Abstract

Many visual tasks can be carried out by using several sources of information. The most accurate estimates of scene properties require the observer to utilize all available information and to combine the information sources in an optimal manner. Two experiments are described that required the observers to judge the relative locations of two texture-defined edges (a vernier task). The edges were signaled by a change across the edge of two texture properties [either frequency and orientation (Experiment 1) or contrast and orientation (Experiment 2)]. The reliability of each cue was controlled by varying the distance over which the change (in frequency, orientation, or contrast) occurred—a kind of “texture blur.” In some conditions, the position of the edge signaled by one cue was shifted relative to the other (“perturbation analysis”). An ideal-observer model, previously used in studies of depth perception and color constancy, was fitted to the data. Although the fit can be rejected relative to some more elaborate models, especially given the large quantity of data, this model does account for most trends in the data. A second, suboptimal model that switches between the available cues from trial to trial does a poor job of accounting for the data.

© 2001 Optical Society of America

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References

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2000 (2)

P. V. McGraw, D. Whitaker, D. R. Badcock, “Localising conflicting visual attributes,” Invest. Ophthalmol. Visual Sci. Suppl. 41, S804 (2000).

N. Prins, A. J. Mussap, “Alignment of orientation-modulated textures,” Vision Res. 40, 3567–3573 (2000).
[CrossRef] [PubMed]

1999 (4)

S. C. Dakin, “Orientation variance as a quantifier of structure in texture,” Spatial Vision 12, 1–30 (1999).
[CrossRef] [PubMed]

E. Brenner, M. S. Landy, “Interaction between the perceived shape of two objects,” Vision Res. 39, 933–945 (1999).
[CrossRef]

I. Fine, R. A. Jacobs, “Modeling the combination of motion, stereo, and vergence angle cues to visual depth,” Neural Comput. 11, 1297–1330 (1999).
[CrossRef] [PubMed]

N. E. Scott-Samuel, M. A. Georgeson, “Does early nonlinearity account for second-order motion?” Vision Res. 39, 2853–2865 (1999).
[CrossRef] [PubMed]

1997 (3)

J. Rivest, I. Boutet, J. Intriligator, “Perceptual learning of orientation discrimination by more than one attribute,” Vision Res. 37, 273–281 (1997).
[CrossRef] [PubMed]

J. R. Li, L. T. Maloney, M. S. Landy, “Combination of consistent and inconsistent depth cues,” Invest. Ophthalmol. Visual Sci. Suppl. 38, S903 (1997).

R. Gray, D. Regan, “Vernier step acuity and bisection acuity for texture-defined form,” Vision Res. 37, 1713–1723 (1997).
[CrossRef]

1996 (1)

J. Rivest, P. Cavanagh, “Localizing contours defined by more than one attribute,” Vision Res. 36, 53–66 (1996).
[CrossRef] [PubMed]

1995 (4)

P. Cavanagh, S. Saida, J. Rivest, “The contribution of color to depth perceived from motion parallax,” Vision Res. 35, 1871–1878 (1995).
[CrossRef] [PubMed]

M. S. Landy, L. T. Maloney, E. B. Johnston, M. J. Young, “Measurement and modeling of depth cue combination: in defense of weak fusion,” Vision Res. 35, 389–412 (1995).
[CrossRef] [PubMed]

S. S. Wolfson, M. S. Landy, “Discrimination of orientation-defined texture edges,” Vision Res. 35, 2863–2877 (1995).
[CrossRef] [PubMed]

F. A. A. Kingdom, D. Keeble, B. Moulden, “Sensitivity to orientation modulation in micropattern-based texture,” Vision Res. 35, 79–91 (1995).
[CrossRef] [PubMed]

1994 (3)

T. Ledgeway, A. T. Smith, “Evidence for separate motion-detecting mechanisms for first- and second-order motion in human vision,” Vision Res. 34, 2727–2740 (1994).
[CrossRef] [PubMed]

