ICCV2009: MAP Inference in Discrete Models: Part 5

Course Program
9.30-10.00 Introduction (Andrew Blake)
10.00-11.00 Discrete Models in Computer Vision (Carsten Rother)
15min Coffee break
11.15-12.30 Message Passing: DP, TRW, LP relaxation (Pawan Kumar)
12.30-13.00 Quadratic pseudo-boolean optimization (Pushmeet Kohli)
1 hour Lunch break
14:00-15.00 Transformation and move-making methods (Pushmeet Kohli)
15:00-15.30 Speed and Efficiency (Pushmeet Kohli)
15min Coffee break
15:45-16.15 Comparison of Methods (Carsten Rother)
16:15-17.30 Recent Advances: Dual-decomposition, higher-order, etc.
(Carsten Rother + Pawan Kumar)

All online material will be online (after conference):
http://research.microsoft.com/en-us/um/cambridge/projects/tutorial/

Comparison of Optimization Methods

Carsten Rother
Microsoft Research Cambridge

Why is good optimization important?

Input: Image sequence

[Data courtesy from Oliver Woodford]

Output: New view

Problem: Minimize a binary 4-connected pair-wise MRF
(choose a colour-mode at each pixel)
[Fitzgibbon et al. ‘03+

Why is good optimization important?

Ground Truth Graph Cut with truncation Belief Propagation ICM, Simulated
[Rother et al. ‘05+
Annealing

QPBOP [Boros ’06, Rother ‘07+
Global Minimum

Comparison papers
• Binary, highly-connected MRFs *Rother et al. ‘07+

• Multi-label, 4-connected MRFs *Szeliski et al. ‘06,‘08+
all online: http://vision.middlebury.edu/MRF/

• Multi-label, highly-connected MRFs *Kolmogorov et al. ‘06+

Random MRFs
o Three important factors:
o Connectivity (av. degree of a node)
o Unary strength: E(x) = w ∑ θi (xi) + ∑ θij (xi,xj)
o Percentage of non-submodular terms (NS)

Computer Vision Problems

perc. unlabeled (sec) Energy (sec)

Conclusions:
• Connectivity is a crucial factor
• Simple methods like Simulated
Annealing sometimes best

Diagram Recognition [Szummer et al ‘04]
71 nodes; 4.8 con.; 28% non-sub; 0.5 unary strength

• 2700 test cases: QPBO solved nearly all
(QPBOP solves all)

Ground truth QPBOP (0sec) - Global Min.
Sim. Ann. E=0 (0.28sec) QPBO: 56.3% unlabeled (0 sec)

BP E=25 (0 sec) GrapCut E= 119 (0 sec) ICM E=999 (0 sec)

Binary Image Deconvolution
50x20 nodes; 80con; 100% non-sub; 109 unary strength

Ground Truth Input

0.2 0.2 0.2 0.2 0.2

0.2 0.2 0.2 0.2 0.2

0.2 0.2 0.2 0.2 0.2

0.2 0.2 0.2 0.2 0.2

0.2 0.2 0.2 0.2 0.2

5x5 blur kernel

MRF: 80 connectivity - illustration

Binary Image Deconvolution
50x20 nodes; 80con; 100% non-sub; 109 unary strength

Ground Truth Input QPBO 80% unlab. (0.1sec)

QPBOP 80% unlab. (0.9sec) ICM E=6 (0.03sec) GC E=999 (0sec)

BP E=71 (0.9sec) QPBOP+BP+I, E=8.1 (31sec) Sim. Ann. E=0 (1.3sec)

Comparison papers
Conclusion: low-connectivity tractable: QPBO(P)

• Multi-label, 4-connected MRFs *Szeliski et al ‘06,‘08+

• Multi-label, highly-connected MRFs *Kolmogorov et al ‘06+

Multiple labels – 4 connected
“Attractive Potentials”

stereo

Panoramic
stitching

Image
Segmentation;
de-noising;
in-painting

[Szelsiki et al ’06,08+

Stereo

image Ground TRW-S image Ground TRW-S
truth truth

Conclusions:
– Solved by alpha-exp. and TRW-S
(within 0.01%-0.9% of lower bound – true for all tests!)
– Expansion-move always better than swap-move

De-noising and in-painting

Ground truth Noisy input TRW-S Alpha-exp.

Conclusion:
– Alpha-expansion has problems with smooth areas
(potential solution: fusion-move *Lempitsky et al. ‘07+)

Panoramic stitching
• Unordered labels are (slightly) more challenging

Comparison papers
Conclusion: low-connectivity tractable (QPBO)

Conclusion: solved by expansion-move; TRW-S
(within 0.01 - 0.9% of lower bound)


Multiple labels – highly connected
Stereo with occlusion:

E(d): {1,…,D}2n → R
Each pixel is connected to D pixels in the other image

*Kolmogorov et al. ‘06+

Multiple labels – highly connected
Tsukuba: 16 labels Cones: 56 labels

• Alpha-exp. considerably better than message passing
Potential reason: smaller connectivity in one expansion-move

Comparison: 4-con. versus highly con.

