrossRef] [PubMed]

H. Fujita, K. Doi, M. L. Giger, “Investigation of basic imaging properties in digital radiography. 6. MTFs of II-TV digital imaging systems,” Med. Phys. 12, 713–719 (1985).
[CrossRef] [PubMed]

Eckstein, M. P.

M. P. Eckstein, J. S. Whiting, J. P. Thomas, “Role of knowledge in human visual temporal integration in spatiotemporal noise,” J. Opt. Soc. Am. A 13, 1960–1968 (1996).
[CrossRef]

N. L. Eigler, M. P. Eckstein, K. N. Mahrer, J. S. Whiting, “Improving detection of coronary morphological features from digital angiograms: effect of stenosis stabilization display,” Circulation 89, 2700–2709 (1994).
[CrossRef] [PubMed]

J. S. Whiting, M. P. Eckstein, C. A. Morioka, N. L. Eigler, “Effect of additive noise, signal contrast, and feature motion on visual detection in structured noise,” in Medical Imaging 1996: Image Perception, H. L. Kundel, ed., Proc. SPIE2712, 26–38 (1996).
[CrossRef]

M. P. Eckstein, J. S. Whiting, J. P. Thomas, “Detection and contrast discrimination of moving signals in uncorrelated gaussian noise,” Medical Imaging 1996: Image Perception, H. L. Kundel, ed., Proc. SPIE2712, 9–25 (1996).
[CrossRef]

Eigler, N.

J. Whiting, D. Honig, E. Carterette, N. Eigler, “Observer performance in dynamic displays: effect of frame rate on visual signal detection in noisy images,” in Human Vision, Visual Processing, and Digital Display II, B. E. Rogowitz, M. H. Brill, J. P. Allebach, eds., Proc. SPIE1453, 165–176 (1991).
[CrossRef]

Eigler, N. L.

N. L. Eigler, M. P. Eckstein, K. N. Mahrer, J. S. Whiting, “Improving detection of coronary morphological features from digital angiograms: effect of stenosis stabilization display,” Circulation 89, 2700–2709 (1994).
[CrossRef] [PubMed]

J. S. Whiting, M. P. Eckstein, C. A. Morioka, N. L. Eigler, “Effect of additive noise, signal contrast, and feature motion on visual detection in structured noise,” in Medical Imaging 1996: Image Perception, H. L. Kundel, ed., Proc. SPIE2712, 26–38 (1996).
[CrossRef]

Fahle, M.

M. Fahle, T. Poggio, “Visual hyperacuity: spatiotemporal interpolation in human vision,” Proc. R. Soc. London 213, 451–477 (1981).
[CrossRef]

M. Fahle, T. Poggio, “Visual hyperacuity: spatiotemporal interpolation in human vision,” in Image Understanding, S. Ullman, W. Richards, eds. (Ablex, Norwood, N.J., 1984), pp. 49–77.

Farrell, J. E.

Fritz, S. L.

S. L. Fritz, S. E. Mirvis, S. O. Pais, S. Roys, “Phantom evaluation of angiographer performance using low frame rate acquisition fluoroscopy,” Med. Phys. 15, 600–603 (1988).
[CrossRef] [PubMed]

Fujita, H.

H. Fujita, K. Doi, M. L. Giger, “Investigation of basic imaging properties in digital radiography. 6. MTFs of II-TV digital imaging systems,” Med. Phys. 12, 713–719 (1985).
[CrossRef] [PubMed]

Ghandeharian, H.

Giger, M. L.

H. Fujita, K. Doi, M. L. Giger, “Investigation of basic imaging properties in digital radiography. 6. MTFs of II-TV digital imaging systems,” Med. Phys. 12, 713–719 (1985).
[CrossRef] [PubMed]

Gilmore, G. C.

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

P. Xue, C. W. Thomas, G. C. Gilmore, D. L. Wilson, “An adaptive reference/test paradigm with application to pulsed fluoroscopy perception,” Behav. Res. Methods Instrum. Comput. (to be published).

Hearty, P. J.

L. B. Stelmach, P. J. Hearty, “Requirements for static and dynamic spatial resolution in advanced television systems: a psychophysical evaluation,” J. Soc. Motion Pict. TV Eng. 100, 5–9 (1991).

Honig, D.

J. Whiting, D. Honig, E. Carterette, N. Eigler, “Observer performance in dynamic displays: effect of frame rate on visual signal detection in noisy images,” in Human Vision, Visual Processing, and Digital Display II, B. E. Rogowitz, M. H. Brill, J. P. Allebach, eds., Proc. SPIE1453, 165–176 (1991).
[CrossRef]

Hubel, D.

