Solving Markov Random Fields using Second Order Cone Programming

1
Solving Markov Random Fields using Second Order
Cone Programming
Andrew Zisserman http//www.robots.ox.ac.uk/vgg

M. Pawan Kumar Philip H.S. Torr http//cms.brookes.ac.uk/computervision

Aim To obtain accurate MAP estimate of Markov
Random Fields
Results
Solving MRFs using SOCP Relaxations
Choice of S
Subgraph Matching

• Desirable to eliminate Y (which squares
  variables) by using slack variables.

• Let ei 0 0 0 1 0 0 0 T

Markov Random Field (MRF)
S (ei ej) (ei ej) T
S ei eiT
S (ei - ej) (ei - ej) T
1
2
1, -1 -1, 1
3
Configuration Vector y
4
G2 (V2,E2)
G1 (V1,E1)
MRF
5
7
5, 2 7, 1
Likelihood Vector l

• 1000 synthetic pairs of graphs
• 5 noise added

Prior Matrix P
- - 0 4
- - 3 0
0 3 - -
4 0 - -
MRF Example
Method Time (sec) Accuracy ()
LP 0.85 6.64
SDP 35.0 93.11
SOCP-A 3.0 92.01
SOCP-B 4.5 94.79
SOCP-C 4.8 96.18
sites S 2 labels L 2
Let A ? B ? Aij Bij
yi2 ? 1
(yi yj)2 ? tij
(yi - yj)2 ? zij
y arg min yT (4l 2P1) - ?ij Pij zij

• Objective function

arg min yT (4l 2P1) P ? Y, Y y yT subject
to ? y(site i) 2 - L
tij zij 4

• Bound on slack variables tij and zij

MAP y
Advantages
( A )

• Converge is guaranteed.

• No restrictions on the MRF.

Second Order Cone Programming (SOCP)
Object Recognition

• Fewer variables, faster than SDP.

• Efficient interior-point algorithms.

Outline
Second Order Cone
u ? t OR u2 ? st
Triangular Inequalities
Additional constraints for better accuracy.
Texture
min yT f subject to Ai y bi ? yT ci di
SOCP

• At least two of yi, yj and yk have the same
  sign.

Yij Yjk Yik ? -1
Part likelihood
Spatial Prior
x2 y2 z2

• Constraints can be specified without using Y.

zij zjk zik ? 8
LBP
( B )
yT q Q ?Y, Y y yT
Convex Relaxations

• Random subset of inequalities used for
  efficiency.

Robust Truncated Prior Model
Truncated for incompatible labels.
GBP
Semidefinite (SDP)
Lift and Project (LP)

• By changing values of prior, P can be made
  sparse.

• Max-k-cut
• MAP - accurate
• Complexity - high

• TRW-S,? -expansion
• MAP - inaccurate
• Complexity - low

SOCP
Reparametrization
Y - y yT ? 0
Y ? -1,1nxn
Prior 0.5 0.5 0.3 0.3 0.5
RTPM Examples
Prior 0 0 -0.2 -0.2 0
Second Order Cone Programming (SOCP)
ROC Curves for 450 ve and 2400 -ve images
Additional Compatibility Constraints
P(yi,yj) lt 0

• More efficient and less accurate than SDP.

• Labels for sites i and j should be compatible

?ij P(yi,yj) zij gt 0

• Relaxation

• S is a set of semidefinite matrices. S U UT ?
  S

( C )

• Choice of S is crucial for accuracy and
  efficiency.

Y ? S - UT y2 ? 0
Code available at http//cms.brookes.ac.uk/staff/P
awanMudigonda/MRFSOCP.zip