Abstract

The effect of inherent location uncertainty on the detection of stationary targets was determined in noisy image sequences. Targets were thick and thin projected cylinders mimicking arteries, catheters, and guide wires in medical imaging x-ray fluoroscopy. With the use of an adaptive forced-choice method, detection contrast sensitivity (the inverse of contrast) was measured both with and without marker cues that directed the attention of observers to the target location. With the probability correct clamped at 80%, contrast sensitivity increased an average of 77% when the marker was added to the thin-cylinder target. There was an insignificant effect on the thick cylinder. The large enhancement with the thin cylinder was obtained even though the target was located exactly in the center of a small panel, giving observers the impression that it was well localized. Psychometric functions consisting of d plotted as a function of the square root of the signal-energy-to-noise-ratio gave a positive x intercept for the case of the thin cylinder without a marker. This x intercept, characteristic of uncertainty in other types of detection experiments, disappeared when the marker was added or when the thick cylinder was used. Inherent location uncertainty was further characterized by using four different markers with varying proximity to the target. Visual detection by human observers increased monotonically as the markers better localized the target. Human performance was modeled as a matched-filter detector with an uncertainty in the placement of the template. The removal of a location cue was modeled by introducing a location uncertainty of ≈0.4 mm on the display device or only 7 μm on the retina, a size on the order of a single photoreceptor field. We conclude that detection is affected by target location uncertainty on the order of cellular dimensions, an observation with important implications for detection mechanisms in humans. In medical imaging, the results argue strongly for inclusion of high-contrast visualization markers on catheters and other interventional devices.

© 2001 Optical Society of America

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1999 (3)

1998 (3)

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

P. Xue, C. W. Thomas, G. C. Gilmore, D. L. Wilson, “An adaptive reference/test paradigm: application to pulsed fluoroscopy perception,” Behav. Res. Methods Instrum. Comput. 30, 332–348 (1998).
[CrossRef]

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

1997 (3)

1996 (3)

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]

R. Hubner, “The efficiency of different cue types for reducing spatial frequency uncertainty,” Vision Res. 36, 401–408 (1996).
[CrossRef]

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

1995 (1)

L.-P. Shiu, H. Pashler, “Spatial attention and vernier acuity,” Vision Res. 35, 337–343 (1995).
[CrossRef] [PubMed]

1991 (1)

H. J. Muller, G. W. Humphreys, “Luminance-increment detection: capacity limited or not?” J. Exp. Psychol. 17, 107–124 (1991).

1989 (1)

K. Nakayama, M. Mackeben, “Sustained and transient components of focal visual attention,” Vision Res. 29, 1631–1647 (1989).
[CrossRef] [PubMed]

1988 (1)

1987 (1)

1985 (3)

1984 (2)

1983 (2)

E. T. Davis, P. Kramer, N. Graham, “Uncertainty about spatial frequency, spatial position or contrast of visual patterns,” Percept. Psychophys. 33, 20–28 (1983).
[CrossRef] [PubMed]

D. J. Tolhurst, J. A. Movshon, A. F. Dean, “The statistical reliability of signals in single neurons in cat and monkey visual cortex,” Vision Res. 23, 775–785 (1983).
[CrossRef] [PubMed]

1981 (4)

J. M. Foley, G. E. Legge, “Contrast detection and near threshold discrimination in human vision,” Vision Res. 21, 1041–1053 (1981).
[CrossRef]

D. J. Lasley, T. E. Cohn, “Why luminance discrimination may be better than detection,” Vision Res. 21, 273–278 (1981).
[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]

R. G. Swensson, P. Judy, “Detection of noisy visual targets: models for the effects of spatial uncertainty and signal-to-noise ratio,” Percept. Psychophys. 29, 521–534 (1981).
[CrossRef] [PubMed]

1975 (1)

1968 (1)

W. A. Wickelgren, “Unidimensional strength theory and component analysis of noise in absolute and comparative judgments,” J. Math. Psychol. 5, 102–122 (1968).
[CrossRef]

1961 (1)

W. P. Tanner, “Physiological implications of psychophysical data,” Ann. N.Y. Acad. Sci. 82, 752–765 (1961).

Ahumada, A. J.

Aufrichtig, R.