M. Livingstone, D. Hubel, “Segregation of form, color, movement, and depth: anatomy, physiology, and perception,” Science 240, 740–749 (1988).
[CrossRef] [PubMed]

Jabri, K. N.

D. L. Wilson, K. N. Jabri, P. Xue, R. Aufrichtig, “Perceived noise versus display noise in temporally filtered image sequences,” J. Electron. Imaging 5, 490–495 (1996).
[CrossRef]

D. L. Wilson, K. N. Jabri, P. Xue, “Modeling human visual detection of low-contrast objects in fluoroscopy image sequences,” in Medical Imaging 1997: Image Perception, H. L. Kundel, ed., Proc. SPIE3036, 21–30 (1997).
[CrossRef]

D. L. Wilson, K. N. Jabri, R. Aufrichtig “Perception of temporally filtered x-ray fluoroscopy images,” submitted to Med. Phys.

P. Xue, K. N. Jabri, D. L. Wilson, “The adaptive reference/test forced-choice method with application to fluoroscopy perception,” Medical Imaging 1997: Image Perception, H. L. Kundel, ed., Proc. SPIE3036, 298–307 (1997).
[CrossRef]

Jennings, R.

A. E. Burgess, R. F. Wagner, R. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93–94 (1981).
[CrossRef] [PubMed]

Kay, S. M.

S. M. Kay, J. S. L. Marple, “Spectrum analysis—a modern perspective,” Proc. IEEE 69, 1380–1419 (1981).
[CrossRef]

Kelly, D. H.

Klein, S. A.

T. Carney, D. A. Silverstein, S. A. Klein, “Vernier acuity during image rotation and translation: visual performance limits,” Vision Res. 35, 1951–1964 (1995).
[CrossRef] [PubMed]

Koenderink, J. J.

Livingstone, M.

M. Livingstone, D. Hubel, “Segregation of form, color, movement, and depth: anatomy, physiology, and perception,” Science 240, 740–749 (1988).
[CrossRef] [PubMed]

Long, G. M.

G. M. Long, D. A. Rourke, “Training effects on the resolution of moving targets—dynamic visual acuity,” Human Factors 31, 443–451 (1989).

Loo, L. D.

L. D. Loo, K. Doi, C. E. Metz, “Investigation of basic imaging properties in digital radiography. 4. Effect of unsharp masking on the detectability of simple patterns,” Med. Phys. 12, 209–214 (1985).
[CrossRef] [PubMed]

Ludvigh, E. J.

J. W. Miller, E. J. Ludvigh, “The effect of relative motion on visual acuity,” Surv. Ophthalmol. 7, 83–116 (1962).
[PubMed]

Luijendijk, H.

H. Luijendijk, “Practical experiments on noise perception in noisy images,” in Medical Imaging 1994: Image Perception, H. L. Kundel, ed., Proc. SPIE2166, 2–8 (1994).
[CrossRef]

Mahrer, K. N.

N. L. Eigler, M. P. Eckstein, K. N. Mahrer, J. S. Whiting, “Improving detection of coronary morphological features from digital angiograms: effect of stenosis stabilization display,” Circulation 89, 2700–2709 (1994).
[CrossRef] [PubMed]

Manjeshwar, R.

R. Manjeshwar, D. L. Wilson, “Role of smooth pursuit eye-movements on the detection of moving objects in x-ray fluoroscopy noise,” in Proceedings of the 19th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1997).

R. Manjeshwar, D. L. Wilson, “Eye-tracking of moving objects in x-ray fluoroscopy enhances detectability,” presented at the Seventh Far West Image Perception Conference, Tucson, Ariz., 1997.

Marple, J. S. L.

S. M. Kay, J. S. L. Marple, “Spectrum analysis—a modern perspective,” Proc. IEEE 69, 1380–1419 (1981).
[CrossRef]

Marrone, M. C.

D. Burr, J. Ross, M. C. Marrone, “Seeing objects in motion,” Proc. R. Soc. London 227, 249–265 (1986).
[CrossRef]

McDonough, R. N.

R. N. McDonough, A. D. Whalen, Detection of Signals in Noise, 2nd ed. (Academic, San Diego, Calif., 1995).

Metz, C. E.

L. D. Loo, K. Doi, C. E. Metz, “Investigation of basic imaging properties in digital radiography. 4. Effect of unsharp masking on the detectability of simple patterns,” Med. Phys. 12, 209–214 (1985).
[CrossRef] [PubMed]

Miller, J. W.

J. W. Miller, E. J. Ludvigh, “The effect of relative motion on visual acuity,” Surv. Ophthalmol. 7, 83–116 (1962).
[PubMed]

Mirvis, S. E.