E. B. Johnston, B. G. Cumming, M. S. Landy, “Integration of stereopsis and motion shape cues,” Vision Res. 34, 2259–2275 (1994).
[CrossRef] [PubMed]

H. R. Wilson, J. Kim, “A model of motion coherence and transparency,” Visual Neurosci. 11, 1205–1220 (1994).
[CrossRef]

1993 (2)

D. W. Massaro, M. M. Cohen, “The paradigm and the fuzzy logical model of perception are alive and well,” J. Exp. Psychol. Gen. 122, 115–124 (1993).
[CrossRef] [PubMed]

M. J. Young, M. S. Landy, L. T. Maloney, “A perturbation analysis of depth perception from combinations of texture and motion cues,” Vision Res. 33, 2685–2696 (1993).
[CrossRef] [PubMed]

1992 (1)

J. E. Cutting, N. Bruno, N. P. Brady, C. Moore, “Selectivity, scope, and simplicity of models: a lesson from fitting judgments of perceived depth,” J. Exp. Psychol. Gen. 121, 364–381 (1992).
[CrossRef] [PubMed]

1991 (1)

M. S. Landy, J. R. Bergen, “Texture segregation and orientation gradient,” Vision Res. 31, 679–691 (1991).
[CrossRef] [PubMed]

1990 (1)

D. W. Massaro, D. Friedman, “Models of integration given multiple sources of information,” Psychol. Rev. 97, 225–252 (1990).
[CrossRef] [PubMed]

1988 (1)

D. W. Massaro, “Ambiguity in perception and experimentation,” J. Exp. Psychol. Gen. 117, 417–421 (1988).
[CrossRef] [PubMed]

1985 (2)

A. M. Derrington, D. R. Badcock, “Separate detectors for simple and complex grating patterns?” Vision Res. 25, 1869–1878 (1985).
[CrossRef] [PubMed]

D. R. Badcock, G. Westheimer, “Spatial location and hyperacuity: the centre/surround localization contribution function has two substrates,” Vision Res. 25, 1259–1267 (1985).
[CrossRef] [PubMed]

1984 (1)

M. S. Landy, Y. Cohen, G. Sperling, “Hips: a unix-based image processing system,” Comput. Vision Graph. Image Process. 25, 331–347 (1984).
[CrossRef]

1981 (1)

1974 (1)

H. Akaike, “A new look at the statistical model identification,” IEEE Trans. Aut. Control AC-19, 716–723 (1974).
[CrossRef]

Akaike, H.

H. Akaike, “A new look at the statistical model identification,” IEEE Trans. Aut. Control AC-19, 716–723 (1974).
[CrossRef]

Aloimonos, J.

J. Aloimonos, D. A. Shulman, Integration of Visual Modules: an Extension of the Marr Paradigm (Academic, New York, 1989).

Badcock, D. R.

P. V. McGraw, D. Whitaker, D. R. Badcock, “Localising conflicting visual attributes,” Invest. Ophthalmol. Visual Sci. Suppl. 41, S804 (2000).

A. M. Derrington, D. R. Badcock, “Separate detectors for simple and complex grating patterns?” Vision Res. 25, 1869–1878 (1985).
[CrossRef] [PubMed]

D. R. Badcock, G. Westheimer, “Spatial location and hyperacuity: the centre/surround localization contribution function has two substrates,” Vision Res. 25, 1259–1267 (1985).
[CrossRef] [PubMed]

Bergen, J. R.

M. S. Landy, J. R. Bergen, “Texture segregation and orientation gradient,” Vision Res. 31, 679–691 (1991).
[CrossRef] [PubMed]

J. R. Bergen, M. S. Landy, “Computational modeling of visual texture segregation,” in Computational Models of Visual Processing, M. S. Landy, J. A. Movshon, eds. (MIT Press, Cambridge, Mass., 1991), pp. 253–271.

Boes, D. C.

A. M. Mood, F. A. Graybill, D. C. Boes, Introduction to the Theory of Statistics, 3rd ed. (McGraw-Hill, New York, 1974).

Boutet, I.

