1、Bayesian Belief Propagation,Reading Group,Overview,Problem Background Bayesian Modelling Markov Random FieldsExamine use of Bayesian Belief Propagation (BBP) in three low level vision applications. Contour Motion Estimation Dense Depth Estimation Unwrapping Phase Images Convergence Issues Conclusion
2、s,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),Bayesian models in low level vision,A statistical description of an estimati
3、on problem. Given data d, we want to estimate unknown parameters uTwo components Prior 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
4、 hidden data u. Combine using Bayes Rule to give the posterior,Markov Random Fields,ui,Pairwise Markov Random Field: Model commonly used to represent images,Contour Motion Estimation,Yair Weiss,Contour Motion Estimation,Estimate the motion of contour using only local information. Less computationall
5、y intensive method than optical flow. Application example: object tracking. Difficult due to the aperture problem.,Contour Motion Estimation,Aperture Problem,Ideal,Actual,Prior Model: ui+1 = ui + nwhere n N(0,sp),Contour Motion Estimation,Brightness Constant Constraint Equation,where Ii = I(xi,yi,t)
6、,1D Belief Propagation,Iterate until message values converge,Results,Contour motion estimation WeissFaster 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
7、velocity components should and do converge to the same value.Interesting to try algorithm on problems where this is not the case Multiple motions within the same contour Rotating contours (requires a new prior model) Only one dimensional problems tackled but extensions to 2D are discussed. Also use
8、of algorithm to solve Direction Of Figure (DOF) problem using convexity (not discussed),Dense Depth Estimation,Richard Szeliski,Depth Estimation,Assume smooth variation in disparity,Define prior using Gibbs Distribution:,Ep(u) is an energy functional:,Depth Estimation,Image T=0,Image T=1,Image T=t,I
9、mage T=t+1,Image T=t+2,Image t=t+3,di,Disparity:,related to correlation metric,i,Where H is a measurement matrix and,Es(u) is an energy functional:,Depth Estimation,E(u) is the overall energy:,Energy function E(u) minimized when u=A-1b,Posterior:,Matrix A-1 is large and expensive to compute,Gauss-Se
10、idel 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:,Results,Dense depth estimation Szeliski Dense (pe
11、r 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. St
12、ochastic Gauss-Seidel relaxation Not 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,Unwrapping Phase Images,Brendan Frey et al,Unwra
13、pping 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 dimension One path through data Use local gradient to estimate shift. For 2D images, the problem is more difficult (NP-hard)
14、 Many paths through the data Shifts along all paths must be consistent,Zero-Curl Constraint,Sensor Data,Estimating relative shift (variables a and b) values -1,0 or 1 between each data point. Use local image gradient as sensor input,Sensor nodes:,Hidden shift nodes:,Gaussian sensor model:,Belief Pro
15、pagation,m4,m5,m5,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 loo
16、ps in the network, messages could cycle leading to incorrect belief estimates. Not guaranteed to converge.,Convergence 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 metho
17、ds such as image pyramids can be used to speed up convergence times (and improve results).,Convergence,Conclusion,BBP 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.
18、 Propagate local information globally to achieve more reliable estimates. Useful for low level vision applications Contour 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.,