D. L. Wilson, K. N. Jabri, R. Aufrichtig, “Perception of temporally filtered x-ray fluoroscopy images,” IEEE Trans. Med. Imaging 18, 22–31 (1999).
[CrossRef] [PubMed]

Balz, G. Z.

G. Z. Balz, H. S. Hock, “The effect of attentional spread on spatial resolution,” Vision Res. 37, 1499–1510 (1997).
[CrossRef] [PubMed]

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]

Burgess, A. E.

Cohn, T. E.

D. J. Lasley, T. E. Cohn, “Why luminance discrimination may be better than detection,” Vision Res. 21, 273–278 (1981).
[CrossRef] [PubMed]

Colborne, B.

Davis, E. T.

E. T. Davis, P. Kramer, N. Graham, “Uncertainty about spatial frequency, spatial position or contrast of visual patterns,” Percept. Psychophys. 33, 20–28 (1983).
[CrossRef] [PubMed]

Dean, A. F.

D. J. Tolhurst, J. A. Movshon, A. F. Dean, “The statistical reliability of signals in single neurons in cat and monkey visual cortex,” Vision Res. 23, 775–785 (1983).
[CrossRef] [PubMed]

Dosher, B. A.

Eckstein, M. P.

Foley, J. M.

J. M. Foley, G. E. Legge, “Contrast detection and near threshold discrimination in human vision,” Vision Res. 21, 1041–1053 (1981).
[CrossRef]

Ghandeharian, H.

Gilmore, G. C.

P. Xue, C. W. Thomas, G. C. Gilmore, D. L. Wilson, “An adaptive reference/test paradigm: application to pulsed fluoroscopy perception,” Behav. Res. Methods Instrum. Comput. 30, 332–348 (1998).
[CrossRef]

Graham, N.

E. T. Davis, P. Kramer, N. Graham, “Uncertainty about spatial frequency, spatial position or contrast of visual patterns,” Percept. Psychophys. 33, 20–28 (1983).
[CrossRef] [PubMed]

Green, D.

D. Green, J. A. Swets, Signal Detection Theory and Psychophysics (Krieger, New York, 1974).

Hock, H. S.

G. Z. Balz, H. S. Hock, “The effect of attentional spread on spatial resolution,” Vision Res. 37, 1499–1510 (1997).
[CrossRef] [PubMed]

Hubner, R.

R. Hubner, “The efficiency of different cue types for reducing spatial frequency uncertainty,” Vision Res. 36, 401–408 (1996).
[CrossRef]

Humphreys, G. W.

H. J. Muller, G. W. Humphreys, “Luminance-increment detection: capacity limited or not?” J. Exp. Psychol. 17, 107–124 (1991).

Jabri, K. N.

D. L. Wilson, K. N. Jabri, R. Aufrichtig, “Perception of temporally filtered x-ray fluoroscopy images,” IEEE Trans. Med. Imaging 18, 22–31 (1999).
[CrossRef] [PubMed]

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]

Judy, P.

R. G. Swensson, P. Judy, “Detection of noisy visual targets: models for the effects of spatial uncertainty and signal-to-noise ratio,” Percept. Psychophys. 29, 521–534 (1981).
[CrossRef] [PubMed]

Judy, P. F.

P. F. Judy, M. F. Kijewski, F. Xiaoyang, R. G. Swensson, “Observer detection efficiency with target size uncertainty,” in Medical Imaging 1995: Image Perception, H. L. Kundel, ed., Proc. SPIE2436, 10–17 (1995).
[CrossRef]

Kersten, D.

Kijewski, M. F.

P. F. Judy, M. F. Kijewski, F. Xiaoyang, R. G. Swensson, “Observer detection efficiency with target size uncertainty,” in Medical Imaging 1995: Image Perception, H. L. Kundel, ed., Proc. SPIE2436, 10–17 (1995).
[CrossRef]

Kramer, P.

E. T. Davis, P. Kramer, N. Graham, “Uncertainty about spatial frequency, spatial position or contrast of visual patterns,” Percept. Psychophys. 33, 20–28 (1983).
[CrossRef] [PubMed]

Kundel, H. L.

