2 Overview Problem Background Bayesian ModellingMarkov Random FieldsExamine use of Bayesian Belief Propagation (BBP) in three low level vision applications.Contour Motion EstimationDense Depth EstimationUnwrapping Phase ImagesConvergence IssuesConclusions
3 Problem Background A problem of probabilistic inference Estimate unknown variables given observed data.For low level vision:Estimate unknown scene properties (e.g. depth) from image properties (e.g. Intensity gradients)
4 Bayesian models in low level vision A statistical description of an estimation problem.Given data d, we want to estimate unknown parameters uTwo componentsPrior Model p(u) – Captures know information about unknown data and is independent of observed data. Distribution of probable solutions.Sensor Model p(d|u) – Describes relationship between sensed measurements d and unknown hidden data u.Combine using Bayes’ Rule to give the posterior
5 Markov Random Fields Image Data Nodes (d) Pairwise Markov Random Field:Model commonly used to represent imagesHidden Scene Nodes (u)Sensor modelPrior modeluiNeighborhood Ni
7 Contour Motion Estimation Estimate the motion of contour using only local information.Less computationally intensive method than optical flow.Application example: object tracking.Difficult due to the aperture problem.
8 Contour Motion Estimation ActualIdealAperture Problem
9 Contour Motion Estimation uiui+1ui-1Brightness Constant Constraint Equationii+1i-1i-2i+2Prior Model: ui+1 = ui + nwhere n ~ N(0,sp)where Ii = I(xi,yi,t)didi-1di+1di-2di+2uiui-1ui+1ui-2ui+2
10 1D Belief Propagation di di-1 di+1 di-2 di+2 ui ui-1 ui+1 ui-2 ui+2 Iterate until message values convergeuiui-1ui+1ui-2ui+2
11 Results Contour motion estimation [Weiss] Faster and more accurate solutions over pre-existing methods such as relaxation.Results after iteration n are optimal given all data within distance of n nodes.Due to the nature of the problem, all velocity components should and do converge to the same value.Interesting to try algorithm on problems where this is not the caseMultiple motions within the same contourRotating contours (requires a new prior model)Only one dimensional problems tackled but extensions to 2D are discussed.Also use of algorithm to solve Direction Of Figure (DOF) problem using convexity (not discussed)
13 Depth Estimation Assume smooth variation in disparity i Depth Zi Disparity ui = 1 / ZiDefine prior using Gibbs Distribution:Ep(u) is an energy functional:
14 Depth Estimation Where H is a measurement matrix and Disparity: related tocorrelation metricdiiImage T=0Image T=1Image T=tImage T=t+1Where H is a measurement matrix andImage T=t+2Image t=t+3Es(u) is an energy functional:
15 Depth Estimation Posterior: E(u) is the overall energy: where and Energy function E(u) minimized when u=A-1bMatrix A-1 is large and expensive to compute
16 Gauss-Seidel Relaxation Minimize energy locally for each node ui keeping all other nodes fixed.Leads to update rule:This is also the estimated mean of the marginal probability distribution p(ui|d) given by Gibbs Sampling.For the 1-D example given by Weiss:
17 Results Dense depth estimation [Szeliski] Dense (per pixel) depth estimation from a sequence of images with known camera motion.Adapted Kalman Filter: estimates of depth from time t-1 are used to improve estimates at time t.Uses multi-resolution technique (image pyramid) to improve convergence times.Uses Gibbs Sampling to sample the posterior.Stochastic Gauss-Seidel relaxationNot guaranteed to converge.Problem can be reformulated to use message passing.Does not account for loops in the network, only recently has belief propagation in networks with loops been fully understood [Yedidia et al]
19 Unwrapping Phase Images Wrapped phase images are produced by devices such as MRI and radar.Unwrapping involves finding shift values between each point.Unwrapping is simple in one dimensionOne path through dataUse local gradient to estimate shift.For 2D images, the problem is more difficult (NP-hard)Many paths through the dataShifts along all paths must be consistent
20 Zero-Curl Constraint Data Point a(x,y) b(x,y) b(x+1,y) a(x,y+1) (x,y) Shift nodeConstraint Node
21 Sensor DataEstimating relative shift (variables a and b) values [-1,0 or 1] between each data point.Use local image gradient as sensor inputSensor nodes:Hidden shift nodes:Gaussian sensor model:Estimate from wrapped image
22 Belief Propagation 1.0 -1 1 Belief a(x,y) Data Point m5 m5 Shift node 1Beliefa(x,y)Data Pointm5m5Shift nodem4m1m2m3Constraint Node
23 Results Unwrapping phase images [Frey et al.] Initialize message to uniform distribution and iterate to convergence.Estimates a solution to an NP-Hard problem in O(n) time in the number of the nodes.Reduction in reconstruction error over relaxation methods.Does not account for loops in the network, messages could cycle leading to incorrect belief estimates.Not guaranteed to converge.
24 ConvergenceConvergence only guaranteed when network is a tree structure and all data is available.In networks with loops, messages can cycle resulting in incorrect belief estimates.Multi-resolution methods such as image pyramids can be used to speed up convergence times (and improve results).
25 ConclusionBBP used to infer marginal posterior distribution of hidden information from observable data.Efficient message passing system is linear in the number of nodes as opposed to exponential.Propagate local information globally to achieve more reliable estimates.Useful for low level vision applicationsContour Motion Estimation [Weiss]Dense Depth Estimation [Szeliski]Unwrapping Phase Images [Frey et al]Improved results over standard relaxation algorithms.Can be used in conjunction with multi-resolution framework to improve convergence times.Need to account for loops to prevent cycling of messages [Yedidia et al].