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Pattern Theory: the Mathematics of Perception

Outline of talk. I.Background: history, motivation, basic definitionsA basic example

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Pattern Theory: the Mathematics of Perception

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    1. Pattern Theory: the Mathematics of Perception Prof. David Mumford Division of Applied Mathematics Brown University International Congress of Mathematics Beijing, 2002

    2. Outline of talk I. Background: history, motivation, basic definitions A basic example – Hidden Markov Models and speech; and extensions The “natural degree of generality” – Markov Random Fields; and vision applications IV. Continuous models: image processing via PDE’s, self-similarity of images and random diffeomorphisms

    3. Some History Is there a mathematical theory underlying intelligence? 40’s – Control theory (Wiener-Pontrjagin), the output side: driving a motor with noisy feedback in a noisy world to achieve a given state 70’s – ARPA speech recognition program 60’s-80’s – AI, esp. medical expert systems, modal, temporal, default and fuzzy logics and finally statistics 80’s-90’s – Computer vision, autonomous land vehicle

    4. Statistics vs. Logic Gauss – Gaussian distributions, least squares ? relocating lost Ceres from noisy incomplete data Control theory – the Kalman-Wiener-Bucy filter AI – Enhanced logics < Bayesian belief networks Vision – Boolean combinations of features < Markov random fields

    5. What you perceive is not what you hear: ACTUAL SOUND The ?eel is on the shoe The ?eel is on the car The ?eel is on the table The ?eel is on the orange PERCEIVED WORDS The heel is on the shoe The wheel is on the car The meal is on the table The peel is on the orange

    6. Why is this old man recognizable from a cursory glance?

    7. The Bayesian Setup, I

    8. The Bayesian Setup, II

    9. A basic example: HMM’s and speech recognition

    10. A basic example: HMM’s and speech recognition

    11. Continuous and discrete variables in perception

    12. A typical stochastic process with jumps

    13. Ex.: daily log-price changes in a sample of stocks

    14. Particle filtering Compiling full conditional probability tables is usually impractical.

    15. Estimating the posterior distribution on optical flow in a movie (from M.Black)

    16. (follow window in red)

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    21. No process is truly Markov Speech has longer range patterns than phonemes: triphones, words, sentences, speech acts, … PCFG’s = “probabilistic context free grammars” = almost surely finite, labeled, random branching processes: Forest of random trees Tn, labels xv on vertices, leaves in 1:1 corresp with observations sm, prob. p1(xvk|xv) on children, p2(sm|xm) on observations.). Unfortunate fact: nature is not so obliging, longer range constraints force context-sensitive grammars. But how to make these stochastic??

    22. Grammar in the parsed speech of Helen, a 2 ˝ year old

    23. Grammar in images (G. Kanisza): contour completion

    24. Markov Random Fields: the natural degree of generality Time ?linear structure of dependencies space/space-time/abstract situations ? general graphical structure of dependencies

    25. A simple MRF: the Ising model

    26. The Ising model and image segmentation

    27. A state-of-the-art image segmentation algorithm (S.-C. Zhu) These results only show the optimal result computed by DDMCMC. One nice thing about DDMCMC is that it has the capability of capturing the global changing intensities on which most the existing algorithms failed. Without the global spline model, the sky will be segmented into several regions.These results only show the optimal result computed by DDMCMC. One nice thing about DDMCMC is that it has the capability of capturing the global changing intensities on which most the existing algorithms failed. Without the global spline model, the sky will be segmented into several regions.

    28. Texture synthesis via MRF’s

    29. Monte Carlo Markov Chains

    30. Bayesian belief propagation and the Bethe approximation

    31. Continuous models I: deblurring and denoising

    32. An example: Bela Bartok enhanced via the Nitzberg-Shiota filter

    33. Continuous models II: images and scaling

    34. Scale invariance has many implications:

    35. Three axioms for natural images

    36. Empirical data on image filter responses

    37. Mathematical models for random images

    38. Continuous models III: random diffeomorphisms

    39. Metrics on Gk, I

    40. Metrics on Gk, II

    41. Geodesics in the quotient space S2

    42. Geodesics in the quotient space of ‘landmark points’ gives a classical mechanical system (Younes)

    43. Outlook for Pattern Theory

    45. A sample of Graunt’s data

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