efficient belief propagation for image restoration qi zhao mar.22,2006
DESCRIPTION
Outline MRF model for Image Restoration Image Restoration using Efficient Belief Propagation Experimental Results(Demo) –Additive noise removal –Image InpaitingTRANSCRIPT
Efficient Belief Propagation for Image Restoration
Qi ZhaoMar.22,2006
References
• Pedro F. Felzenszwalb and Daniel P. Huttenlocher. Efficient Belief Propagation for Early Vision. To appear in the International Journal of Computer Vision.
• Y.Weiss and W.T. Freeman. On the optimality of solutions of themax-product belief propagation algorithm in arbitrary graphs. IEEE Transactions on Information Theory, 47(2):723–735, 2001.
• L.I. Rudin, S. Osher, and E. Fatemi, "NONLINEAR TOTAL VARIATION BASED NOISE REMOVAL ALGORITHMS", PHYSICA D 60 (1-4): 259-268 Nov. 1, 1992.
Outline
• MRF model for Image Restoration• Image Restoration using Efficient Belief
Propagation• Experimental Results(Demo)
– Additive noise removal– Image Inpaiting
Markov Model
• Motivation– Markov random field models provide a robust and
unified framework for early vision problems.
MRF Models for Image Restoration
• : set of pixels in an image • : a finite set of labels, which correspond to the underlying
intensities of the pixels.• E.g., , where• Objective: Finding a labeling that minimizes the energy
corresponds to the MAP estimation problem for the defined MRF.– Neighborhood system: 4-neighborhood system– Prior: pair-wise potential cliques– Likelihood energy:
e.g. ,where
– Posterior energy:
gf
i i ig f n 2(0, )in N
2( , ) min{( ) , }, ( , )i i i i i iV f f f f f f
1 2
1( | ) ( | ) exp( )( 2 )
Ni ii N
p g f p g f V
2 2( ) ( ) / 2i i ii i
V V f f g ( ) ( | ) ( ) ( , )i i ii iE f E f g V f V f f
Loopy Belief Propagation: Max-Product
• Let be the message that node sends to a neighboring node at iteration , we have
• Finally, the label that maximizes is individually selected for each node.
ti im
i i
t0 0;i im
1min ( ( ) ( , ) ), ( )i
t ti i f i i i i im V f V f f m i i i
( )( ) ( ) ti i i i ii N ib f V f m
*if ( )i ib f
2( )O nk T
Speed Up Techniques (1)
• Computing a Message Update in Linear Time– Computing , where
• Potts Model:
( ) min ( ( , ) ( ))i
ti i i f i i im f V f f h f
1( ) ( ) ( )t
i i i i ih f V f m f 2( )O k ( )O k
2( ) min ( ( ) ( ))i
ti i i f i i im f c f f h f
( ) min( ( ),min ( ) )i
ti i i i f im f h f h f d
0, 0( )
, 0x
V xd x
• Firstly, compute the lower envelope of the parabolas;
• Secondly, fill in the value of by checking the height of the lower envelope at each grid location .
( )im f
Speed Up Techniques (2)
• BP on the Grid Graph
– The grid graph is bipartite– Two groups of nodes: A & B– Time t:
• Msg from Nodes A -> Nodes B– Time t+1:
• Msg from Nodes B -> Nodes A
Speed Up Techniques (3)
• Multi-Grid BP– Problem in BP: it takes many iterations for information to
flow over large distances in the grid graph. – Basic Idea: to perform BP in a coarse-to-fine manner, so that
long range interactions between pixels can be captured by short paths in coarse graphs.
Experiments (1)
• Noise Removal2( ) min(( ) , )i i i i iV f f f f d 2( ) (( ) )i i i iV f g f
0.05,20,1
TL
20 Original Image
BP Restored Image
• Parameters
Experiments (2)
20
0.05 1 0.2 0.01
20,1
TL
Original Image
Experiments (3)
• Comparisons
0.05,20,1
TL
20
0.2,100T
TV Restored Image
BP Restored Image
Experiments (4)
• Image Impainting– For pixel in the masked region,( ) 0iV f i
(a) Noised, masked image (b) L=1,T=25 (c) L=1,T=14 (d) L=5,T=5
Efficiency Improved by the Coarse-to-Fine Technique!
Thank You!