A Probabilistic Appraoch to Nonlinear Diffusion: Feature-Driven Filtering Hamid Krim ECE Dept. NCSU, Raleigh, NC Joint Work with Y. Bao, G. Bozkurt and A. Yezzi

2 Outline IntroductionIntroduction Diffusion and Stochastic ProcessesDiffusion and Stochastic Processes Stochastic Interpretation of Heat equation and Perona-Malik equationStochastic Interpretation of Heat equation and Perona-Malik equation Shape driven diffusionShape driven diffusion Numerical resultsNumerical results

3 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

4 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

5 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

6 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

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

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

9 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

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

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

12 Brownian Motion as a random walk 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 Discretize spatial variable and time variable Brownian motion in 2-D space can be viewed as a symmetric random walk

13 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 The solution to the heat equation may then be formulated as a Markov chain

14 Sample paths

15 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

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

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

18 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 The intuitive appeal of above equation is in the dependence of the random walk on geometric features: e.g. gradients at positions P-M diffusion is a controlled diffusion

19 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:

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

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22 Infrared noisy imageFiltered image

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27 Sinusoid Image Filtered Sinusoid Image

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

31 FORMULATION Heat Equation  Gaussian smoothing Image u(x,y): a collection of iso-intensity contours  direction normal to the contour  direction tangent to the contour Idea: less smoothing across image features (  : more smoothing along image features ( 

32 Isotropic diffusion : Anisotropic diffusion [Perona-Malik]: –P-M in terms of u  and u  [Carmona]: –u t = u  Geometric Heat Eqn. (GHE) RELATED WORK

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

34 Question: Given the operator A h 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 STOCHASTIC FORMULATION

35 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(  ) which reflects specific desired goals Corresponding SDE: NEW CLASS OF FLOWS

36 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 NEW CLASS OF FLOWS

37RESULTS Initial set of shapes FlowFlowFlow

38RESULTS (a) Noisy Checkerboard (b) P-M Flow (usual discretization) (c) P-M Flow [Carmona] (d) Proposed stochastic framework for nonlinear diffusionProposed stochastic framework for nonlinear diffusion

39RESULTS (a) Clean building image (b) Noisy building image (c) P-M flow (usual discretization) (d) P-M flow [Carmona] (e) Flow (f) Flow

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