Efficient Inference in Fully Connected CRFs with Gaussian Edge Potentials
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1 Efficient Inference in Fully Connected CRFs with Gaussian Edge Potentials by Phillip Krahenbuhl and Vladlen Koltun Presented by Adam Stambler
2 Multi-class image segmentation Assign a class label to each pixel in the image
3 Super pixels are hard to make meanshift ncut-10 ncut-30 Don t make super pixels
4 Operate On Pixels Directly Pixel wise classification- texture/local shape features
5 Consistency with MRF/CRF
6 36 hours!
7 Efficient CRF s results: 0.2 s
8 Solving MRFs and CRFs Each Clique Modeled as Gibbs Distribution Unary Potentials and Pairwise potential + (optional higher order terms) Maximum-a-posteriori solutions are NP-Hard Message Passing algorithms : belief propagation Move Making Algorithms : α-expansion, αβ-swap
9 Graph connections 1.Adjacent pixels are connected Textonboost CRF approach 2.Adjacent pixels are connected + super-pixels consistent Robust Pn CRF 3.All pixels are fully connected Efficient CRF (this paper)
10 Unary Potential : Texton Boost Responsible for most of the accuracy in all of the papers
11 Texton Boosting just pixel crf-no color crf-full TextonBoost: Joint Appearance, Shape and Context Modeling for Multi-Class Object Recognition and Segmentation Each pixel is only connected to its adjacent neighbors Jointly model the texture and shape a single feature
12 Adjacency CRF models E(x) = i ψ u (x i ) unary term + i ψ p (x i, x j ) j N i pairwise term Efficient inference 1 second for variables Limited expressive power Only local interactions Excessive smoothing of object boundaries Shrinking bias P. Krähenbühl (Stanford) Efficient Inference in Fully Connected CRFs December / 29
13 P. Krähenbühl (Stanford) Efficient Inference in Fully Connected CRFs December / 29 Adjacency CRF models E(x) = i ψ u (x i ) unary term + i ψ p (x i, x j ) j N i pairwise term Efficient inference 1 second for variables Limited expressive power Only local interactions Excessive smoothing of object boundaries Shrinking bias
14 P. Krähenbühl (Stanford) Efficient Inference in Fully Connected CRFs December / 29 Adjacency CRF models E(x) = i ψ u (x i ) unary term + i ψ p (x i, x j ) j N i pairwise term Efficient inference 1 second for variables Limited expressive power Only local interactions Excessive smoothing of object boundaries Shrinking bias
15 Operate On Pixels + Super-pixels meanshifts textonboost super pixel based higher order terms ground truth Robust Higher Order Potentials for Enforcing Label Consistency - Koli et al.
16 Operate on Super-pixels + Pixels higher order potentials defined on super pixels to enforce regional consistency soft label constraints using super-pixel consistency potentials unary pairwise super-pixel Super pixel term also models consistency within super pixel
17 Model definition E(x) = i Gaussian edge potentials ψ u (x i ) unary term + i ψ p (x i, x j )=µ(x i, x j ) Label compatibility function µ K m=1 Linear combination of Gaussian kernels j>i ψ p (x i, x j ) pairwise term w (m) k (m) (f i, f j ) k (m) (f i, f j )=exp( 1 2 (f i f j )Σ (m) (f i f j )) Arbitrary feature space f i
18 P. Krähenbühl (Stanford) Efficient Inference in Fully Connected CRFs December / 29 Model definition E(x) = i Gaussian edge potentials ψ u (x i ) unary term + i ψ p (x i, x j )=µ(x i, x j ) Label compatibility function µ K m=1 Linear combination of Gaussian kernels j>i ψ p (x i, x j ) pairwise term w (m) k (m) (f i, f j ) k (m) (f i, f j )=exp( 1 2 (f i f j )Σ (m) (f i f j )) Arbitrary feature space f i
