1. Probabilities of binocular half occlusions in 3D cluttered scenes
Michael Langer
School of Computer Science
McGill University

2. Occlusions
When a binocular observer looks at a 3D scene, there will typically be many points that are not
Visible to each eye. Points that are hidden are said to be OCCLUDED.
For example, consider the simple scene shown here which consists of a foreground surface in front of a background surface.
The foreground surface is said to occlude the background surface.

3. Half Occlusions monocular left right
When we consider the points that are visible to each eye, we find that there
Are points on the background surface that are visible to one eye but not
The other. Such points are said to be MONOCULAR points, or they are
Said to be HALF occluded. Here we see points that are visible to the left
Eye but not the right, and points that are visible to the right eye but noth
The left.

4. Half Occlusions “da Vinci Stereopsis”
monocular
left
right
When we consider the points that are visible to each eye, we find that there
Are points on the background surface that are visible to one eye but not
The other. Such points are said to be MONOCULAR points, or they are
Said to be HALF occluded. Here we see points that are visible to the left
Eye but not the right, and points that are visible to the right eye but noth
The left.
“da Vinci Stereopsis”
Nakayama & Shimojo, Vision Research 1990

5. Natural 3D Cluttered Scene
Today I am going to address a much more complicated situation in which half occlusions arise. This is the case of 3D cluttered scenes such as shown here. 3D cluttered scenes consist of hundreds or small surfaces distributed over a volume in space.

6. Synthetic 3D cluttered scenes

7. Two Questions
Monocular: How are visible surfaces distributed over depth?
Binocular: What percentage of visible surfaces at each depth are half occluded (binocular) ?

8. Assumptions of Model
observer
density h

9. Monocular Visibility
Take surface point at depth Z. Is it visible?

10. Monocular Visibility
A point at depth Z is visible if and only if no sphere center lies within a distance R from the line of sight.

11. Poisson model V

12. Poisson model

13. Probability of monocular visibility
depth Z
depth Z

14. Two Main Questions How are visible surfaces distributed over depth ?
What percentage of points at each depth are half occluded ?

15. Probability of binocular visibility

16. Probability of binocular visibility

17. Conditional probability of binocular visibility
Given that a point at depth Z is visible
to the right eye, what is the probability
that it is also visible to the left eye ?

18. Conditional probability of visibility
left occluded
left visible
right visible at depth Z

19. Conditional probability of visibility
left occluded
left visible
right visible at depth Z

20. Conclusion
The conditional probability that a surface point is visible to one eye, given that it is visible to the other eye, is an (approximately) exponentially decreasing function of depth.
The decay rate depends on the size and density of the surfaces present.

21. For more details, see paper:
“Surface visibility probabilities in 3D cluttered scenes”
M.S. Langer
10th European Conference on Computer Vision (ECCV)
Marseilles FR, Oct. 2008

22. Computational Theory (Bayes)
binocular vs monocular
half occluded

23. Poisson model