H. L. Kundel, C. F. Nodine, L. Toto, S. Lauver, “A circle cue enhances detection of simulated masses on mammogram backgrounds,” in Medical Imaging: Image Perception, H. L. Kundel, ed., Proc. SPIE3036, 81–84 (1997).
[CrossRef]

Lasley, D. J.

D. J. Lasley, T. E. Cohn, “Why luminance discrimination may be better than detection,” Vision Res. 21, 273–278 (1981).
[CrossRef] [PubMed]

Lauver, S.

H. L. Kundel, C. F. Nodine, L. Toto, S. Lauver, “A circle cue enhances detection of simulated masses on mammogram backgrounds,” in Medical Imaging: Image Perception, H. L. Kundel, ed., Proc. SPIE3036, 81–84 (1997).
[CrossRef]

Lavie, N.

Legge, G. E.

G. E. Legge, D. Kersten, A. E. Burgess, “Contrast discrimination in noise,” J. Opt. Soc. Am. A 4, 391–404 (1987).
[CrossRef] [PubMed]

J. M. Foley, G. E. Legge, “Contrast detection and near threshold discrimination in human vision,” Vision Res. 21, 1041–1053 (1981).
[CrossRef]

Lu, Z.-L.

Mackeben, M.

K. Nakayama, M. Mackeben, “Sustained and transient components of focal visual attention,” Vision Res. 29, 1631–1647 (1989).
[CrossRef] [PubMed]

Manjeshwar, R. M.

D. L. Wilson, R. M. Manjeshwar, “Role of phase information and eye pursuit in the detection of moving objects in noise,” J. Opt. Soc. Am. A 16, 669–678 (1999).
[CrossRef]

R. M. 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 (IEEE, Piscataway, N.J., 1997), pp. 792–794.

McDonough, R. N.

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

McKee, S. P.

Morgan, M. J.

Movshon, J. A.

D. J. Tolhurst, J. A. Movshon, A. F. Dean, “The statistical reliability of signals in single neurons in cat and monkey visual cortex,” Vision Res. 23, 775–785 (1983).
[CrossRef] [PubMed]

Muller, H. J.

H. J. Muller, G. W. Humphreys, “Luminance-increment detection: capacity limited or not?” J. Exp. Psychol. 17, 107–124 (1991).

Nakayama, K.

K. Nakayama, M. Mackeben, “Sustained and transient components of focal visual attention,” Vision Res. 29, 1631–1647 (1989).
[CrossRef] [PubMed]

Nodine, C. F.

H. L. Kundel, C. F. Nodine, L. Toto, S. Lauver, “A circle cue enhances detection of simulated masses on mammogram backgrounds,” in Medical Imaging: Image Perception, H. L. Kundel, ed., Proc. SPIE3036, 81–84 (1997).
[CrossRef]

Pashler, H.

L.-P. Shiu, H. Pashler, “Spatial attention and vernier acuity,” Vision Res. 35, 337–343 (1995).
[CrossRef] [PubMed]

Pelli, D. G.

Shiu, L.-P.

L.-P. Shiu, H. Pashler, “Spatial attention and vernier acuity,” Vision Res. 35, 337–343 (1995).
[CrossRef] [PubMed]

Solomon, J. A.

Swensson, R. G.

R. G. Swensson, P. Judy, “Detection of noisy visual targets: models for the effects of spatial uncertainty and signal-to-noise ratio,” Percept. Psychophys. 29, 521–534 (1981).
[CrossRef] [PubMed]

P. F. Judy, M. F. Kijewski, F. Xiaoyang, R. G. Swensson, “Observer detection efficiency with target size uncertainty,” in Medical Imaging 1995: Image Perception, H. L. Kundel, ed., Proc. SPIE2436, 10–17 (1995).
[CrossRef]

Swets, J. A.

D. Green, J. A. Swets, Signal Detection Theory and Psychophysics (Krieger, New York, 1974).

Tanner, W. P.

W. P. Tanner, “Physiological implications of psychophysical data,” Ann. N.Y. Acad. Sci. 82, 752–765 (1961).

Thomas, C. W.

P. Xue, C. W. Thomas, G. C. Gilmore, D. L. Wilson, “An adaptive reference/test paradigm: application to pulsed fluoroscopy perception,” Behav. Res. Methods Instrum. Comput. 30, 332–348 (1998).
[CrossRef]

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,” in Medical Imaging 1996: Image Perception, H. L. Kundel, ed., Proc. SPIE2712, 9–25 (1996).
[CrossRef]

Tolhurst, D. J.