S. L. Fritz, S. E. Mirvis, S. O. Pais, S. Roys, “Phantom evaluation of angiographer performance using low frame rate acquisition fluoroscopy,” Med. Phys. 15, 600–603 (1988).
[CrossRef] [PubMed]

Morgan, M. J.

M. J. Morgan, S. Benton, “Motion-deblurring in human vision,” Nature (London) 340, 385–386 (1989).
[CrossRef]

Morioka, C. A.

J. S. Whiting, M. P. Eckstein, C. A. Morioka, N. L. Eigler, “Effect of additive noise, signal contrast, and feature motion on visual detection in structured noise,” in Medical Imaging 1996: Image Perception, H. L. Kundel, ed., Proc. SPIE2712, 26–38 (1996).
[CrossRef]

Morrison, T. R.

T. R. Morrison, “A review of dynamic visual acuity,” Monograph 28 (Naval Aerospace Medical Research Laboratory, Pensacola, Fla., 1980).

Nas, H.

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, Discrete-Time Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

Pais, S. O.

S. L. Fritz, S. E. Mirvis, S. O. Pais, S. Roys, “Phantom evaluation of angiographer performance using low frame rate acquisition fluoroscopy,” Med. Phys. 15, 600–603 (1988).
[CrossRef] [PubMed]

Poggio, T.

M. Fahle, T. Poggio, “Visual hyperacuity: spatiotemporal interpolation in human vision,” Proc. R. Soc. London 213, 451–477 (1981).
[CrossRef]

M. Fahle, T. Poggio, “Visual hyperacuity: spatiotemporal interpolation in human vision,” in Image Understanding, S. Ullman, W. Richards, eds. (Ablex, Norwood, N.J., 1984), pp. 49–77.

Robson, J. G.

Rose, A.

A. Rose, Vision: Human and Electronic (Plenum, New York, 1973).

Ross, J.

D. Burr, J. Ross, M. C. Marrone, “Seeing objects in motion,” Proc. R. Soc. London 227, 249–265 (1986).
[CrossRef]

Rourke, D. A.

G. M. Long, D. A. Rourke, “Training effects on the resolution of moving targets—dynamic visual acuity,” Human Factors 31, 443–451 (1989).

Roys, S.

S. L. Fritz, S. E. Mirvis, S. O. Pais, S. Roys, “Phantom evaluation of angiographer performance using low frame rate acquisition fluoroscopy,” Med. Phys. 15, 600–603 (1988).
[CrossRef] [PubMed]

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, Discrete-Time Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

Shope, T. B.

T. B. Shope, “Radiation-induced skin injuries from fluoroscopy,” Radiology 197(P), 209 (1995) (abstract).

Silverstein, D. A.

T. Carney, D. A. Silverstein, S. A. Klein, “Vernier acuity during image rotation and translation: visual performance limits,” Vision Res. 35, 1951–1964 (1995).
[CrossRef] [PubMed]

Stelmach, L. B.

L. B. Stelmach, P. J. Hearty, “Requirements for static and dynamic spatial resolution in advanced television systems: a psychophysical evaluation,” J. Soc. Motion Pict. TV Eng. 100, 5–9 (1991).

Tapiovaara, M. J.

M. J. Tapiovaara, “Efficiency of low-contrast detail detectability in fluoroscopic imaging,” Med. Phys. 24, 655–664 (1997).
[CrossRef] [PubMed]

Thomas, C.

Thomas, C. W.

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

P. Xue, C. W. Thomas, G. C. Gilmore, D. L. Wilson, “An adaptive reference/test paradigm with application to pulsed fluoroscopy perception,” Behav. Res. Methods Instrum. Comput. (to be published).

Thomas, J. P.

M. P. Eckstein, J. S. Whiting, J. P. Thomas, “Role of knowledge in human visual temporal integration in spatiotemporal noise,” J. Opt. Soc. Am. A 13, 1960–1968 (1996).
[CrossRef]

M. P. Eckstein, J. S. Whiting, J. P. Thomas, “Detection and contrast discrimination of moving signals in uncorrelated gaussian noise,” Medical Imaging 1996: Image Perception, H. L. Kundel, ed., Proc. SPIE2712, 9–25 (1996).
[CrossRef]

Trees, H. L. V.

H. L. V. Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1968).

van Nes, F.

Wagner, R. F.

R. F. Wagner, D. G. Brown, “Unified SNR analysis of medical imaging systems,” Phys. Med. Biol. 30, 489–518 (1985).
[CrossRef]

A. E. Burgess, R. F. Wagner, R. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93–94 (1981).
[CrossRef] [PubMed]

Watson, A. B.

Whalen, A. D.

