Geometric Inputs to Visual Depth Perception

A translating observer viewing a rigid environment experiences "motion parallax," the relative movement upon the observer's retina of variously positioned objects in the scene. This retinal movement of images provides a cue to the relative depth of objects in the environment, however retinal motion alone cannot mathematically determine relative depth of the objects.

We recently discovered a formula for relative depth in terms of the ratio of the retinal motion rate over the smooth pursuit eye tracking rate called the motion/pursuit law (M/PL), Vision Res. 49, p.1969, 2009. The inclusion of the extra-retinal information from the pursuit eye movement system for the perception of depth is now well established for central vision under lateral motion. Both human psychophysics [NJ, N2003, & M/PL] and primate neural recordings [NAA, NNAA] confirm the use of both cues in depth perception. Work is ongoing for peripheral vision and oblique motion.

Mathematica played a role in this discovery and is very helpful in explaining both the new dynamic formula and the old static formula for depth from convergence and disparity.  There is a remarkable relation between the new and old formulas.  This collection of demonstration NoteBooks explains the new and old formulas interactively.   The "Motion/Pursuit Law" is the most important computation, but the other interactive computations help you understand how that formula works and how it compares with static inputs to depth perception.  By downloading the free Mathematica Player program, you will be able to do the computations yourself.

Motion/Pursuit Law: The 1-D case

1) Interactive computation of the M/PL in 1-D Motion/Pursuit Law in 1D (Visual Depth Perception 1)
2) Interactive computation of the M/PL in 1-D showing pursuit and motion on the axes. Motion, Pursuit, Fixate & Distraction (Visual Depth Perception 2)

Motion/Pursuit Law: The 2-D case

3) Interactive computation of the M/PL t=0 and peak in 2-D with mouse-movable distractor Motion/Pursuit Law in 2 D (Visual Depth Perception 3)
4) Interactive computation of the M/PL t=0 and peak in 2-D around time zero invariant circles Motion/Pursuit Law on Invariant Circles (Visual Depth Perception 4)

The horizontal fixation plane

5) A basic program to show what "fixation" means. Fixation & Distraction (Visual Depth Perception 5)

Eye parameters

6) A program to show variable interocular distance, eye radius, node percent, base point, head aim. Eye Parameters (Visual Depth Perception 6)

Binocular and retinal disparity

7) Interactive computaton of binocular and retinal disparity with variable node percent. Binocular Disparity (Visual Depth Perception 7)
8) The invariant circles (incl. "Vieth-Meuller circle" = "geometric horopter") for binocular disparity. Vieth-Müller Circles (Visual Depth Perception 8)

Static inputs to depth

9) A program showing a curve of constant B.D. with variable depth.  In other words, disparity alone does NOT determine depth. Binocular Disparity vs Depth (Visual Depth Perception 9)
10) The depth and relative depth formula using both B.D. and convergence. Disparity, Convergence & Depth (Visual Depth Perception 10)

Motion parallax and depth

11) A basic program showing the tracking and separation angles animated in time. Tracking & Separation (Visual Depth Perception 11)
12) A program showing a curve of constant motion and variable depth. Motion Parallax vs Depth, 2D (Visual Depth Perception 12)
13) A program showing a 3D curve of constant motion and variable depth and pursuit. Motion Parallax vs Depth, 3D (Visual Depth Perception 13)

Comparison between static and dynamic inputs

14) Interactive computation showing the asymptotic approximation of static and dynamic quantities. Dynamic Approximation of Static Quantities (Visual Depth Perception 14)

Knowledge of speed determines absolute depth

If an observer could determine speed from visual cues, d and f could be determined rather than only d/f
15) f & d from speed and motion Speed, Motion, Pursuit & Depth (Visual Depth Perception 15)
16) f & d from speed and M/P Speed, Pursuit, M/P Ratio & Depth (Visual Depth Perception 16)

The Motion/Pursuit Ratio and the Motion/Pursuit Law

17) Under Esitmate from Ratio Alone 1D Motion/Pursuit Ratio & Depth in 1D (Visual Depth Perceptio 17)
18) Under Esitmate from Ratio Alone 2D Motion/Pursuit Ratio & Depth in 2D(Visual Depth Perception 18)

References

[NJ] M. Nawrot and L. Joyce, "The pursuit theory of motion parallax," Vision Res. 46, 4709-4725 (2006).

[NAA] J. W. Nadler, D. E. Angelaki and G. C. DeAngelis,"A neural representation of depth from motion parallax in macaque visual cortex," Nature 452, 642-645 (2008).

[NNAA] J. W. Nadler, M. Nawrot, M., D. E. Angelaki, and G. C. DeAngelis, G.C., "MT neurons combine visual motion with a smooth eye movement signal to code depth sign from motion parallax, " Neuron, 63, 523-532.

[SN] K. Stroyan and M. Nawrot, "Visual Depth from Motion Parallax and Eye Pursuit," (submitted).

[S1] K. Stroyan (2009). "Motion Parallax is Asymptotic to Binocular Disparity," arXiv

[S2] K. Stroyan (2010). "Optic Flow with Fixation," (in preparation)

 

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