Multi-label MRF Optimization via a Least Squares s - t Cut
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Citations
Local optimization based segmentation of spatially-recurring, multi-region objects with part configuration constraints.
Exhaustive family of energies minimizable exactly by a graph cut
References
Efficient shape matching via graph cuts
A Simple Algorithm for the Planar Multiway Cut Problem
A new graph cut-based multiple active contour algorithm without initial contours and seed points
Landmark-based non-rigid registration via graph cuts
Efficient Algorithms for k-Terminal Cuts on Planar Graphs
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Frequently Asked Questions (16)
Q2. What future works have the authors mentioned in the paper "Multi-label mrf optimization via a least squares s − t cut" ?
More elaborate analysis of the algorithm ( e. g. error bounds, value of the minimized energy, computational complexity, running times ) and comparison with state-of-the-art approaches on standard benchmarks is left for future work.
Q3. What is the LS error in edge weights?
The LS error in edge weights induces error in the s − t cut or binary labeling, which is decoded into a suboptimal solution to the multi-label problem.
Q4. What is the way to label a graph?
Many visual computing tasks can be formulated as graph labeling problems, e.g. segmentation and stereo-reconstruction [1], in which one out of k labels is assigned to each graph vertex.
Q5. What is the simplest way to decode labels?
The authors perform a single (non-iterative and initialization-independent) s − t cut to obtain a “Gray” binary encoding, which is then unambiguously decoded into the k labels.
Q6. how much is a severing edge in g2?
the cost of severing intra-links in G2 to assign li to vertex vi in G isDintrai (li) =b∑m=1b∑n=m+1(lim ⊕ lin) wvim,vin (3)where ⊕ denotes binary XOR.
Q7. what is the solution to the original multi-label problem?
every sequence of b binary labels (vij)bj=1 is decoded to a decimal label li ∈ Lk = {l0, l1, ..., lk−1}, ∀vi ∈ V , i.e. the solution to the original multi-label MRF problem.
Q8. What is the noise level of the edge weights?
The authors then construct GLSE , a noisy version of G, by adding uniformly distributed noise with support [0, noise level] to the edge weights.
Q9. What is the purpose of this paper?
the authors are exploring the use of non-negative least squares (e.g. Chapter 23 in [16]) to guarantee non-negative edge weights as well as quantifying the benefits of the Gray encoding.
Q10. what is the way to label a graph?
The Markov random field (MRF) formulation captures this desired label interaction via an energy ξ(l) to be minimized with respect to the vertex labels l.ξ(l) = ∑vi∈V Di(li) + λ∑(vi,vj)∈E Vij(li, lj , di, dj) (1)where Di(li) penalizes labeling vi with li, and Vij , aka prior, penalizes assigning labels (li, lj) to neighboring vertices1.
Q11. what is the simplest solution to the multi-label problem?
If Vij(li, lj , di, dj) = Vij(di, dj), i.e. label-independent, the authors can simply ignore the outcome of li ⊕ lj by setting it to a constant.
Q12. What is the way to minimize the LS error in a linear system of equations?
The calculated edge weights are optimal in the sense that they minimize the least squares (LS) error when solving a linear system of equations capturing the original MRF penalties.
Q13. what is the cost of severing a tlink in g2?
(4)The interaction penalty Vij(li, lj, di, dj) for assigning li to vi and lj to neighboring vj in G must equal the cost of assigning a sequence of binary labels (lim)bm=1 to (vim)bm=1 and (ljn) b n=1 to (vin) b n=1 in G2.
Q14. what is the cost of severing t-links in g2?
the cost of severing t-links in G2 to assign li to vertex vi in G is calculated asDtlinksi (li) =b∑j=1lijwvij ,s + l̄ijwvij ,t (2)where l̄ij denotes the unary complement (NOT) of lij .
Q15. how many pairs of labels can be substituted?
For b = 2, (5) simplifies toVij(li, lj , di, dj) = (li1 ⊕ lj1)wvi1,vj1 + (li1 ⊕ lj2)wvi1,vj2+ (li2 ⊕ lj1)wvi2,vj1 + (li2 ⊕ lj2)wvi2,vj2 (15)The authors can now substitute all possible 2b2b = 22b = 16 combinations of the pairs of interacting labels (li, lj)∈{l0, l1, l2, l3}×{l0, l1, l2, l3}, or equivalently, ((li)2, (lj)2) ∈ {00, 01, 10, 11}× {00, 01, 10, 11}.
Q16. What is the solution to the multi-label problem?
in the general case when Vij depends on the labels li and lj of the neighboring vertices vi and vj , a single edge weight is insufficient to capture such elaborate label interactions, intuitively, because wi,j needs to take on a different value for every pair of labels.