Ecological Statistics of Good Continuation: Multiscale Markov Models for Contours

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Ecological Statistics of Good ContinuationMulti-
scale Markov Models for Contours

• Xiaofeng Ren and Jitendra Malik

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Good Continuation

• Wertheimer 23
• Kanizsa 55
• von der Heydt, Peterhans Baumgartner 84
• Kellman Shipley 91
• Field, Hayes Hess 93
• Kapadia, Westheimer Gilbert 00
• Parent Zucker 89
• Heitger von der Heydt 93
• Mumford 94
• Williams Jacobs 95

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Approach Ecological Statistics

• E. Brunswick
• Ecological validity of perceptual cues
  characteristics of perception match to
  underlying statistical properties of the
  environment

• Brunswick Kamiya 53
• Ruderman 94
• Huang Mumford 99
• Martin et. al. 01

• Gibson 66
• Olshausen Field 96
• Geisler et. al. 01

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Human-Segmented Natural Images
D. Martin et. al., ICCV 2001 1,000 images,
gt14,000 segmentations
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More Examples
D. Martin et. al. ICCV 2001
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Segmentations are Consistent
Perceptual organization forms a tree
Image
BG
L-bird
R-bird
bush
far
grass
beak
body
beak
body
head
eye
eye
head
Two segmentations are consistent when they can
be explained by the same segmentation tree (i.e.
they could be derived from a single perceptual
organization).

• A,C are refinements of B
• A,C are mutual refinements
• A,B,C represent the same percept
• Attention accounts for differences

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Outline of Experiments

• Prior model of contours in natural images
• First-order Markov model
• Test of Markov property
• Multi-scale Markov models
• Information-theoretic evaluation
• Contour synthesis
• Good continuation algorithm and results

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Contour Geometry

• First-Order Markov Model
• ( Mumford 94, Williams Jacobs 95 )
• Curvature white noise ( independent from
  position to position )
• Tangent t(s) random walk
• Markov property the tangent at the next
  position, t(s1), only depends on the current
  tangent t(s)

t(s1)
s1
t(s)
s
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Test of Markov Property
Segment the contours at high-curvature positions
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Prediction Exponential Distribution

• If the first-order Markov property holds
• At every step, there is a constant probability p
  that a high curvature event will occur
• High curvature events are independent from step
  to step
• ? Then the probability of finding a segment of
  length k with no high curvature is (1-p)k

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Empirical Distribution
Exponential ?
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Empirical Distribution Power Law
Probability density
Contour segment length
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Power Laws in Nature

• Power Law widely exists in nature
• Brightness of stars
• Magnitude of earthquakes
• Population of cities
• Word frequency in natural languages
• Revenue of commercial corporations
• Connectivity in Internet topology
• Usually characterized by self-similarity and
  multi-scale phenomena

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Multi-scale Markov Models

• Assume knowledge of contour orientation at
  coarser scales

t(s1)
s1
2nd Order Markov P( t(s1) t(s) , t(1)(s1)
) Higher Order Models P( t(s1) t(s) ,
t(1)(s1), t(2)(s1), )
t(s)
s
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Information Gain in Multi-scale
14.6
of total entropy ( at order 5 )
H( t(s1) t(s) , t(1)(s1), t(2)(s1), )
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Contour Synthesis
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Multi-scale Contour Completion

• Coarse-to-Fine
• Coarse-scale completes large gaps
• Fine-scale detects details
• Completed contours at coarser scales are used in
  the higher-order Markov models of contour prior
  for finer scales
• P( t(s1) t(s) , t(1)(s1), )

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Multi-scale Example
coarse scale
fine scale w/o multi-scale
fine scale w/ multi-scale
input
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Comparison same number of edge pixels
Our result
Canny
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Comparison same number of edge pixels
Our result
Canny
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Conclusion

• Contours are multi-scale in nature the
  first-order Markov property does not hold for
  contours in natural images.
• Higher-order Markov models explicitly model the
  multi-scale nature of contours. We have shown
• The information gain is significant
• Synthesized contours are smooth and rich in
  structure
• Efficient good continuation algorithm has
  produced promising results

Ren Malik, ECCV 2002
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Thank You