D. J. Tolhurst, J. A. Movshon, A. F. Dean, “The statistical reliability of signals in single neurons in cat and monkey visual cortex,” Vision Res. 23, 775–785 (1983).
[CrossRef] [PubMed]

Toto, L.

H. L. Kundel, C. F. Nodine, L. Toto, S. Lauver, “A circle cue enhances detection of simulated masses on mammogram backgrounds,” in Medical Imaging: Image Perception, H. L. Kundel, ed., Proc. SPIE3036, 81–84 (1997).
[CrossRef]

Wagner, R. F.

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.

Westheimer, G.

Whalen, A. D.

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

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]

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

Wickelgren, W. A.

W. A. Wickelgren, “Unidimensional strength theory and component analysis of noise in absolute and comparative judgments,” J. Math. Psychol. 5, 102–122 (1968).
[CrossRef]

Wilson, D. L.

D. L. Wilson, R. M. Manjeshwar, “Role of phase information and eye pursuit in the detection of moving objects in noise,” J. Opt. Soc. Am. A 16, 669–678 (1999).
[CrossRef]

D. L. Wilson, K. N. Jabri, R. Aufrichtig, “Perception of temporally filtered x-ray fluoroscopy images,” IEEE Trans. Med. Imaging 18, 22–31 (1999).
[CrossRef] [PubMed]

P. Xue, C. W. Thomas, G. C. Gilmore, D. L. Wilson, “An adaptive reference/test paradigm: application to pulsed fluoroscopy perception,” Behav. Res. Methods Instrum. Comput. 30, 332–348 (1998).
[CrossRef]

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

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

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

R. M. 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 (IEEE, Piscataway, N.J., 1997), pp. 792–794.

Xiaoyang, F.

P. F. Judy, M. F. Kijewski, F. Xiaoyang, R. G. Swensson, “Observer detection efficiency with target size uncertainty,” in Medical Imaging 1995: Image Perception, H. L. Kundel, ed., Proc. SPIE2436, 10–17 (1995).
[CrossRef]

Xue, P.

P. Xue, C. W. Thomas, G. C. Gilmore, D. L. Wilson, “An adaptive reference/test paradigm: application to pulsed fluoroscopy perception,” Behav. Res. Methods Instrum. Comput. 30, 332–348 (1998).
[CrossRef]

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

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

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

Ann. N.Y. Acad. Sci. (1)

W. P. Tanner, “Physiological implications of psychophysical data,” Ann. N.Y. Acad. Sci. 82, 752–765 (1961).

Behav. Res. Methods Instrum. Comput. (1)

P. Xue, C. W. Thomas, G. C. Gilmore, D. L. Wilson, “An adaptive reference/test paradigm: application to pulsed fluoroscopy perception,” Behav. Res. Methods Instrum. Comput. 30, 332–348 (1998).
[CrossRef]

IEEE Trans. Med. Imaging (1)

D. L. Wilson, K. N. Jabri, R. Aufrichtig, “Perception of temporally filtered x-ray fluoroscopy images,” IEEE Trans. Med. Imaging 18, 22–31 (1999).
[CrossRef] [PubMed]

J. Exp. Psychol. (1)

H. J. Muller, G. W. Humphreys, “Luminance-increment detection: capacity limited or not?” J. Exp. Psychol. 17, 107–124 (1991).

J. Math. Psychol. (1)

W. A. Wickelgren, “Unidimensional strength theory and component analysis of noise in absolute and comparative judgments,” J. Math. Psychol. 5, 102–122 (1968).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (13)

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]

D. L. Wilson, R. M. Manjeshwar, “Role of phase information and eye pursuit in the detection of moving objects in noise,” J. Opt. Soc. Am. A 16, 669–678 (1999).
[CrossRef]

Z.-L. Lu, B. A. Dosher, “Characterizing human perceptual inefficiencies with equivalent internal noise,” J. Opt. Soc. Am. A 16, 764–778 (1999).
[CrossRef]

