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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 Malik. Good Continuation. Wertheimer ’23 Kanizsa ’55 von der Heydt, Peterhans & Baumgartner ’84 Kellman & Shipley ’91 Field, Hayes & Hess ’93 Kapadia, Westheimer & Gilbert ’00

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

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  1. Ecological Statistics of Good Continuation:Multi-scale Markov Models for Contours Xiaofeng Ren and Jitendra Malik

  2. 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 … …

  3. Brunswick & Kamiya ’53 Ruderman ’94 Huang & Mumford ’99 Martin et. al. ’01 E. Brunswick Ecological validity of perceptual cues: characteristics of perception match to underlying statistical properties of the environment Approach: Ecological Statistics • Gibson ’66 • Olshausen & Field ’96 • Geisler et. al. ’01 • … …

  4. Human-Segmented Natural Images D. Martin et. al., ICCV 2001 1,000 images, >14,000 segmentations

  5. More Examples D. Martin et. al. ICCV 2001

  6. A B C Segmentations are Consistent Perceptual organization forms a tree: Image BG L-bird R-bird bush far grass body beak body beak eye head 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

  7. 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

  8. 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(s+1), only depends on the current tangent t(s) t(s+1) s+1 t(s) s

  9. Test of Markov Property Segment the contours at high-curvature positions

  10. 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

  11. NO Empirical Distribution Exponential ?

  12. Empirical Distribution: Power Law Probability density Contour segment length

  13. 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

  14. t(1)(s+1) s+1 Multi-scale Markov Models • Assume knowledge of contour orientation at coarser scales t(s+1) 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), … ) t(s) s

  15. Information 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), … )

  16. First-Order Markov Multi-scale Markov Contour Synthesis

  17. 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(s+1) | t(s) , t(1)(s+1), … )

  18. Multi-scale: Example coarse scale fine scale w/ multi-scale fine scale w/o multi-scale input

  19. Comparison: same number of edge pixels Canny Our result

  20. Comparison: same number of edge pixels Canny Our result

  21. 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

  22. Thank You

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