1. Why is Spatial Stereoacuity so Poor?
Martin S. Banks
School of Optometry, Dept. of Psychology
UC Berkeley
Sergei Gepshtein
Vision Science Program
UC Berkeley
Michael S. Landy
Dept. of Psychology, Center for Neural Science
NYU
Supported by NIH

2. Depth Perception

3. How precise is the depth map generated from disparity?
Depth Perception
How precise is the depth map generated from disparity?

4. Precision of Stereopsis
from Tyler (1977)
Stereo precision measured in various ways
A: Precision of detecting depth change on line of sight
D: Precision of detecting spatial variation in depth

5. Precision of Stereopsis
from Tyler (1977)
Stereo precision measured in various ways
A: Detect depth change on line of sight
D: Precision of detecting spatial variation in depth

6. Precision of Stereopsis
from Tyler (1977)
Stereo precision measured in various ways
A: Detect depth change on line of sight
D: Detect spatial variation in depth

7. Spatial Stereoacuity
Modulate disparity sinusoidally creating corrugations in depth.
Least disparity required for detection as a function of spatial frequency of corrugations: “Disparity MTF”.
Index of precision of depth map.

8. Disparity MTF
Disparity modulation threshold as a function of spatial frequency of corrugations.
Bradshaw & Rogers (1999).
Horizontal & vertical corrugations.
Disparity MTF: acuity = 2-3 cpd; peak at 0.3 cpd.

9. Luminance Contrast Sensitivity & Acuity
Luminance contrast sensitivity function (CSF): contrast for detection as function of spatial frequency.
Proven useful for characterizing limits of visual performance and for understanding optical, retinal, & post-retinal processing.
Highest detectable spatial frequency (grating acuity): c/deg.

10. Disparity MTF
Spatial stereoacuity more than 1 log unit lower than luminance acuity.

11. Disparity MTF
Spatial stereoacuity more than 1 log unit lower than luminance acuity.
Why is spatial stereoacuity so low?

12. Likely Constraints to Spatial Stereoacuity
Sampling constraints in the stimulus: Stereoacuity measured using random-element stereograms. Discrete sampling limits the highest spatial frequency one can reconstruct.
Disparity gradient limit: With increasing spatial frequency, the disparity gradient increases. If gradient approaches 1.0, binocular fusion fails.
Spatial filtering at the front end: Optical quality & retinal sampling limit acuity in other tasks, so probably limits spatial stereoacuity as well.
The correspondence problem: Manner in which binocular matching occurs presumably affects spatial stereoacuity.

13. Likely Constraints to Spatial Stereoacuity
Sampling constraints in the stimulus: Stereoacuity measured using random-element stereograms. Discrete sampling limits the highest spatial frequency one can reconstruct.
Disparity gradient limit: With increasing spatial frequency, the disparity gradient increases. If gradient approaches 1.0, binocular fusion fails.
Spatial filtering at the front end: Optical quality & retinal sampling limit acuity in other tasks, so probably limits spatial stereoacuity as well.
The correspondence problem: Manner in which binocular matching occurs presumably affects spatial stereoacuity.

14. Spatial Sampling Limit: Nyquist Frequency
Signal reconstruction from discrete samples.
At least 2 samples required per cycle.
In 1d, highest recoverable spatial frequency is Nyquist frequency:
where N is number of samples per unit distance.

15. Spatial Sampling Limit: Nyquist Frequency
Signal reconstruction from 2d discrete samples.
In 2d, Nyquist frequency is:
where N is number of samples in area A.

16. Methodology
Random-dot stereograms with sinusoidal disparity corrugations.
Corrugation orientations: +/-20 deg (near horizontal).
Observers identified orientation in 2-IFC psychophysical procedure; phase randomized.
Spatial frequency of corrugations varied according to adaptive staircase procedure.
Spatial stereoacuity threshold obtained for wide range of dot densities.
Duration = 600 msec; disparity amplitude = 16 minarc.

