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Outline. Introduction Diffusion and Stochastic Processes Stochastic Interpretation of Heat equation and Perona-Malik equation Shape driven diffusion Numerical results. Motivation. View Nonlinear Diffusion Equations as average laws

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  1. Outline • Introduction • Diffusion and Stochastic Processes • Stochastic Interpretation of Heat equation and Perona-Malik equation • Shape driven diffusion • Numerical results

  2. Motivation • View Nonlinear Diffusion Equations as average laws • Average laws involve macroscopic variables such as temperature, heat flux • Address performance variability and its dependence upon initial data • Statistical approach aims at accepting NDE as such, and at refining description by seeking probabilistic laws for variables

  3. Motivation • Approach entails a more detailed study of equations • Interacting particle systems are not new in physics • Techniques tying macroscopic to microscopic description abound in statistical mechanics • Use notions as inspiration to investigate nonlinear evolution equations applied to images

  4. Strategy • Correlation structure of images naturally captured by nonlinearities are modeled by particle interaction • An image is viewed as a density of particles to be evolved • Motion of particle achieving diffusion assumes a probabilistic model • Simplest model is well known for linear diffusion

  5. Introduction • Multiscale filtering has played an increasingly important role in signal and image analysis • A nonlinear PDE (Perona- Malik first proposed equation) for feature(edge) - driven smoothing

  6. Diffusion and Random Processes • Motion of particles in as a result of diffusion is captured by a stochastic process Where is a d-dimensional Brownian Motion. drift and diffusion coefficients. • Denote as the transition probability function • of , i.e., • pace is a continuous Markov process with infinitesimal operator

  7. Infinitesimal operator of this diffusion: • It can be shown that satisfies PDE • where • The solution to PDE e.q.(3) can be written as

  8. An image is a 2-D function, • Let be image data at scale s=0. • PDE-based diffusion on this image • a filtered image at scale t

  9. Heat Equation: • First consider the Heat diffusion • In e.q. (3), let • Namely infinitesimal operator is Laplacian • Heat equation: --- original noisy image --- image at scale

  10. Stochastic Solution of a Heat Equation • Resulting solution is written as --- Standard Brownian motion • --Gaussian probability transition • density function

  11. Brownian Motion as a random walk • Brownian motion in 2-D spacecan be viewed as a • symmetric random walk • Discretize spatial variable and time variable • Simple random walk converges to a Brownian Motion as • A particle on such trajectory will move to its four nearest neighbors with equal probability of 1/4

  12. The solution to the heat equation may then be formulated as a Markov chain • A random walk with equal probability(1/4) moving to its neighbors. Brownian motion corresponds to constant diffusion coefficient, i.e., homogenous evolution which smoothes away noise and sharp features

  13. Sample paths

  14. Nonlinear Stochastic Diffusion Perona-Malik equation • One possible choice is where K determines the rate of decay • Discretizing scale space, yields the following evolution equation

  15. is the transition probability of a Markov chain from state to its four neighbors.

  16. A south moving walk takes place with probability • A north moving walk takes place with probability • A random walk as a self loop takes place with probability • A east moving walk takes place with probability • A west moving walk takes place with probability

  17. The intuitive appeal of above equation is in the • dependence of the random walk on geometric • features: e.g. gradients at positions • Motion towards the region of smaller gradient is favored by the transition probability • The diffusion corresponding to this random walk is • nonlinear with diffusion coefficient varying • with positions • Underlying diffusion of P-M equation is non-homogenous • P-M diffusion is a controlled diffusion

  18. A Subgradient Driven Stochastic Diffusion • Our proposed technique is a Markov chain with a transition probability based on subgradients • Using Markov chain formalism, we can model the following transition probabilities:

  19. To yield • Note: Stability of the filter is reached when the domain • of attraction (staircase function) • Note: For very small subgradients, the motion of the particle is reduced to a simple random walk

  20. Infrared noisy image Filtered image

  21. Sinusoid Image Filtered Sinusoid Image

  22. EXTENSIONSTO CURVE EVOLUTION Evolution of level sets of 2-d fn. via PDE’s: • Scale-space analysis • Feature-driven progressive smoothing • Extracting desired features in noise

  23. FORMULATION • Heat Equation  Gaussian smoothing • Image u(x,y): a collection of iso-intensity contours • h: direction normal to the contour x: direction tangent to the contour • Idea: less smoothing across image features (h) : more smoothing along image features (x)

  24. RELATED WORK • Isotropic diffusion : • Anisotropic diffusion [Perona-Malik]: • P-M in terms of uxx and uhh [Carmona]: • ut = uxxGeometric Heat Eqn. (GHE)

  25. STOCHASTIC FORMULATION • Evolution eqn (PDE) corresponds to an infinitesimal generator of a Stochastic Differential Eqn. (SDE) • Ito Diffusion Xt ; A mathematical model for the position Xt of a small particle suspended in a moving liquid at time t • A 2nd order partial differential operator A can be associated to an Ito diffusion Xt as the generator of the process

  26. STOCHASTIC FORMULATION • Question: Given the operator Ah governing the geometric heat flow, can we obtain a corresponding SDE of the underlying diffusion of the individual pixels? • Explains • GHE generates a tangential diffusion of individual pixels along the contour • Iso-intensity contours are smoothed maximally

  27. NEW CLASS OF FLOWS • Convergence to a point of every contour subjected to a GHE will not preserve features of a level curve [Grayson] • A generalization of GHE to feature/shape adapted flow: construct an SDE with chosen functional h(q) which reflects specific desired goals • Corresponding SDE:

  28. NEW CLASS OF FLOWS • One class of functional whose flow leads level curves to 2-n-gones : • E.g. • Heuristically: • stop diffusion at 4 diagonal orientations • allow maximal diffusion at 4 horizontal/vertical orientations of a level curve • Denoise/Enhance square-like features in an image by driving its level curves to squares

  29. RESULTS Initial set of shapes Flow Flow Flow

  30. RESULTS • Proposed stochastic framework for nonlinear diffusion (a) Noisy Checkerboard (b) P-M Flow (usual discretization) (c) P-M Flow [Carmona] (d)

  31. RESULTS • (a) Clean building image (b) Noisy building image (c) P-M flow (usual discretization) • (d) P-M flow [Carmona] (e) Flow (f) Flow

  32. CONCLUSIONS • Clarification of dependence on initial data • A new class of flows which can produce polygonal-invariant shapes that can be useful in various shape recognition tasks • Exploiting joint stochastic and geometric viewpoint, a general feature/shape adapted flow for image enhancement/segmentation/feature extraction problems • No prior knwoledgeof stopping time required

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