ecological statistics of good continuation: multi-scale markov models for contours
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Computer Vision GroupUniversity of California Berkeley
Ecological Statistics of Good Continuation:Multi-scale Markov Models for Contours
Ecological Statistics of Good Continuation:Multi-scale Markov Models for Contours
Xiaofeng Ren and Jitendra MalikXiaofeng Ren and Jitendra Malik
Computer Vision GroupUniversity of California Berkeley
Good Continuation 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
… …
• 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
… …
Computer Vision GroupUniversity of California Berkeley
Approach: Ecological StatisticsApproach: Ecological Statistics
• Brunswick & Kamiya ’53• Ruderman ’94• Huang & Mumford ’99• Martin et. al. ’01
• Brunswick & Kamiya ’53• Ruderman ’94• Huang & Mumford ’99• Martin et. al. ’01
E. BrunswickEcological validity of perceptual cues:
characteristics of perception match to underlying statistical properties of the environment
E. BrunswickEcological validity of perceptual cues:
characteristics of perception match to underlying statistical properties of the environment
• Gibson ’66• Olshausen & Field ’96• Geisler et. al. ’01
… …
• Gibson ’66• Olshausen & Field ’96• Geisler et. al. ’01
… …
Computer Vision GroupUniversity of California Berkeley
Human-Segmented Natural ImagesHuman-Segmented Natural Images
D. Martin et. al., ICCV 20011,000 images, >14,000 segmentations
Computer Vision GroupUniversity of California Berkeley
More ExamplesMore Examples
D. Martin et. al.ICCV 2001
Computer Vision GroupUniversity of California Berkeley
Segmentations are ConsistentA
B C
• A,C are refinements of B• A,C are mutual refinements • A,B,C represent the same percept
• Attention accounts for differences
Image
BG L-bird R-bird
grass bush
headeye
beakfar body
headeye
beak body
Perceptual organization forms a tree:
Two segmentations are consistent when they can beexplained by the samesegmentation tree (i.e. theycould be derived from a single perceptual organization).
Computer Vision GroupUniversity of California Berkeley
Outline of ExperimentsOutline 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
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
Computer Vision GroupUniversity of California Berkeley
Contour GeometryContour 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(s+1), only depends on the current tangent t(s)
• 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(s+1), only depends on the current tangent t(s)
t(s)
t(s+1)
s
s+1
Computer Vision GroupUniversity of California Berkeley
Test of Markov PropertyTest of Markov Property
Segment the contours at high-curvature positions
Computer Vision GroupUniversity of California Berkeley
Prediction: Exponential DistributionPrediction: 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
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
Computer Vision GroupUniversity of California Berkeley
Exponential
?
Empirical DistributionEmpirical Distribution
NO
Computer Vision GroupUniversity of California Berkeley
Empirical Distribution: Power LawEmpirical Distribution: Power Law
Contour segment length
Probability
density
62.1)length(
1.Prob
Computer Vision GroupUniversity of California Berkeley
Power Laws in NaturePower 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
• 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
Computer Vision GroupUniversity of California Berkeley
Multi-scale Markov ModelsMulti-scale Markov Models
• Assume knowledge of contour orientation at coarser scales
• Assume knowledge of contour orientation at coarser scales
t(s)
t(s+1)
2nd Order Markov:
P( t(s+1) | t(s) , t(1)(s+1) )
Higher Order Models:
P( t(s+1) | t(s) , t(1)(s+1), t(2)(s+1), … )
s+1
s
t(1)(s+1)
s+1
Computer Vision GroupUniversity of California Berkeley
Information Gain in Multi-scaleInformation Gain in Multi-scale
14.6%
of total entropy ( at order 5 )
H( t(s+1) | t(s) , t(1)(s+1), t(2)(s+1), … )
00.5
11.5
22.5
3
1 2 3 4 5 6
Order of Markov model
Co
nd
itio
na
l E
ntr
op
y
Computer Vision GroupUniversity of California Berkeley
Contour SynthesisContour Synthesis
Multi-scale Markov
First-Order Markov
Computer Vision GroupUniversity of California Berkeley
Multi-scale Contour CompletionMulti-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(s+1) | t(s) , t(1)(s+1), … )
• 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(s+1) | t(s) , t(1)(s+1), … )
Computer Vision GroupUniversity of California Berkeley
Multi-scale: ExampleMulti-scale: Example
input coarse scale fine scalew/o multi-scale
fine scalew/ multi-scale
Computer Vision GroupUniversity of California Berkeley
Our resultCanny
Comparison: same number of edge pixels
Computer Vision GroupUniversity of California Berkeley
Our resultCanny
Comparison: same number of edge pixels
Computer Vision GroupUniversity of California Berkeley
ConclusionConclusion• 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
• 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
Computer Vision GroupUniversity of California Berkeley
Thank You
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