J. Rivest, I. Boutet, J. Intriligator, “Perceptual learning of orientation discrimination by more than one attribute,” Vision Res. 37, 273–281 (1997).
[CrossRef] [PubMed]

Boynton, R. M.

Brady, N. P.

J. E. Cutting, N. Bruno, N. P. Brady, C. Moore, “Selectivity, scope, and simplicity of models: a lesson from fitting judgments of perceived depth,” J. Exp. Psychol. Gen. 121, 364–381 (1992).
[CrossRef] [PubMed]

Brenner, E.

E. Brenner, M. S. Landy, “Interaction between the perceived shape of two objects,” Vision Res. 39, 933–945 (1999).
[CrossRef]

M. S. Landy, E. Brenner, “Motion-disparity interaction and the scaling of stereoscopic disparity,” in Vision and Attention, L. R. Harris, M. R. M. Jenkin, eds. (Springer- Verlag, New York, 2001), Chap. 7.

Bruno, N.

J. E. Cutting, N. Bruno, N. P. Brady, C. Moore, “Selectivity, scope, and simplicity of models: a lesson from fitting judgments of perceived depth,” J. Exp. Psychol. Gen. 121, 364–381 (1992).
[CrossRef] [PubMed]

Buck, S. L.

Bülthoff, H. H.

A. L. Yuille, H. H. Bülthoff, “Bayesian decision theory and psychophysics,” in Perception as Bayesian Inference, D. C. Knill, W. Richards, eds. (Cambridge U. Press, Cambridge, UK, 1996), pp. 123–161.

Cavanagh, P.

J. Rivest, P. Cavanagh, “Localizing contours defined by more than one attribute,” Vision Res. 36, 53–66 (1996).
[CrossRef] [PubMed]

P. Cavanagh, S. Saida, J. Rivest, “The contribution of color to depth perceived from motion parallax,” Vision Res. 35, 1871–1878 (1995).
[CrossRef] [PubMed]

Clark, J. J.

J. J. Clark, A. L. Yuille, Data Fusion for Sensory Information Processing Systems (Kluwer Academic, Boston, 1990).

Cohen, M. M.

D. W. Massaro, M. M. Cohen, “The paradigm and the fuzzy logical model of perception are alive and well,” J. Exp. Psychol. Gen. 122, 115–124 (1993).
[CrossRef] [PubMed]

Cohen, Y.

M. S. Landy, Y. Cohen, G. Sperling, “Hips: a unix-based image processing system,” Comput. Vision Graph. Image Process. 25, 331–347 (1984).
[CrossRef]

Cumming, B. G.

E. B. Johnston, B. G. Cumming, M. S. Landy, “Integration of stereopsis and motion shape cues,” Vision Res. 34, 2259–2275 (1994).
[CrossRef] [PubMed]

Cutting, J. E.

J. E. Cutting, N. Bruno, N. P. Brady, C. Moore, “Selectivity, scope, and simplicity of models: a lesson from fitting judgments of perceived depth,” J. Exp. Psychol. Gen. 121, 364–381 (1992).
[CrossRef] [PubMed]

Dakin, S. C.

S. C. Dakin, “Orientation variance as a quantifier of structure in texture,” Spatial Vision 12, 1–30 (1999).
[CrossRef] [PubMed]

Derrington, A. M.

A. M. Derrington, D. R. Badcock, “Separate detectors for simple and complex grating patterns?” Vision Res. 25, 1869–1878 (1985).
[CrossRef] [PubMed]

Fine, I.

I. Fine, R. A. Jacobs, “Modeling the combination of motion, stereo, and vergence angle cues to visual depth,” Neural Comput. 11, 1297–1330 (1999).
[CrossRef] [PubMed]

Friedman, D.

D. W. Massaro, D. Friedman, “Models of integration given multiple sources of information,” Psychol. Rev. 97, 225–252 (1990).
[CrossRef] [PubMed]

Frome, F. S.

Georgeson, M. A.