17. Stimuli

18. Spatial Stereoacuity as a function of Dot Density
Acuity proportional to dot density squared.
Scale invariance!
Asymptote at high density. 1
0.1
Spatial Stereoacuity (c/deg) 1
0.1 1
0.1 10
100 1
0.1 10
100
Dot Density (dots/deg2)

19. Spatial Stereoacuity & Nyquist Limit
Calculated Nyquist frequency for our displays. 1
0.1
Spatial Stereoacuity (c/deg) 1
0.1 1
0.1 10
100 1
0.1 10
100
Dot Density (dots/deg2)

20. Spatial Stereoacuity & Nyquist Limit
Nyquist frequency
Calculated Nyquist frequency for our displays.
Acuity approx. equal to Nyquist frequency except at high densities. 1
0.1
Spatial Stereoacuity (c/deg) 1
0.1 1
0.1 10
100 1
0.1 10
100
Dot Density (dots/deg2)

21. Types of Random-element Stereograms
Jittered-lattice: dots displaced randomly from regular lattice
Sparse random: dots positioned randomly

22. Spatial Sampling Limit: Nyquist Frequency
Same acuities with jittered-lattice and sparse random stereograms.
Both follow Nyquist limit at low densities.

23. Likely Constraints to Spatial Stereoacuity
Sampling constraints in the stimulus: Stereoacuity measured using random-element stereograms. Discrete sampling limits the highest spatial frequency one can reconstruct.
Disparity gradient limit: With increasing spatial frequency, the disparity gradient increases. If gradient approaches 1.0, binocular fusion fails.
Spatial filtering at the front end: Optical quality & retinal sampling limit acuity in other tasks, so probably limits spatial stereoacuity as well.
The correspondence problem: Manner in which binocular matching occurs presumably affects spatial stereoacuity.

24. Disparity Gradient P1 P2 aL aR
Disparity gradient = disparity / separation
= (aR – aL) / [(aL + aR)/2] aL aR

25. Disparity Gradient P1 P2 P1 & P2 on horopter aR = aL, so disparity = 0

26. Disparity Gradient P1 P2 P1 & P2 on cyclopean line of sight
aR = -aL, so separation = 0 P2
Disparity gradient =

27. Disparity Gradient P1 P2 (left & right eyes)
Disparity gradient for different directions. P1
separation
P2 (left & right eyes)
horizontal
disparity

28. Disparity Gradient Limit
Burt & Julesz (1980): fusion as function of disparity, separation, & direction (tilt).
Set direction & horizontal disparity and found smallest fusable separation. P1
separation
direction
P2 (left & right eyes)
disparity

29. Disparity Gradient Limit
Fusion breaks when disparity gradient reaches constant value.
Critical gradient = ~1.
“Disparity gradient limit”.
Limit same for all directions.

30. Disparity Gradient Limit

31. Disparity Gradient Limit
Panum’s fusion area (hatched).
Disparity gradient limit means that fusion area affected by nearby objects (A).
Forbidden zone is conical (isotropic).

32. Horizontal Position (deg)
Disparity Gradient & Spatial Frequency
Disparity gradient for sinusoid is indeterminant.
But for fixed amplitude, gradient proportional to spatial frequency.
We may have approached disparity gradient limit.
Tested by reducing disparity amplitude from 16 to 4.8 minarc.
highest gradient
peak-trough gradient
Disparity (deg)
Horizontal Position (deg)

33. Spatial Stereoacuity & Disparity Gradient Limit
Reducing disparity amplitude increases acuity at high dot densities (where DG is high).
Lowers acuity slightly at low densities (where DG is low). 1
Spatial Stereoacuity (c/deg)
0.1 1
0.1 1
0.1 10
100 1
0.1 10
100
Dot Density (dots/deg2)

34. Likely Constraints to Spatial Stereoacuity
Sampling constraints in the stimulus: Stereoacuity measured using random-element stereograms. Discrete sampling limits the highest spatial frequency one can reconstruct.
Disparity gradient limit: With increasing spatial frequency, the disparity gradient increases. If gradient approaches 1.0, binocular fusion fails.
Spatial filtering at the front end: Optical quality & retinal sampling limit acuity in other tasks, so probably limits spatial stereoacuity as well.
The correspondence problem: Manner in which binocular matching occurs presumably affects spatial stereoacuity.

35. Stereoacuity & Front-end Spatial Filtering
Low-pass spatial filtering at front-end of visual system determines high-frequency roll-off of luminance CSF.
Tested similar effects on spatial stereoacuity by:
Decreasing retinal image size of dots by increasing viewing distance.
Measuring stereoacuity as a function of retinal eccentricity.
Measuring stereoacuity as a function of blur.