R. N. McDonough, A. D. Whalen, Detection of Signals in Noise, 2nd ed. (Academic, San Diego, Calif., 1995).

Whiting, J.

J. Whiting, D. Honig, E. Carterette, N. Eigler, “Observer performance in dynamic displays: effect of frame rate on visual signal detection in noisy images,” in Human Vision, Visual Processing, and Digital Display II, B. E. Rogowitz, M. H. Brill, J. P. Allebach, eds., Proc. SPIE1453, 165–176 (1991).
[CrossRef]

Whiting, J. S.

M. P. Eckstein, J. S. Whiting, J. P. Thomas, “Role of knowledge in human visual temporal integration in spatiotemporal noise,” J. Opt. Soc. Am. A 13, 1960–1968 (1996).
[CrossRef]

N. L. Eigler, M. P. Eckstein, K. N. Mahrer, J. S. Whiting, “Improving detection of coronary morphological features from digital angiograms: effect of stenosis stabilization display,” Circulation 89, 2700–2709 (1994).
[CrossRef] [PubMed]

J. S. Whiting, M. P. Eckstein, C. A. Morioka, N. L. Eigler, “Effect of additive noise, signal contrast, and feature motion on visual detection in structured noise,” in Medical Imaging 1996: Image Perception, H. L. Kundel, ed., Proc. SPIE2712, 26–38 (1996).
[CrossRef]

M. P. Eckstein, J. S. Whiting, J. P. Thomas, “Detection and contrast discrimination of moving signals in uncorrelated gaussian noise,” Medical Imaging 1996: Image Perception, H. L. Kundel, ed., Proc. SPIE2712, 9–25 (1996).
[CrossRef]

Wiede, G.

E. Ammann, G. Wiede, “Generators and tubes in interventional radiology,” in Syllabus: A Categorial Course in Physics, Physical and Technical Aspects of Angiography and Interventional Angiography, S. Balter, T. B. Shope, eds., (RSNA, Oak Brook, Ill., 1995), pp. 59–74.

Wilson, D. L.

P. Xue, D. L. Wilson, “Pulsed fluoroscopy detectability from interspersed adaptive forced choice measurements,” Med. Phys. 23, 1833–1843 (1996).
[CrossRef] [PubMed]

D. L. Wilson, K. N. Jabri, P. Xue, R. Aufrichtig, “Perceived noise versus display noise in temporally filtered image sequences,” J. Electron. Imaging 5, 490–495 (1996).
[CrossRef]

R. Aufrichtig, D. L. Wilson, “X-ray fluoroscopy spatio-temporal filtering with object detection,” IEEE Trans. Med. Imaging 14, 733–746 (1995).
[CrossRef] [PubMed]

R. Aufrichtig, C. Thomas, P. Xue, D. L. Wilson, “A model for perception of pulsed fluoroscopy image sequences,” J. Opt. Soc. Am. A 11, 3167–3176 (1994).
[CrossRef]

D. L. Wilson, P. Xue, R. Aufrichtig, “Perception of fluoroscopy last-image-hold,” Med. Phys. 21, 1875–1883 (1994).
[CrossRef] [PubMed]

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

P. Xue, D. L. Wilson, “Effects of motion blurring in x-ray fluoroscopy,” submitted to Med. Phys.

D. L. Wilson, K. N. Jabri, R. Aufrichtig “Perception of temporally filtered x-ray fluoroscopy images,” submitted to Med. Phys.

D. L. Wilson, K. N. Jabri, P. Xue, “Modeling human visual detection of low-contrast objects in fluoroscopy image sequences,” in Medical Imaging 1997: Image Perception, H. L. Kundel, ed., Proc. SPIE3036, 21–30 (1997).
[CrossRef]

R. Manjeshwar, D. L. Wilson, “Role of smooth pursuit eye-movements on the detection of moving objects in x-ray fluoroscopy noise,” in Proceedings of the 19th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1997).

P. Xue, K. N. Jabri, D. L. Wilson, “The adaptive reference/test forced-choice method with application to fluoroscopy perception,” Medical Imaging 1997: Image Perception, H. L. Kundel, ed., Proc. SPIE3036, 298–307 (1997).
[CrossRef]

R. Manjeshwar, D. L. Wilson, “Eye-tracking of moving objects in x-ray fluoroscopy enhances detectability,” presented at the Seventh Far West Image Perception Conference, Tucson, Ariz., 1997.

P. Xue, C. W. Thomas, G. C. Gilmore, D. L. Wilson, “An adaptive reference/test paradigm with application to pulsed fluoroscopy perception,” Behav. Res. Methods Instrum. Comput. (to be published).

Xue, P.