N. E. Scott-Samuel, M. A. Georgeson, “Does early nonlinearity account for second-order motion?” Vision Res. 39, 2853–2865 (1999).
[CrossRef] [PubMed]

Gray, R.

R. Gray, D. Regan, “Vernier step acuity and bisection acuity for texture-defined form,” Vision Res. 37, 1713–1723 (1997).
[CrossRef]

Graybill, F. A.

A. M. Mood, F. A. Graybill, D. C. Boes, Introduction to the Theory of Statistics, 3rd ed. (McGraw-Hill, New York, 1974).

Hon, A. K.

A. K. Hon, L. T. Maloney, M. S. Landy, “The influence function for visual interpolation,” in Human Vision and Electronic Imaging II, B. E. Rogowitz, T. N. Pappas, eds., Proc. SPIE3016, 409–419 (1997).
[CrossRef]

Intriligator, J.

J. Rivest, I. Boutet, J. Intriligator, “Perceptual learning of orientation discrimination by more than one attribute,” Vision Res. 37, 273–281 (1997).
[CrossRef] [PubMed]

Jacobs, R. A.

I. Fine, R. A. Jacobs, “Modeling the combination of motion, stereo, and vergence angle cues to visual depth,” Neural Comput. 11, 1297–1330 (1999).
[CrossRef] [PubMed]

Johnston, E. B.

M. S. Landy, L. T. Maloney, E. B. Johnston, M. J. Young, “Measurement and modeling of depth cue combination: in defense of weak fusion,” Vision Res. 35, 389–412 (1995).
[CrossRef] [PubMed]

E. B. Johnston, B. G. Cumming, M. S. Landy, “Integration of stereopsis and motion shape cues,” Vision Res. 34, 2259–2275 (1994).
[CrossRef] [PubMed]

Kaufman, L.

L. Kaufman, Sight and Mind (Oxford, New York, 1974).

Keeble, D.

F. A. A. Kingdom, D. Keeble, B. Moulden, “Sensitivity to orientation modulation in micropattern-based texture,” Vision Res. 35, 79–91 (1995).
[CrossRef] [PubMed]

Kendall, M. K.

M. K. Kendall, A. Stuart, The Advanced Theory of Statistics: Vol. 2. Inference and Relationship, 4th ed. (Macmillan, New York, 1979).

Kim, J.

H. R. Wilson, J. Kim, “A model of motion coherence and transparency,” Visual Neurosci. 11, 1205–1220 (1994).
[CrossRef]

Kingdom, F. A. A.

F. A. A. Kingdom, D. Keeble, B. Moulden, “Sensitivity to orientation modulation in micropattern-based texture,” Vision Res. 35, 79–91 (1995).
[CrossRef] [PubMed]

Knill, D. C.

D. C. Knill, W. Richards, Perception as Bayesian Inference (Cambridge U. Press, Cambridge, UK, 1996).

Landy, M. S.

E. Brenner, M. S. Landy, “Interaction between the perceived shape of two objects,” Vision Res. 39, 933–945 (1999).
[CrossRef]

J. R. Li, L. T. Maloney, M. S. Landy, “Combination of consistent and inconsistent depth cues,” Invest. Ophthalmol. Visual Sci. Suppl. 38, S903 (1997).

M. S. Landy, L. T. Maloney, E. B. Johnston, M. J. Young, “Measurement and modeling of depth cue combination: in defense of weak fusion,” Vision Res. 35, 389–412 (1995).
[CrossRef] [PubMed]

S. S. Wolfson, M. S. Landy, “Discrimination of orientation-defined texture edges,” Vision Res. 35, 2863–2877 (1995).
[CrossRef] [PubMed]

E. B. Johnston, B. G. Cumming, M. S. Landy, “Integration of stereopsis and motion shape cues,” Vision Res. 34, 2259–2275 (1994).
[CrossRef] [PubMed]

M. J. Young, M. S. Landy, L. T. Maloney, “A perturbation analysis of depth perception from combinations of texture and motion cues,” Vision Res. 33, 2685–2696 (1993).
[CrossRef] [PubMed]