36. Stereoacuity & Front-end Spatial Filtering
Low-pass spatial filtering at front-end of visual system determines high-frequency roll-off of luminance CSF.
Tested similar effects on spatial stereoacuity by:
Decreasing retinal image size of dots by increasing viewing distance.
Measuring stereoacuity as a function of retinal eccentricity.
Measuring stereoacuity as a function of blur.

37. Spatial Stereoacuity at Higher Densities. 1 J M A M S B p a t i l e r o c u y ( / d g ) N y q u i s t f r e q u e n c y
Monocular artifacts at high dot densities.
Reduce dot size to test upper limit.
Increase viewing distance from cm.
Acuity still levels off, but at higher value. 1 M o d u l a t i o n V i e w i n g A m p l i t u d e D i s t a n c e 4 . 8 m i n 3 9 c m 4 . 8 m i n 1 5 4 c m
Spatial Stereoacuity (c/deg)
0.1 D M V T M G D o t e n s i y ( d / g 2 ) . 1 1 1 .
0.1 . 1 1
0.1 10
100 1
0.1 10
100 1 1
Dot Density (dots/deg2)

38. Stereoacuity & Front-end Spatial Filtering
Low-pass spatial filtering at front-end of visual system determines high-frequency roll-off of luminance CSF.
Tested similar effects on spatial stereoacuity by:
Decreasing retinal image size of dots by increasing viewing distance.
Measuring stereoacuity as a function of retinal eccentricity.
Measuring stereoacuity as a function of blur.

39. Spatial Stereoacuity & Retinal Eccentricity
Elliptical patch with sinusoidal corrugation.
Patch centered at one of three eccentricities (subject dependent).
Eccentricity random; duration = 250 ms.
Same task as before.
Again vary dot density.
fixation point
eccentricity
4 deg
8 deg

40. Spatial Stereoacuity & Retinal Eccentricity

41. Spatial Stereoacuity & Retinal EccentricityM G Y H H S S G 1 1 .
Spatial Stereoacuity (c/deg)
Retinal eccentricity
0 deg
0 deg
0 deg
0.1 . 1
5.2
6.2
6.8
10.4
12.4
13.6
0.1 1 10
100 1 10
100 1 10
100 . 1 1 . 1 1 1 . 1 1 1 . 1 1
Dot Density (dots/deg2)
Same acuities at low dot densities; Nyquist.
Asymptote varies significantly with retinal eccentricity.

42. Stereoacuity & Front-end Spatial Filtering
Low-pass spatial filtering at front-end of visual system determines high-frequency roll-off of luminance CSF.
Tested similar effects on spatial stereoacuity by:
Decreasing retinal image size of dots by increasing viewing distance.
Measuring stereoacuity as a function of retinal eccentricity.
Measuring stereoacuity as a function of blur.

43. Spatial Stereoacuity & Blur
We examined effect of blur on foveal spatial stereoacuity.
Three levels of blur introduced with diffusion plate:
no blur (s = 0 deg)
low blur (s = 0.12)
high blur (s = 0.25)

44. Spatial Stereoacuity & Blur
We examined effect of blur on foveal spatial stereoacuity.
Three levels of blur introduced with diffusion plate:
no blur (s = 0 deg)
low blur (s = 0.12)
high blur (s = 0.25)

45. Spatial Stereoacuity & Blur
Same acuities at low dot densities; Nyquist.
Asymptote varies significantly with spatial lowpass filtering.

46. Likely Constraints to Spatial Stereoacuity
Sampling constraints in the stimulus: Stereoacuity measured using random-element stereograms. Discrete sampling limits the highest spatial frequency one can reconstruct.
Disparity gradient limit: With increasing spatial frequency, the disparity gradient increases. If gradient approaches 1.0, binocular fusion fails.
Spatial filtering at the front end: Optical quality & retinal sampling limit acuity in other tasks, so probably limits spatial stereoacuity as well.
The correspondence problem: Manner in which binocular matching occurs presumably affects spatial stereoacuity.

47. Binocular Matching by Correlation
Binocular matching by correlation: basic and well-studied technique for obtaining depth map from binocular images.
Computer vision: Kanade & Okutomi (1994); Panton (1978)
Physiology: Ohzawa, DeAngelis, & Freeman (1990); Cumming & Parker (1997)
2. We developed a cross-correlation algorithm for binocular matching & compared its properties to the psychophysics.