19 Model definition E(x) = i Gaussian edge potentials ψ u (x i ) unary term + i ψ p (x i, x j )= µ(x i, x j ) K m=1 j>i ψ p (x i, x j ) pairwise term w (m) k (m) (f i, f j ) Label compatibility function µ Linear combination of Gaussian kernels k (m) (f i, f j )=exp( 1 2 (f i f j )Σ (m) (f i f j )) Arbitrary feature space f i
20 P. Krähenbühl (Stanford) Efficient Inference in Fully Connected CRFs December / 29 Model definition E(x) = i Gaussian edge potentials ψ u (x i ) unary term + i ψ p (x i, x j )=µ(x i, x j ) Label compatibility function µ K m=1 Linear combination of Gaussian kernels j>i ψ p (x i, x j ) pairwise term w (m) k (m) ( f i, f j ) k (m) (f i, f j )=exp( 1 2 (f i f j )Σ (m) (f i f j )) Arbitrary feature space f i
21 P. Krähenbühl (Stanford) Efficient Inference in Fully Connected CRFs December / 29 Model definition E(x) = i Gaussian edge potentials ψ u (x i ) unary term + i ψ p (x i, x j )=µ(x i, x j ) Label compatibility function µ K m=1 Linear combination of Gaussian kernels j>i ψ p (x i, x j ) pairwise term w (m) k (m) ( f i, f j ) Convolution is key to efficiency k (m) (f i, f j )=exp( 1 2 (f i f j )Σ (m) (f i f j )) Arbitrary feature space f i
22 P. Krähenbühl (Stanford) Efficient Inference in Fully Connected CRFs December / 29 Detailed model definition ψ p (x i, x j )=µ(x i, x j ) w (1) exp( p i p j 2θ 2 α w (2) exp( p i p j 2θ 2 γ I i I j ) + ) 2θ 2 β Label compatibility Potts model: µ(xi, x j )=1 [xi =x j ] Semi-metric model: µ(x i, x j ) learned from data Appearance kernel Color-sensitive model Local smoothness Discourages pixel level noise
23
24
25 Message Passing Uses Mean Field Approximation to minimize KL-divergence Efficiency through signal theory low pass filtering Separable low-pass Gaussian filters propagate information over permutohedral lattice
26 Message Passing by high dimensional filtering Initialize graph with Unary potentials While not converged Pass messages from each node to all other nodes Messages consist of the pairwise blur weighting Update node using compatibility transform
27 P. Krähenbühl (Stanford) Efficient Inference in Fully Connected CRFs December / 29 High-dimensional filtering [Paris & Durand 09] High Dimensional Filtering Downsample input signal Q j (l) Blur the sampled signal Upsample to reconstruct the filtered signal Q j (l)
28 P. Krähenbühl (Stanford) Efficient Inference in Fully Connected CRFs December / 29 High-dimensional filtering [Paris & Durand 09] Downsample input signal Q j (l) Blur the sampled signal Upsample to reconstruct the filtered signal Q j (l)
29 P. Krähenbühl (Stanford) Efficient Inference in Fully Connected CRFs December / 29 High-dimensional filtering [Paris & Durand 09] Downsample input signal Q j (l) Blur the sampled signal Upsample to reconstruct the filtered signal Q j (l)
30 High-dimensional filtering [Paris & Durand 09] Downsample input signal Q j (l) Blur the sampled signal Upsample to reconstruct the filtered signal Q j (l)
31 Permutohedral Lattice
32 Message Passing
33 Learned Parameters Unary Potentials learned using Joint Boost Allows classes to share boundaries and improves generalization Weights for pairwise filtering found via L-BGFS using expectation maximization Kernel Bandwidths hard to learn; Grid search used to pick best value
34
35
36 Really Cool Fine feature segmentation in 0.2 seconds
37 Reported Failures
38 Replicating Results ground truth unary crf
39 Replicating Results : overlay
40 Replicating Results gt unary crf
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