D. L. Wilson, K. N. Jabri, P. Xue, R. Aufrichtig, “Perceived noise versus display noise in temporally filtered image sequences,” J. Electron. Imaging 5, 490–495 (1996).
[CrossRef]

P. Xue, D. L. Wilson, “Pulsed fluoroscopy detectability from interspersed adaptive forced choice measurements,” Med. Phys. 23, 1833–1843 (1996).
[CrossRef] [PubMed]

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

D. L. Wilson, P. Xue, R. Aufrichtig, “Perception of fluoroscopy last-image-hold,” Med. Phys. 21, 1875–1883 (1994).
[CrossRef] [PubMed]

R. Aufrichtig, C. Thomas, P. Xue, D. L. Wilson, “A model for perception of pulsed fluoroscopy image sequences,” J. Opt. Soc. Am. A 11, 3167–3176 (1994).
[CrossRef]

P. Xue, C. W. Thomas, G. C. Gilmore, D. L. Wilson, “An adaptive reference/test paradigm with application to pulsed fluoroscopy perception,” Behav. Res. Methods Instrum. Comput. (to be published).

P. Xue, K. N. Jabri, D. L. Wilson, “The adaptive reference/test forced-choice method with application to fluoroscopy perception,” Medical Imaging 1997: Image Perception, H. L. Kundel, ed., Proc. SPIE3036, 298–307 (1997).
[CrossRef]

D. L. Wilson, K. N. Jabri, P. Xue, “Modeling human visual detection of low-contrast objects in fluoroscopy image sequences,” in Medical Imaging 1997: Image Perception, H. L. Kundel, ed., Proc. SPIE3036, 21–30 (1997).
[CrossRef]

P. Xue, D. L. Wilson, “Effects of motion blurring in x-ray fluoroscopy,” submitted to Med. Phys.

Yassa, F.

A. Abdel-Malek, F. Yassa, J. Bloomer, “An adaptive gating approach for x-ray dose reduction during interventional procedures,” IEEE Trans. Med. Imaging 13, 2–12 (1994).
[CrossRef]

Circulation

N. L. Eigler, M. P. Eckstein, K. N. Mahrer, J. S. Whiting, “Improving detection of coronary morphological features from digital angiograms: effect of stenosis stabilization display,” Circulation 89, 2700–2709 (1994).
[CrossRef] [PubMed]

Human Factors

G. M. Long, D. A. Rourke, “Training effects on the resolution of moving targets—dynamic visual acuity,” Human Factors 31, 443–451 (1989).

IEEE Trans. Med. Imaging

A. Abdel-Malek, F. Yassa, J. Bloomer, “An adaptive gating approach for x-ray dose reduction during interventional procedures,” IEEE Trans. Med. Imaging 13, 2–12 (1994).
[CrossRef]

R. Aufrichtig, D. L. Wilson, “X-ray fluoroscopy spatio-temporal filtering with object detection,” IEEE Trans. Med. Imaging 14, 733–746 (1995).
[CrossRef] [PubMed]

J. Electron. Imaging

D. L. Wilson, K. N. Jabri, P. Xue, R. Aufrichtig, “Perceived noise versus display noise in temporally filtered image sequences,” J. Electron. Imaging 5, 490–495 (1996).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Soc. Motion Pict. TV Eng.

L. B. Stelmach, P. J. Hearty, “Requirements for static and dynamic spatial resolution in advanced television systems: a psychophysical evaluation,” J. Soc. Motion Pict. TV Eng. 100, 5–9 (1991).

Med. Phys.

A. E. Burgess, “Comparison of receiver operating characteristic and forced choice observer performance measurement methods,” Med. Phys. 22, 643–655 (1995).
[CrossRef] [PubMed]

H. Fujita, K. Doi, M. L. Giger, “Investigation of basic imaging properties in digital radiography. 6. MTFs of II-TV digital imaging systems,” Med. Phys. 12, 713–719 (1985).
[CrossRef] [PubMed]

L. D. Loo, K. Doi, C. E. Metz, “Investigation of basic imaging properties in digital radiography. 4. Effect of unsharp masking on the detectability of simple patterns,” Med. Phys. 12, 209–214 (1985).
[CrossRef] [PubMed]

M. J. Tapiovaara, “Efficiency of low-contrast detail detectability in fluoroscopic imaging,” Med. Phys. 24, 655–664 (1997).
[CrossRef] [PubMed]

D. L. Wilson, P. Xue, R. Aufrichtig, “Perception of fluoroscopy last-image-hold,” Med. Phys. 21, 1875–1883 (1994).
[CrossRef] [PubMed]

P. Xue, D. L. Wilson, “Pulsed fluoroscopy detectability from interspersed adaptive forced choice measurements,” Med. Phys. 23, 1833–1843 (1996).
[CrossRef] [PubMed]

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

S. L. Fritz, S. E. Mirvis, S. O. Pais, S. Roys, “Phantom evaluation of angiographer performance using low frame rate acquisition fluoroscopy,” Med. Phys. 15, 600–603 (1988).
[CrossRef] [PubMed]

Nature (London)

M. J. Morgan, S. Benton, “Motion-deblurring in human vision,” Nature (London) 340, 385–386 (1989).
[CrossRef]

Phys. Med. Biol.

