3D%20Shape%20Inference

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3D Shape Inference

• Computer Vision
• No.2-1

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Pinhole Camera Model
the camera center
Principal axis
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Perspective Projection
the camera center
the optical axis
the image plane
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Orthographic Projection
the camera center
the optical axis
the image plane
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Weak Perspective Projection
the camera center
the optical axis
the image plane
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Para Perspective Projection
the camera center
the optical axis
the image plane
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Orthographic Projection
the camera center
the optical axis
the image plane
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Obtain a 3D Information form Line Drawing

• Given
• Line drawing(2D)
• Find
• 3D object that projects to given lines
• Find
• How do you think its a cube, not a painted
  pancake?

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Line Labeling

• Significance
• Provides 3D interpretation(within limits)
• Illustrates successful(but incomplete)approach
• Introduces constraints satisfaction
• Pioneers
• Roberts(1976)
• Guzman(1969)
• HuffmanClows (1971)
• Waltz (1972)

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Outline

• Types of lines
• types of vertices
• Junction Dictionary
• Labeling by constraint propagation
• Discussion

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Line Types
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Labeling a Line Drawing
Easy to label lines for this solid ?Now invert
this in order to understand shape
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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)
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Vertex Types

• Divide junctions into categories

Need some constraints to reduce junction types
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Restrictions

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

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Fewer Vertex Types
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Vertex Labeling

• Three planes divide space into octants

• Trihedral vertex
• at intersection of 3 planes

• Enumerate all possibilities
• (Some full, some empty)

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

• 3octants full
• upper behind L
• right above L
• left above L
• straight above ARROW
• straight below FORK

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Enumeration(3)
1octant--Seven viewing octants supply
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HuffmanClows Junction Dictionary

• Any other arrangements cannot arise
• Have reduced configuration from 144 to 12

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Constraints on Labeling

• Without constraints-- 250,000possibilities
• Consider constraints

?3x3x3x6x6x6x5 29,000possibilities

• We can reduce more by
• coherency/consistency along line.

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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

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Example of Labeling
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Ambiguity

• Line drawing can have multiple labelings

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Necker Reversal(1)

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

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Necker Reversal(2)
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Impossible Objects

• No consistent labeling
• But some do have a consistent labeling
• Whats wrong here?

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Limitations of Line Labeling

• Only qualitativeonly gets topology
• Something wrong

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Summary(1)
Preliminary 3D analysis of shape 1. Identify 3D
constraint 2. Determine how constraint affects
images 3. Develop algorithm to exploit
constraint --gt General method for 3D
vision Toolconstraint propagation/satisfaction
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Summary(2)
Problems 1. Significant ambiguity possible 2.
Assumes perfect segmentation 3. Can be fooled
without quantitative analysis
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Gradient Space

• Computer Vision
• No. 2-2

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Gradient Space and Line Labeling

• Last time line labeling by constraint
  propagation
• Use gradient space to represent surface
  orientation

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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

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Surface Normal

• Normal of a plane
• Rewrite

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Surface Gradient

• Gradient of surface is
• Gradient of plane

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Surface Gradient
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Relationship of Normal to Gradient
Normal Vector
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Polyhedron in Gradient Space
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Vector on a Surface

• Suppose vector on surface with
  gradient
• Under orthography, vector in scene projects to
• is surface normal vector, so

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Vector on Two Surfaces

• Suppose vector on boundary between
  two surfaces
• Surfaces have gradients and
• If , then

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Ordering of Points Along Gradient Line
Perpendicular to Connect Edge
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
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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
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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 AB) (line 2)
  impossible. So line 2 occluding.

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Line Labeling using Gradient Space

• 5. Suppose lines 3 and 4 are connect edges
• 6. and so forth can get multiple interpretations

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Another Payoff Detect Inconsistencies
L1
L2
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Summary

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

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References

• M.B. Clowes, On seeing things, Artificial
  Intelligence, Vol.2, pp.79-116, 1971
• D.A. Huffman, Impossible objects as nonsense
  sentences, Machine Intelligence, Vol.6,
  pp.295-323, 1971
• A.K.Mackworth, On reading sketch maps, 5th
  IJCAI, pp.598-606, 1977