22
Computer Vision Group University of California Berkeley Ecological Statistics of Good Continuation: Multi-scale Markov Models for Contours Xiaofeng Ren and Jitendra Malik

Ecological Statistics of Good Continuation: Multi-scale Markov Models for Contours

  • Upload
    jola

  • View
    28

  • Download
    0

Embed Size (px)

DESCRIPTION

Ecological Statistics of Good Continuation: Multi-scale Markov Models for Contours. Xiaofeng Ren and Jitendra Malik. Good Continuation. Wertheimer ’23 Kanizsa ’55 von der Heydt, Peterhans & Baumgartner ’84 Kellman & Shipley ’91 Field, Hayes & Hess ’93 Kapadia, Westheimer & Gilbert ’00 - PowerPoint PPT Presentation

Citation preview

Page 1: Ecological Statistics of Good Continuation: Multi-scale Markov Models for Contours

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

Page 2: Ecological Statistics of Good Continuation: Multi-scale Markov Models for Contours

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

… …

Page 3: Ecological Statistics of Good Continuation: Multi-scale Markov Models for Contours

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

… …

Page 4: Ecological Statistics of Good Continuation: Multi-scale Markov Models for Contours

Computer Vision GroupUniversity of California Berkeley

Human-Segmented Natural ImagesHuman-Segmented Natural Images

D. Martin et. al., ICCV 20011,000 images, >14,000 segmentations

Page 5: Ecological Statistics of Good Continuation: Multi-scale Markov Models for Contours

Computer Vision GroupUniversity of California Berkeley

More ExamplesMore Examples

D. Martin et. al.ICCV 2001

Page 6: Ecological Statistics of Good Continuation: Multi-scale Markov Models for Contours

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).

Page 7: Ecological Statistics of Good Continuation: Multi-scale Markov Models for Contours

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

Page 8: Ecological Statistics of Good Continuation: Multi-scale Markov Models for Contours

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

Page 9: Ecological Statistics of Good Continuation: Multi-scale Markov Models for Contours

Computer Vision GroupUniversity of California Berkeley

Test of Markov PropertyTest of Markov Property

Segment the contours at high-curvature positions

Page 10: Ecological Statistics of Good Continuation: Multi-scale Markov Models for Contours

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

Page 11: Ecological Statistics of Good Continuation: Multi-scale Markov Models for Contours

Computer Vision GroupUniversity of California Berkeley

Exponential

?

Empirical DistributionEmpirical Distribution

NO

Page 12: Ecological Statistics of Good Continuation: Multi-scale Markov Models for Contours

Computer Vision GroupUniversity of California Berkeley

Empirical Distribution: Power LawEmpirical Distribution: Power Law

Contour segment length

Probability

density

62.1)length(

1.Prob

Page 13: Ecological Statistics of Good Continuation: Multi-scale Markov Models for Contours

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

Page 14: Ecological Statistics of Good Continuation: Multi-scale Markov Models for Contours

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

Page 15: Ecological Statistics of Good Continuation: Multi-scale Markov Models for Contours

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

Page 16: Ecological Statistics of Good Continuation: Multi-scale Markov Models for Contours

Computer Vision GroupUniversity of California Berkeley

Contour SynthesisContour Synthesis

Multi-scale Markov

First-Order Markov

Page 17: Ecological Statistics of Good Continuation: Multi-scale Markov Models for Contours

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), … )

Page 18: Ecological Statistics of Good Continuation: Multi-scale Markov Models for Contours

Computer Vision GroupUniversity of California Berkeley

Multi-scale: ExampleMulti-scale: Example

input coarse scale fine scalew/o multi-scale

fine scalew/ multi-scale

Page 19: Ecological Statistics of Good Continuation: Multi-scale Markov Models for Contours

Computer Vision GroupUniversity of California Berkeley

Our resultCanny

Comparison: same number of edge pixels

Page 20: Ecological Statistics of Good Continuation: Multi-scale Markov Models for Contours

Computer Vision GroupUniversity of California Berkeley

Our resultCanny

Comparison: same number of edge pixels

Page 21: Ecological Statistics of Good Continuation: Multi-scale Markov Models for Contours

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

Page 22: Ecological Statistics of Good Continuation: Multi-scale Markov Models for Contours

Computer Vision GroupUniversity of California Berkeley

Thank You