R. F. Wagner, D. G. Brown, “Unified SNR analysis of medical imaging systems,” Phys. Med. Biol. 30, 489–518 (1985).
[CrossRef]

Proc. IEEE

S. M. Kay, J. S. L. Marple, “Spectrum analysis—a modern perspective,” Proc. IEEE 69, 1380–1419 (1981).
[CrossRef]

Proc. R. Soc. London

M. Fahle, T. Poggio, “Visual hyperacuity: spatiotemporal interpolation in human vision,” Proc. R. Soc. London 213, 451–477 (1981).
[CrossRef]

D. Burr, J. Ross, M. C. Marrone, “Seeing objects in motion,” Proc. R. Soc. London 227, 249–265 (1986).
[CrossRef]

Radiology

T. B. Shope, “Radiation-induced skin injuries from fluoroscopy,” Radiology 197(P), 209 (1995) (abstract).

Science

M. Livingstone, D. Hubel, “Segregation of form, color, movement, and depth: anatomy, physiology, and perception,” Science 240, 740–749 (1988).
[CrossRef] [PubMed]

A. E. Burgess, R. F. Wagner, R. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93–94 (1981).
[CrossRef] [PubMed]

Surv. Ophthalmol.

J. W. Miller, E. J. Ludvigh, “The effect of relative motion on visual acuity,” Surv. Ophthalmol. 7, 83–116 (1962).
[PubMed]

Vision Res.

T. Carney, D. A. Silverstein, S. A. Klein, “Vernier acuity during image rotation and translation: visual performance limits,” Vision Res. 35, 1951–1964 (1995).
[CrossRef] [PubMed]

Other

A. Rose, Vision: Human and Electronic (Plenum, New York, 1973).

P. Xue, D. L. Wilson, “Effects of motion blurring in x-ray fluoroscopy,” submitted to Med. Phys.

D. L. Wilson, K. N. Jabri, P. Xue, “Modeling human visual detection of low-contrast objects in fluoroscopy image sequences,” in Medical Imaging 1997: Image Perception, H. L. Kundel, ed., Proc. SPIE3036, 21–30 (1997).
[CrossRef]

M. Fahle, T. Poggio, “Visual hyperacuity: spatiotemporal interpolation in human vision,” in Image Understanding, S. Ullman, W. Richards, eds. (Ablex, Norwood, N.J., 1984), pp. 49–77.

T. R. Morrison, “A review of dynamic visual acuity,” Monograph 28 (Naval Aerospace Medical Research Laboratory, Pensacola, Fla., 1980).

Unlike in some new experiments with a high-contrast fixation marker that moves with the object,38,39 subjects do not report eye motion during the initial detection process. However, they do report that, with easy-to-see targets, they detect and possibly track the target cylinder for a few cycles, leading to enhanced confidence in detection.

R. Manjeshwar, D. L. Wilson, “Role of smooth pursuit eye-movements on the detection of moving objects in x-ray fluoroscopy noise,” in Proceedings of the 19th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1997).

R. Manjeshwar, D. L. Wilson, “Eye-tracking of moving objects in x-ray fluoroscopy enhances detectability,” presented at the Seventh Far West Image Perception Conference, Tucson, Ariz., 1997.

H. L. V. Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1968).

R. N. McDonough, A. D. Whalen, Detection of Signals in Noise, 2nd ed. (Academic, San Diego, Calif., 1995).

National Council on Radiation Protection and Measurement, rep. 100, Exposure of the U.S. Population From Diagnostic Medical Radiation (National Council on Radiation Protection and Measurement, Bethesda, Md., 1989).