M. S. Landy, J. R. Bergen, “Texture segregation and orientation gradient,” Vision Res. 31, 679–691 (1991).
[CrossRef] [PubMed]

M. S. Landy, Y. Cohen, G. Sperling, “Hips: a unix-based image processing system,” Comput. Vision Graph. Image Process. 25, 331–347 (1984).
[CrossRef]

A. K. Hon, L. T. Maloney, M. S. Landy, “The influence function for visual interpolation,” in Human Vision and Electronic Imaging II, B. E. Rogowitz, T. N. Pappas, eds., Proc. SPIE3016, 409–419 (1997).
[CrossRef]

J. R. Bergen, M. S. Landy, “Computational modeling of visual texture segregation,” in Computational Models of Visual Processing, M. S. Landy, J. A. Movshon, eds. (MIT Press, Cambridge, Mass., 1991), pp. 253–271.

L. T. Maloney, M. S. Landy, “A statistical framework for robust fusion of depth information,” in Visual Communications and Image Processing IV, W. A. Pearlman, ed., Proc. SPIE1199, 1154–1163 (1989).
[CrossRef]

M. S. Landy, E. Brenner, “Motion-disparity interaction and the scaling of stereoscopic disparity,” in Vision and Attention, L. R. Harris, M. R. M. Jenkin, eds. (Springer- Verlag, New York, 2001), Chap. 7.

M. S. Landy, “Combining multiple cues for texture edge localization,” in Human Vision, Visual Processing, and Digital Display IV, J. P. Allebach, B. E. Rogowitz, eds., Proc. SPIE1913, 506–517 (1993).
[CrossRef]

Ledgeway, T.

T. Ledgeway, A. T. Smith, “Evidence for separate motion-detecting mechanisms for first- and second-order motion in human vision,” Vision Res. 34, 2727–2740 (1994).
[CrossRef] [PubMed]

Li, J. R.

J. R. Li, L. T. Maloney, M. S. Landy, “Combination of consistent and inconsistent depth cues,” Invest. Ophthalmol. Visual Sci. Suppl. 38, S903 (1997).

Maloney, L. T.

J. R. Li, L. T. Maloney, M. S. Landy, “Combination of consistent and inconsistent depth cues,” Invest. Ophthalmol. Visual Sci. Suppl. 38, S903 (1997).

M. S. Landy, L. T. Maloney, E. B. Johnston, M. J. Young, “Measurement and modeling of depth cue combination: in defense of weak fusion,” Vision Res. 35, 389–412 (1995).
[CrossRef] [PubMed]

M. J. Young, M. S. Landy, L. T. Maloney, “A perturbation analysis of depth perception from combinations of texture and motion cues,” Vision Res. 33, 2685–2696 (1993).
[CrossRef] [PubMed]

L. T. Maloney, M. S. Landy, “A statistical framework for robust fusion of depth information,” in Visual Communications and Image Processing IV, W. A. Pearlman, ed., Proc. SPIE1199, 1154–1163 (1989).
[CrossRef]

L. T. Maloney, “Physics-based approaches to modeling surface color perception,” in Color Vision: From Genes to Perception, K. R. Gegenfurtner, L. T. Sharpe, eds. (Cambridge U. Press, Cambridge, UK, 1999), pp. 387–422.

L. T. Maloney, J. N. Yang, “The illuminant estimation hypothesis and surface color perception,” in Colour Vision:From Light to Object, R. Mausfeld, D. Heyer, eds. (Oxford U. Press, Oxford, UK, to be published).

A. K. Hon, L. T. Maloney, M. S. Landy, “The influence function for visual interpolation,” in Human Vision and Electronic Imaging II, B. E. Rogowitz, T. N. Pappas, eds., Proc. SPIE3016, 409–419 (1997).
[CrossRef]

Marr, D.

D. Marr, Vision (Freeman, San Francisco, Calif., 1982).

Massaro, D. W.

