Probabilistic Relaxation Labelling by Fokker-Planck Diffusion on a Graph Hongfang Wang and Edwin R. Hancock Department of Computer Science University of York

Outline Introduction Spectral graph theory Probabilistic relaxation labelling Diffusion processes Probabilistic relaxation by diffusion Experiments Discussion

Overview The aim is to exploit the similarities between diffusion processes and relaxation labelling to develop a new iterative process for consistent labelling. A compositional graph structure is used to represent the compatibilities between object-label assignments. Evidence combination and label probability update are controlled by the Fokker-Planck equation. We evaluate our new development on the problem of data classification.

Probabilistic Relaxation Labelling Classical object labelling technique first introduced by Rosenfeld, Hummel and Zucker in early 1970’s.

Relaxation Labelling It aims at assigning consistent and unambiguous labels to a given set of objects. Relies on contextual information provided by topology of object arrangement and sets of compatibility relations on object-configutations. Involves evidence combination to update label probabilities and propagate label consistency constraints. Requires initial label probability assignment.

Relaxation Labelling Support functions for evidence combination: Hummer & Zucker: Kittler & Hancock: Probability update formula:

Graph theoretical setting for relaxation labelling. Formulate relaxation labelling as a random walk on a support graph.

Graph spectral relaxation labeling Set up a support graph where nodes represent possible object-label assignments. Edges represent label compatibility. Set up a random walk on the graph. Probability of visiting node is probability of object-label assignment (state-vector). Evolve state-vector of random with time to update probabilities and combine evidence.

Support graph Node-set Cartesian product of object-set X and label-set Adjacency matrix encodes object proximity and label compatibility: where: W(xi ,xj): the edge weight in the object graph; ωi , ωj: object labels; R(ωi , ωj ): compatibility value between the two labels. Compatiblities assigned using prior konwledge of label domain. Proximity weight set using inter-point distance measure (usually Gaussian).

Example

Fokker-Planck Diffusion Model random walk on graph using Fokker-Planck equation.

Diffusion Processes Diffusion processes are Markov random variables that are correlated and indexed by time. Markov property states that ``for a collection of random variables indexed by time t, given the current value of the variable, the future is independent of the variable’s past’’ Examples of using Markov property: image restoration [e.g., Geman & Geman 1985], Texture [Zhu,Wu & Mumford 1998], and object tracking [Isard & Blake 1996, Hua & Wu 2004, Yang, Duraiswami & Davis 2005]. Here use diffusion process to model random walk on support graph. Local evidence collection and propagation. … short time behaviour [Zhu, Wu & Mumford 1998]: FRAME: Filters, Random field And Maximum Entropy: --- Towards a Unified Theory for Texture Modeling, IJCV 27(2) pp.1-20 [Isard & Blake 1996]: Contour tracking by stochastic propagation of conditional density, ECCV 1996 [Hua & Wu 2004]: Multi-scale Visual Tracking by Sequential Belief Propagation, CVPR 2004 [Yang, Duraiswami & Davis 2005]: Fast Multiple Object Tracking via a Hierarchical Particle Filter, ICCV 2005

Probabilistic Relaxation Labelling by Diffusion Diffusion: global propagation and local evidence combination Diffusion:m we derive the diffusion operator from discrete random walks on graph [Feller Vol.II, 1970]. State space of the process: represented by the support graph. Justification of using diffusion processes on graphs: given sufficient data points, graph is capable of representing the manifold underlying the given data [Belkin&Niyogi 2002, Singer 2006, Sochen2001] Where: F is the Fokker-Planck operator [Feller 1970]: W. Feller, An introduction to probability theory and its applications, Vol.II [Belkin & Niyogi 2002]: Laplacian Eigenmaps and Spectral Techniques for Embedding and Clustering. NIPS 2001 [Singer 2006]: From graph to manifold {L}aplacian: The convergence rate. Applied and Computational Harmonic Analysis, Vol. 21, pp.128-134, 2006 [Sochen2001]: Stochastic Processes in Vision: From Langevin to Beltrami. ICCV 2001.

Solve Fokker-Planck equation using spectrum of operator. Spectral Graph Theory Solve Fokker-Planck equation using spectrum of operator.

Spectral Graph Theory Adjacency matrix A: Laplacian matrix L (weighted): Transition matrix P : Where: E is the graph’s edge set, Wij is the weight between nodes vi and vj ,and degi is the degree of node vi .

Graph-spectral solution of FP Eqn. Transition matrix P and the Fokker Planck operator F: Label probability update formula: where F has the eigen-decomposition: An iteration is carried by setting the newly obtained label probabilities as the current initial label probabilities p0. Haven’t given the relation bet’n Q and F…

Experiments The development is applied to tasks of data classification; Five synthetic and two real world data-sets have been used; Small time steps are used to exploit the short time behaviour of the diffusion process; Compatibility values between labels: Short time behaviour; t is set to a range between 1 and 10. …?

Experiments Five synthetic data-sets: I. Two rings (R2) II. Ring-Gaussian (RG) III. Three Gaussian (G3) V. Four Gaussian G4_2) Top to bottom, left to right: i) syn. Data with two class labels (R2); ii) Three class labels (RG); iii) Three class labels, a mixture of three gaussians (G3); iv) A mixture sample of points from four Gaussian distributions (G4_1); v) A mixture sample of points from four Gaussian distributions (G4_2). IV. Four Gaussian (G4_1)

Experimental Results Resutls of synthetic data-sets: II. Results of Ring-Gaussian data (RG) I. Results of Three Gaussian data (G3) Some data-sets from UCI Repository of machine learning databases: Iris and wine III. Results of Four Gaussian data (G4-1)

Experimental Results Real world data (from UCI machine learning data repository) III. Results of Iris data-set IV. Results of Wine data

Discussion New development of probabilistic relaxation labelling in a graph setting; The diffusion process is used on the graph to collect evidences locally and propagate them globally; The process can also be viewed from a kernel methods perspective; Experiments on data classification tasks are successful; Can be applied to other tasks: Image segmentation; Speaker recognition; Other applications. I’d like to relate these concepts together…possibility of discovering new world.

Muchas grasias!