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
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Efficient Belief Propagation for Image Restoration
Qi ZhaoMar.22,2006
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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.
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Outline
• MRF model for Image Restoration• Image Restoration using Efficient Belief
Propagation• Experimental Results(Demo)
– Additive noise removal– Image Inpaiting
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Markov Model
• Motivation– Markov random field models provide a robust and
unified framework for early vision problems.
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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
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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
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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
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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
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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.
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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
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• Parameters
Experiments (2)
20
0.05 1 0.2 0.01
20,1
TL
Original Image
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Experiments (3)
• Comparisons
0.05,20,1
TL
20
0.2,100T
TV Restored Image
BP Restored Image
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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!
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Thank You!