D. W. Massaro, M. M. Cohen, “The paradigm and the fuzzy logical model of perception are alive and well,” J. Exp. Psychol. Gen. 122, 115–124 (1993).
[CrossRef] [PubMed]

D. W. Massaro, D. Friedman, “Models of integration given multiple sources of information,” Psychol. Rev. 97, 225–252 (1990).
[CrossRef] [PubMed]

D. W. Massaro, “Ambiguity in perception and experimentation,” J. Exp. Psychol. Gen. 117, 417–421 (1988).
[CrossRef] [PubMed]

McGraw, P. V.

P. V. McGraw, D. Whitaker, D. R. Badcock, “Localising conflicting visual attributes,” Invest. Ophthalmol. Visual Sci. Suppl. 41, S804 (2000).

Mood, A. M.

A. M. Mood, F. A. Graybill, D. C. Boes, Introduction to the Theory of Statistics, 3rd ed. (McGraw-Hill, New York, 1974).

Moore, C.

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D. Marr, Vision (Freeman, San Francisco, Calif., 1982).

J. Aloimonos, D. A. Shulman, Integration of Visual Modules: an Extension of the Marr Paradigm (Academic, New York, 1989).

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A. L. Yuille, H. H. Bülthoff, “Bayesian decision theory and psychophysics,” in Perception as Bayesian Inference, D. C. Knill, W. Richards, eds. (Cambridge U. Press, Cambridge, UK, 1996), pp. 123–161.

L. T. Maloney, “Physics-based approaches to modeling surface color perception,” in Color Vision: From Genes to Perception, K. R. Gegenfurtner, L. T. Sharpe, eds. (Cambridge U. Press, Cambridge, UK, 1999), pp. 387–422.

L. T. Maloney, J. N. Yang, “The illuminant estimation hypothesis and surface color perception,” in Colour Vision:From Light to Object, R. Mausfeld, D. Heyer, eds. (Oxford U. Press, Oxford, UK, to be published).

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Figures (11)

Fig. 1
Fig. 1

Example stimuli from Experiment 1. (a) An edge cued by a change in spatial frequency from 3 to 6 cpd with texture blur σf=9 min and vertically oriented filtered noise. (b) σf=36 min, horizontal noise. (c) Orientation-defined edge: f=3 cpd, σo=9 min. (d) f=6 cpd, σo=36 min. (e) A typical stimulus from a two-cue experiment. Upper texture, a consistent-cues stimulus; lower texture, Δcue=13.5 min (i.e., the edge cued by spatial frequency is 13.5 min to the right of the orientation-defined edge). Both stimuli have σf=σo=9 min. Observers judged whether the upper edge was to the right or left of the lower edge.

Fig. 2
Fig. 2

Results from single-cue trials for Experiment 1 for observer MJY. Data from each of the four single-cue conditions are plotted. The proportion of times the upper edge was seen as lying to the right of the lower edge is plotted as a function of the distance of actual vernier shift. Symbol area is proportional to the number of trials contributing to each point. The same data are plotted in the two panels. (a) The solid curves are ML fits of cumulative Gaussian functions to individual psychometric functions. (b) The solid curves are the ML fits of the optimal linear cue combination model to the entire Experiment 1 data set for this observer.

Fig. 3
Fig. 3

Results from two-cue trials for Experiment 1 for observer MJY. Data from each of the 21 two-cue conditions are plotted. The proportion of times the consistent-cues edge was seen as lying to the right of the inconsistent-cues edge is plotted as a function of the distance of actual vernier shift between the orientation-defined edges. The separate data sets in each plot are for different values of Δcue: the shift of the frequency-defined edge relative to the orientation-defined edge in the inconsistent-cues stimulus. The symbol area is proportional to the number of trials contributing to each point. The same data are plotted in each pair of panels. Each row of panels corresponds to a different cue reliability condition (indicated in each figure). (a), (c), (e) The curves are ML fits of cumulative Gaussian functions to individual psychometric functions. (b), (d), (f) The curves are ML fits of the optimal linear cue combination model to the entire Experiment 1 data set for this observer.

