1. 3D Shape Inference
Computer Vision
No.2-1

2. Pinhole Camera Model
the image plane
the camera center
Principal axis

3. Perspective Projection
the camera center
the optical axis
Focal length
the image plane

4. Orthographic Projection
the camera center
the optical axis
the image plane

5. Weak Perspective Projection
the reference plane
the camera center
the optical axis
the image plane

6. Para Perspective Projection
the reference plane
the camera center
the optical axis
the image plane

7. Orthographic Projection
the camera center
the optical axis
the image plane

8. Obtain a 3D Information form Line Drawing
Given
Line drawing(2D)
Find
3D object that projects to given lines
How do you think it’s a cube, not a painted pancake?

9. Line Labeling Significance Pioneers
Provides 3D interpretation(within limits)
Illustrates successful(but incomplete)approach
Introduces constraints satisfaction
Pioneers
Roberts(1976)
Guzman(1969)
Huffman&Clows (1971)
Waltz (1972)

10. Outline Types of lines types of vertices Junction Dictionary
Labeling by constraint propagation
Discussion

11. Line Types
concave
convex
occluding
occluding

12. Labeling a Line Drawing
Easy to label lines for this solid
→Now invert this in order to understand shape

13. Enumerating Possible Line Labeling without Constraints
9 lines
4 labels each
→4ｘ4ｘ4ｘ4ｘ4ｘ4ｘ4ｘ4ｘ4= 250,000 possibilities
We want just one reality
must reduce surplus possibilities
→Need constraints　(by 3D relationship)

14. Vertex Types Divide junctions into categories
Need some constraints to reduce junction types

15. Restrictions No shadows, no cracks Non-singular views
At most three faces meet at vertex

16. Fewer Vertex Types

17. Vertex Labeling Three planes divide space into octants
Trihedral vertex
at intersection of 3 planes
Enumerate all possibilities　
(Some full, some empty)

18. Enumerating Possible Vertex Labeling(1)
0or8octants full--no vertex
2,4,6 octants full
singular view
7octants full
1FORK
5octants full
2L,1ARROW

19. 3octants full Enumeration(2) upper behind L right above L left above L
straight above ARROW
straight below FORK

20. １octant--Seven viewing octants supply
Enumeration(3)
１octant--Seven viewing octants supply

21. Huffman&Clows Junction Dictionary
Any other arrangements cannot arise
Have reduced configuration from 144 to 12

22. Constraints on Labeling
Without constraints ,000possibilities
Consider constraints
→3ｘ3ｘ3ｘ6ｘ6ｘ6ｘ5＝ 29,000possibilities
We can reduce more by
coherency/consistency along line.

23. Labeling by Constraint Propagation
“Waltz filtering”
By coherence rule, line label constrains neighbors
Propagate constraint through common vertex
Usually begin on boundary
May need to backtrack

24. Example of Labeling

25. Line drawing can have multiple labelings
Ambiguity
Line drawing can have multiple labelings

26. Wire-frame cube Necker Reversal(1)
Human perception flips from one to the other
(After Necker 1832,Swiss naturalist)

27. Necker Reversal(2)

29. Impossible Objects No consistent labeling
But some do have a consistent labeling
What’s wrong here?

30. Limitations of Line Labeling
Only qualitative;only gets topology
Something wrong

31. Preliminary 3D analysis of shape
Summary(1)
Preliminary 3D analysis of shape
1. Identify 3D constraint
2. Determine how constraint affects images
3. Develop algorithm to exploit constraint
--> General method for 3D vision
Tool:constraint propagation/satisfaction

32. Summary(2) Problems 1. Significant ambiguity possible
2. Assumes perfect segmentation
3. Can be fooled without quantitative analysis

33. Gradient Space
Computer Vision
No. 2-2

34. Gradient Space and Line Labeling
Last time: line labeling by constraint propagation
Use gradient space to represent surface orientation － ＋

35. Review of Line Labeling
Problem
Given a line drawing, label all the lines with one of 4 symbols
＋ convex edge
－ concave edge
←→ occluding edges
Approach
Narrow down the number of possible labels with a vertex catalog ＋ －

36. Surface Normal
Normal of a plane
Rewrite
Normal vector (A,B,C)

37. Surface Gradient
Gradient of surface is
Gradient of plane

38. Surface Gradient p1 p3 p2 y q p

39. Relationship of Normal to Gradient
(p,q) 1 p q x y
Normal Vector x p1 p4 p5 q p p1 p3 p2 y

40. Polyhedron in Gradient SpaceF E D C B I A ＋ － x y A’ D’ C’ B’ I’ H’ G’ F’ E’ p q
Top view of polyhedron
A ∥ x-y plane
Same order as left

41. Vector on a Surface Suppose vector on surface with gradient
Under orthography, vector in scene projects to
is surface normal vector, so

42. Vector on Two Surfaces Suppose vector on boundary between two surfaces
Surfaces have gradients and
If , then p q G1 G2

43. Ordering of Points Along Gradient Line Perpendicular to Connect Edge
B1’
B2’
B3’ A p q B1 B3 B2 S T A
If connect edge ST convex, then points on gradient space
maintain same order (left-right) as A and Bi in image
If ST concave, then order switches

44. How does this gradient space stuff help us to label lines?＋
L is a “connect edge” (vector on two surface)
Assume orthography
Line in gradient space connecting R1 and R2 must be
perpendicular to line L

45. Line Labeling using Gradient Space
1. Assign arbitrary gradient (0,0) to A
2. Consider B lines 1,2 may be connect edges or may be occluding edges
3. Suppose line 1 a connect edge
4. Suppose line 2 a connect edge, then (line A’B’) (line 2) impossible. So line 2 occluding. B A C 1 2 3 4 5 B’ A’ p q

46. Line Labeling using Gradient Space
5. Suppose lines 3 and 4 are connect edges
6. and so forth can get multiple interpretations B A C 1 2 3 4 5 B’ A’ p q C’ C ＋ －

47. Another Payoff: Detect InconsistenciesL2 L1 L1 L2

48. Summary Can use gradient space to represent surface orientation
detect inconsistent line labels
constraint labeled line drawings
establish line labels without the vertex catalog

49. References
M.B. Clowes, “On seeing things,” Artificial Intelligence, Vol.2, pp , 1971
D.A. Huffman, “Impossible objects as nonsense sentences,” Machine Intelligence, Vol.6, pp , 1971
A.K.Mackworth, “On reading sketch maps,” 5th IJCAI, pp , 1977