J. Whiting, D. Honig, E. Carterette, N. Eigler, “Observer performance in dynamic displays: effect of frame rate on visual signal detection in noisy images,” in Human Vision, Visual Processing, and Digital Display II, B. E. Rogowitz, M. H. Brill, J. P. Allebach, eds., Proc. SPIE1453, 165–176 (1991).
[CrossRef]

H. Luijendijk, “Practical experiments on noise perception in noisy images,” in Medical Imaging 1994: Image Perception, H. L. Kundel, ed., Proc. SPIE2166, 2–8 (1994).
[CrossRef]

J. S. Whiting, M. P. Eckstein, C. A. Morioka, N. L. Eigler, “Effect of additive noise, signal contrast, and feature motion on visual detection in structured noise,” in Medical Imaging 1996: Image Perception, H. L. Kundel, ed., Proc. SPIE2712, 26–38 (1996).
[CrossRef]

M. P. Eckstein, J. S. Whiting, J. P. Thomas, “Detection and contrast discrimination of moving signals in uncorrelated gaussian noise,” Medical Imaging 1996: Image Perception, H. L. Kundel, ed., Proc. SPIE2712, 9–25 (1996).
[CrossRef]

E. Krestel, ed., Imaging Systems for Medical Diagnostics (Siemens, Berlin, 1991).

P. Xue, C. W. Thomas, G. C. Gilmore, D. L. Wilson, “An adaptive reference/test paradigm with application to pulsed fluoroscopy perception,” Behav. Res. Methods Instrum. Comput. (to be published).

D. L. Wilson, K. N. Jabri, R. Aufrichtig “Perception of temporally filtered x-ray fluoroscopy images,” submitted to Med. Phys.

P. Xue, K. N. Jabri, D. L. Wilson, “The adaptive reference/test forced-choice method with application to fluoroscopy perception,” Medical Imaging 1997: Image Perception, H. L. Kundel, ed., Proc. SPIE3036, 298–307 (1997).
[CrossRef]

E. Ammann, G. Wiede, “Generators and tubes in interventional radiology,” in Syllabus: A Categorial Course in Physics, Physical and Technical Aspects of Angiography and Interventional Angiography, S. Balter, T. B. Shope, eds., (RSNA, Oak Brook, Ill., 1995), pp. 59–74.

A. V. Oppenheim, R. W. Schafer, Discrete-Time Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

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Figures (10)

Fig. 1
Fig. 1

Human observer model. A noisy, x-ray fluoroscopy image sequence is acquired and displayed. The observer model consists of a spatiotemporal filter, a spatiotemporal template filter, an optional internal noise source, and a threshold detector (see text for details).

Fig. 2
Fig. 2

Signals and spectra at various steps in the calculations. (a) A thin projected cylinder moving back and forth across the screen at a constant speed gives a 2D signal in space and time (a). Taking the DFT, we get the spatiotemporal frequency domain result in (b). A space–time continuous signal is multiplied by the temporal window function (c). The DFT of the windowed signal in (c) gives the result in (d). (e) A gray-scale rendition of the visual system sensitivity function is shown in (e). The power spectrum of the noise is shown in (f). To produce this acquisition rate less than the display rate, we replicate the pixel row 256/16=16 times, giving the dashed effect in panel (a). 2D data are rendered by use of a logarithmic conversion to gray. Frequency axes have been carefully calibrated in terms of cycles and cycles/deg. (1-pixel-diameter cylinder, 256 pixels/s, 16acq/s, 256 frames/s display, τw=0.5 s.)

Fig. 3
Fig. 3

Absolute detectability u plotted as a function of velocity. Left and right columns are pulsed-16 and pulsed-32, respectively. Cylinder diameter is 1, 5, or 21 pixels, as indicated. To show trends, straight-line segments connect data points for each subject. Data are from four subjects: ×, BA; *, JK; °, PX; Δ, SC.

Fig. 4
Fig. 4

Effects of cylinder size and velocity for 16 and 32 acq/s. Data (symbols) and theory (continuous curves) are shown. Ellipses enclose data points for the four subjects. Normalized data are u/u0, where u0 is the measurement from a stationary cylinder 1 pixel in diameter at 32 acq/s. This normalization clusters the measurements better than in Fig. 3. Model predictions are also normalized to give SNR/SNR0, where SNR0 is the prediction for a stationary cylinder 1 pixel in diameter at 32 acq/sec.

Fig. 5
Fig. 5

For pulsed-16 compared with pulsed-32, measured EPDR values (white bars) are plotted with model predictions (gray bars). (a) Data averaged over four subjects and three cylinder sizes are plotted as a function of velocity. (b) EPDR averaged over four subjects and three velocities are plotted as a function of cylinder diameter. Uncertainties are average standard errors.

Fig. 6
Fig. 6

For three models, predictions of SNR/SNR0 plotted as a function of velocity. In the case of continuous, unidirectional motion the model can be simplified to yield Eq. (A7). This simplified theory is plotted by the thin continuous curve. The thick curve labeled continuous template is the result with a suboptimal, continuous template. The remaining curve labeled discrete template is the prediction for the case when the template, S1(fx, ft), is exactly what one calculates for the signal at a reduced acquisition rate. The ellipses visually link appropriate data points. In all calculations the cylinder diameter is 1 pixel and the display rate is 32 acq/s. Data and models are all normalized to values for a stationary cylinder 1 pixel in diameter at 32 acq/s (Fig. 4).