48. Binocular Matching by Correlation
Compute cross-correlation between eyes’ images.
Window in left eye’s image moved orthogonal to signal.
For each position in left eye, window in right eye’s image moved horizontally & cross-correlation computed.
Left eye’s image
Right eye’s image

49. Binocular Matching by Correlation
Compute cross-correlation between eyes’ images.
Window in left eye’s image moved orthogonal to signal.
For each position in left eye, window in right eye’s image moved horizontally & cross-correlation computed.
Left eye’s image
Right eye’s image

50. Binocular Matching by Correlation
Compute cross-correlation between eyes’ images.
Window in left eye’s image moved orthogonal to signal.
For each position in left eye, window in right eye’s image moved horizontally & cross-correlation computed.
Left eye’s image
Right eye’s image

51. Position in right eye’s image Position in left eye’s image
Binocular Matching by Correlation
X axis
Y axis
Plot correlation as a function of position in left eye (red arrow) & relative position in right eye (blue arrow); disparity.
Correlation (gray value) high where images similar & low where dissimilar.
Position in right eye’s image
Position in left eye’s image

52. Disparity waveform evident in output
Examples of Output 1
Spatial Frequency
0.1
0.1 1 10
100
Dot Density
Dot density: 16 dots/deg2
Spatial frequency: 1 c/deg
Window size: 0.2 deg
Disparity waveform evident in output

53. Effect of Window Size & Dot Density

54. Effect of Window Size & Dot Density
Correlation window must be large enough to contain sufficient luminance variation to find correct matches

55. Effect of Window Size & Spatial Frequency

56. Effect of Window Size & Spatial Frequency
When significant depth variation is present in a region, window must be small enough to respond differentially

57. Window Size, # Samples, & Spatial Frequency
From two constraints:
then substitute for w, take log:

58. Effect of Disparity Gradient
Fix spatial frequency, dot density, & window size.
Increase disparity amplitude (which also increases disparity gradient: 0.21, 0.59, 1.77).
As approach 1.0, disparity estimation becomes poor. Images too dissimilar in two eyes.
Matching by correlation yields piecewise frontal estimates.

59. Effect of Disparity Gradient
Fix spatial frequency, dot density, & window size.
Increase disparity amplitude (which also increases disparity gradient: 0.21, 0.59, 1.77).
As approach 1.0, disparity estimation becomes poor. Images too dissimilar in two eyes.
Matching by correlation yields piecewise frontal estimates.

60. Effect of Low-pass Spatial Filtering
Amount of variation in image dependent on spatial-frequency content.
If s proportional to w and inversely proportional to , variation constant in cycles/window.
Algorithm yields similar outputs for these images.
For each s, there’s a window just large enough to yield good disparity estimates.
Highest detectable spatial frequency inversely proportional to s.

61. Effect of Low-pass Spatial Filtering
Spatial stereoacuity for different amounts of blur.
s: all filtering elements: dots, optics, diffusion screen.
Horizontal lines: predictions for asymptotic acuities.
Asymptotic acuity limited by filtering before binocular combination.

62. Summary of Matching Effects
Correlation algorithm reveals two effects.
1. Disparity estimation is poor when there’s insufficient intensity variation within correlation window.
a. when window too small for presented dot density
b. when spatial-frequency content is too low.
c. employ a larger window (or receptive field).
2. Disparity estimation is poor when correlation window is too large in direction of maximum disparity gradient.
a. when window width greater than half cycle of stimulus.
b. employ a smaller window (or receptive field).

63. Summary
Sampling constraints in the stimulus: Stereoacuity follows Nyquist limit for all but highest densities. Occurs in peripheral visual field and in fovea with blur.
Disparity gradient limit: Stereoacuity reduced at high gradients.
Spatial filtering at the front end: Low-pass filtering before binocular combination determines asymptotic acuity.
The correspondence problem: Binocular matching by correlation requires sufficient information in correlation window & thereby reduces highest attainable acuity. Visual system measures disparity in piecewise frontal fashion.

64. Depth Map from Disparity
depth-varying scene
Disparity estimates are piecewise frontal.
Only one perceived depth per direction.