Fig. 4
Fig. 4

PSEs as a function of Δcue. PSEs are estimated as the 50% points of individual cumulative Gaussian fits to the 21 two-cue conditions in each experiment (e.g., left-hand panels of Fig. 3). Error bars shown on a subset of the data are ±2 standard error of the mean, computed via the matrix of second derivatives of the log-likelihood function with respect to the fit parameters.40,43 Lines are from the fits of the optimal linear cue combination model to the entire data set for each observer and experiment.

Fig. 5
Fig. 5

Example stimuli from Experiment 2. (a) Orientation-defined edge with 100% contrast and σo=36 min. (b) Contrast-defined edge with vertical line segments and σc=9 min. (c) A typical stimulus from a two-cue experiment. Lower texture, consistent-cues stimulus; upper texture, Δcue=-13.5 min (i.e., the edge cued by contrast is 13.5 min to the left of the orientation-defined edge). Both stimuli have σc=σo=9 min.

Fig. 6
Fig. 6

Results from single-cue trials for Experiment 2 for observer MSL. Data from each of the four single-cue conditions are plotted. Plotting conventions and fits are identical to those of Fig. 2. (a) Independent cumulative Gaussian fits. (b) Fits of the optimal linear cue combination model to the entire Experiment 2 data set for this observer.

Fig. 7
Fig. 7

Results from two-cue trials in Experiment 2 for observer MSL. Data from each of the 21 two-cue conditions are plotted. Plotting conventions and fit curves are as in Fig. 3. (a), (c), (e) The curves are ML fits of cumulative Gaussian functions to individual psychometric functions. (b), (d), (f) The curves are ML fits of the optimal linear cue combination model to the entire Experiment 2 data set for this observer.

Fig. 8
Fig. 8

Values of the four σcue parameters from fits of the optimal linear cue combination model to the data for each observer in each experiment. The fit values of σcue increased with increasing texture blur. A logarithmic scale was employed to accommodate the huge value of σcue(o, 36) for observer MJY in Experiment 2.

Fig. 9
Fig. 9

Fits to the two-cue data sets from Experiment 2 for observer MJY. Fits to each of the 21 two-cue conditions are plotted. Plotting conventions and curves are as in Fig. 3, but with the data points omitted. (a), (c), (e) The curves are fits of cumulative Gaussian functions to individual psychometric functions. (b), (d), (f) The curves are fits of the optimal linear cue combination model to the entire Experiment 2 data set for this observer.

Fig. 10
Fig. 10

Scatterplots of the parameters of the individual psychometric function fits to each of the 100 psychometric functions (25 per experiment, 2 experiments, 2 observers) versus those predicted by the four fits of the optimal linear cue combination model to the data from each experiment and observer. (a) PSEs, (b) standard deviations.

Fig. 11
Fig. 11

Histograms of the ratios of the estimated standard deviations from the individual fits for the two-cue conditions to the smaller of the two standard deviations from the corresponding one-cue conditions.

Tables (2)

Tables Icon

Table 1 Relative Log-Likelihood Values and Nested Hypothesis Tests for Various Models Fitted to the Entire Data Set for the Two Experiments and Two Observers a

Tables Icon

Table 2 Relative Log-Likelihood Values and Nested Hypothesis Tests for Various Models Fitted to the Two-Cue Data for the Two Experiments and Two Observers a

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

G(x, y)=1fexp(-f2r2)sin{2πf [x cos(θ)+y sin(θ)]},
p(o, 9, 7, 2)=Pr{N[7.2, σcue2(o, 9)]>N[0, σcue2(o, 9)]},
 
wf=1/σcue2(f, 9)[1/σcue2(f, 9)]+(1/σcue2(o, 9)],
wo=1-wf.
σopt2(f, 9, o, 9)=σcue2(f, 9)σcue2(o, 9)σcue2(f, 9)+σcue2(o, 9).
p(f, 9, o, 9, Δcue, Δx)=Pr[N(Δx, σopt2)>N(wfΔcue, σopt2)].

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