A. E. Burgess, H. Ghandeharian, “Visual signal detection. I. Ability to use phase information,” J. Opt. Soc. Am. A 1, 900–905 (1984).
[CrossRef] [PubMed]

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Figures (5)

Fig. 1
Fig. 1

Display in the four-alternative forced-choice experiment. A presentation is made of 64 noisy frames of 400×400 pixels in a continuous cine loop. A target cylinder is placed randomly in one of the four noisy panels. The four noise-free panels on the side are reference displays for the subject. The high-contrast, black markers are 3×3-pixel squares placed as shown. In an experiment, the black markers are easily seen, whereas the cylinder contrast is adapted near the threshold for detection. In the figure, the target cylinder contrast is increased so that it can be more easily seen.

Fig. 2
Fig. 2

Four different marker configurations. In (A) the marker is a pair of 3×3-pixel squares placed directly above and below the cylinder. In (B) the marker is a pair of horizontal black bars of height 3 pixels and width 75 pixels centered above and below the cylinder. In (C) the marker is a set of four black 3×3-pixel squares placed at the corners of a rectangle with a distance of 75 pixels on a side. In (D) the distance between the two pairs of marker squares is increased to 145 pixels. The target cylinder is always centered on the markers and the field. The display is 400 pixels across, as shown in Fig. 1.  

Fig. 3
Fig. 3

Detection data for (a) 1-pixel-diameter cylinder and (b) 21-pixel-diameter cylinder with and without location marker A (see Fig. 2). Filled and unfilled symbols represent the data for three observers with and without the location markers, respectively. The signal-to-noise ratios (SNR’s) plotted along the horizontal axis are computed from the final contrasts for 80%, 60%, and 40% probability of correct detection. The values of d are fixed by the probability correct. The standard errors of the SNR’s are 17%, 8%, and 6% for the 40%, 60%, and 80% threshold levels, respectively. (a) For the 1-pixel-diameter cylinder, an analysis of variance gives a significant effect (p<0.05) of adding the location marker. A linear fit to data with the location marker gives a slope and an x intercept of 0.23±0.004 and 0.14±0.12, respectively. Without the marker, the parameters are 0.14±0.005 and 0.96±0.42, respectively. (b) For the 21-pixel-diameter cylinder, there is no significant effect of the marker (p>0.05). A linear fit to the data with the marker gives slopes and intercepts of 0.25±0.004 and 0.09±0.09, respectively. A linear fit to the data without the marker gives slopes and intercepts of 0.24±0.004 and 0.10±0.11, respectively.

Fig. 4
Fig. 4

Monte Carlo predictions of the effect of location error on the detection of (a) a 1-pixel-diameter cylinder. The index of detectability d is plotted as a function of SNR for the ideal observer and the suboptimal observer with and without a location uncertainty of 1 pixel. Both observer models show a significant effect of location uncertainty. For the suboptimal observer, the internal noise is 0.8 times the image noise. Sampling efficiencies are 0.08 and 0.03 with and without location uncertainty, respectively. Model estimates for the 21-pixel-diameter cylinder are plotted in (b). Internal noise and uncertainty values are the same as those in (a). Sampling efficiencies of the suboptimal observer are 0.10 and 0.09 with and without location uncertainty, respectively.

Fig. 5
Fig. 5

Contrast sensitivities for a 1-pixel-diameter cylinder are plotted as a function of the type of marker in (a). Labels A–D refer to similarly labeled markers in Fig. 2. The label NM refers to no marker present. Estimates and standard errors of the 1/C measurements are plotted for two subjects, JK and RM. An analysis of variance shows a significant effect (p<0.05) of markers on 1/C for the 1-pixel-diameter cylinder. Except for markers B and C, the Student–Neuman–Kuel test at α=0.05 shows that the effect of each marker is significantly different from that of the others. Contrast sensitivities for the 21-pixel-diameter cylinder are plotted in (b). An analysis of variance does not show a significant effect of any marker on 1/C (p>0.05).

Equations (4)

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dMF=(E/σext2)1/2,
dSO=JEσext2+K2σext2+σint21/2=Jσext2(1+K2)σext2+σint21/2dMF,
L=Hj=1Pexp1σext2xytSj(x, y, t)D(x, y, t),
C=gb-gt,

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