Fig. 7
Fig. 7

Model SNR/SNR0 plotted as a function of diameter and velocity. For all calculations the acquisition rate is 32 acq/s and the display rate is 256 frames/s. SNR0 is the prediction for a stationary cylinder 1 pixel in diameter at 32 acq/s.

Fig. 8
Fig. 8

Effect of velocity on detectability of large and small objects. Visual system spatial filters are extracted and plotted for (a) a stationary object (v=0 pixels/s) and (b) motion at 256 pixels/s. For convenience these curves are normalized by the peak of the spatial frequency curve at v=0 pixel/s. (c) A large cylinder projection (21 pixels in diameter). (d) Fourier spectrum of the projected cylinder (thick curve) and its product with the filter for v=0 and for v=256 pixels/s. (e) Small cylinder projection (1 pixel in diameter). (f) Fourier spectrum of the projected cylinder (thick curve) and its product with the filter for v=0 and for v=256 pixels/s. See text for interpretation.

Fig. 9
Fig. 9

A projected cylinder moves across the screen at a constant velocity. (a) At a given y location on the display screen the input signal acquired at 256 acq/s is shown where the position of the cylinder moves from left to right with increasing time going down the page. (b) The signal pattern with the acquisition rate reduced to 16 acq/s. The signal in (a) is approximately continuous, whereas the signal in (b) is a low acquisition rate that exhibits discrete sampling properties. Taking the DFT yields the spatial and temporal frequency domain results in (c) and (d) for (a) and (b), respectively. The discrete, low-acquisition-rate input signal (b) gives a periodic result in the frequency domain (d). In both (c) and (d) there are small-magnitude values over much of the 2D surface. In these linearly rendered, gray-scale displays of magnitude, only the dominant components are seen. Cylinder width, 1 pixel; velocity, 256 pixels/s; display rate, 256 frames/s.

Fig. 10
Fig. 10

(a) Visual system, spatiotemporal contrast-sensitivity function as a contour plot. Motion at a constant velocity gives a line in the spatiotemporal frequency domain, fx=ft/v. Dashed line, the particular case of motion at 1 deg/s. This curve is then projected to the vertical dashed line in (b), where the x axis is now velocity. Numbers in the figures are contour values after the logarithm of the sensitivity function is taken.

Equations (12)

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d=uCD1/2,
μl(x)=2µd24-x21/2,-d2xd2,
C=|gb-gp|gb+gp.
SNR=Sst(Nst2+Ni2)1/2= S(fx, fy, ft)|Vst(fx, fy, ft)|2[S1(fx, fy, ft)**Wt(ft)*]dfxdfydft{|[S1(fx, fy, ft)*Wt(ft)]Vst(fx, fy, ft)|2|Vst(fx, fy, ft)|2×Pn(fx, fy, ft)+|[S1(fx, fy, ft)*Wt(ft)]Vst(fx, fy, ft)|2Pi}dfxdfydft1/2.
SNR=AyS(fx, ft)|Vst(fx, ft)|2×[S1(fx, ft)**Wt(ft)*]dfxdft|[S1(fx, ft)*Wt(ft)]|2|Vst(fx, ft)|4×Pn(fx, ft)dfxdft1/2.
V(fx, ft)=[6.1+7.3|log(ft/3fx)|3]4π2ftfx×exp[-4π(ft+2 fx)/45.9].
s(x, t)=s(x-vt).
|S(fx, ft)|2=limT 1T-T/2T/2-s(x-vt)×exp[-j2π(xfx+tft)]dxdt2=limT 1T-T/2T/2-s(z)exp[-j2π(zfx+vtfx+tft)]dzdt2=|S(fx)|2δ(vfx+ft),
SNR=Ay|S(fx)Vst(fx,-vfx)|2dfx|S(fx)Vst(fx,-vfx)|2|Vst(fx,-vfx)|2×σe2dfx1/2.
SNR=Ay|S(fx)V(fx, v)|2dfx|S(fx)V(fx, v)|2|V(fx, v)|2σe2dfx1/2.
SNR=AyS(fx, ft)|Vst(fx, ft)|2×[S1(fx, ft)**Wt(ft)*]ΔfxΔft|[S1(fx, ft)*Wt(ft)]|2×|Vst(fx, ft)|4Pn(fx, ft)ΔfxΔft1/2.
Pn(fx, ft)=sin(kπftΔt)sin(πftΔt)2 σ2kΔtΔx,

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