Tsukuba (E) Map (E) Venus (E)
highly-con. 103.09% 103.28% 102.26%
4-con. 100.004% 100.056% 100.014%
Lower-bound scaled to 100%

Conclusion:
• highly connected graphs are harder to optimize

Comparison papers
• binary, highly-connected MRFs *Rother et al. ‘07+
Conclusion: low-connectivity tractable (QPBO)

Conclusion: solved by alpha-exp.; TRW
(within 0.9% to lower bound)

Conclusion: challenging optimization (alpha-exp. best)

How to efficiently optimize general highly-connected
(higher-order) MRFs is still an open question

Advanced Topics –
Optimizing Higher-Order MRFs
Carsten Rother
Microsoft Research Cambridge

Challenging Optimization Problems
• How to solve higher-order MRFs:

• Possible Approaches:

- Convert to Pairwise MRF (Pushmeet has explained)

- Branch & MinCut (Pushmeet has explained)

- Add global constraint to LP relaxation

- Dual Decomposition

Add global constraints to LP
Basic idea:

T

∑ Xi ≥ 1
i Є T
References:
[K. Kolev et al. ECCV’ 08] silhouette constraint
[Nowizin et al. CVPR ‘09+ connectivity prior
[Lempitsky et al ICCV ‘09+ bounding box prior (see talk on Thursday)
See talk on Thursday:
[Lempitsky et al ICCV ‘09+ bounding box prior

Dual Decomposition

• Well known in optimization community [Bertsekas ’95, ‘99+

• Other names: “Master-Slave” [Komodiakis et al. ‘07, ’09+

• Examples of Dual-Decomposition approaches:
– Solve LP of TRW [Komodiakis et al. ICCV ‘07+
– Image segmentation with connectivity prior [Vicente et al CVPR ‘08+
– Feature Matching [Toressani et al ECCV ‘08+
– Optimizing Higher-Order Clique MRFs [Komodiakis et al CVPR ‘09+
– Marginal Probability Field *Woodford et al ICCV ‘09+
– Jointly optimizing appearance and Segmentation [Vicente et al ICCV 09]

Dual Decomposition
Hard to optimize Possible to optimize Possible to optimize

min E(x) = min [ E1(x) + θTx + E2(x) – θTx ]
x x
≥ min [E1(x1) + θTx1] + min [E2(x2) - θTx2] = L(θ)
x1 x2
“Lower bound”

• θ is called the dual vector (same size as x)

• Goal: max L(θ) ≤ min E(x)
θ x

• Properties:
• L(θ) is concave (optimal bound can be found)
• If x1=x2 then problem solved (not guaranteed)

Why is the lower bound a concave function?

L(θ) = min [E1(x1) + θTx1] + min [E2(x2) - θTx2] L(θ) : Rn -> R
x1 x2
L1(θ) L2(θ)

θTx’1
L1(θ)
θTx’’1

θTx’’’1

θ

L(θ) concave since a sum of concave functions

How to maximize the lower bound?
If L(θ) were to be differentiable use gradient ascent
L(θ) not diff. … so subgradient approach [Shor ‘85+

How to maximize the lower bound?
If L(θ) were to be differentiable use gradient ascent
L(θ) not diff. so subgradient approach [Shor ‘85+

L(θ) = min [E1(x1) + θTx1] + min [E2(x2) - θTx2] L(θ) : Rn -> R
x1 x2
L1(θ) L2(θ)
θTx’1
L1(θ)
θTx’’1

Subgradient g θTx’’’1

Θ’ Θ’’ Θ
Θ’’ = Θ’ + λ g = Θ’ + λ x’1 Θ’’ = Θ’ + λ (x1-x2)

Dual Decomposition
L(θ) = min [E1(x1) + θTx1] + min [E2(x2) - θTx2]
x1 x2
Subgradient Optimization:
“Master”
Θ = Θ + λ(x*-x2)
1
*

x1* x2
*

Θ Θ subgradient

Subproblem 1 Subproblem 2
x* = min [E1(x1) + θTx1]
1 x2 = min [E2(x2) + θTx2]
* “Slaves”
x1 x2

• Guaranteed to converge to optimal bound L(θ)
• Choose step-width λ correctly ([Bertsekas ’95])
• Pick solution x as the best of x1 or x2
• E and L can in- and decrease during optimization
• Each step: θ gets close to optimal θ* Example optimization

Why can the lower bound go down?
Lower envelop of planes in 3D:

L(θ)

L(θ’)

L(θ’) ≤ L(θ)

Analyse the model
push x1p towards 0
1 push x2p towards 1
0

L(θ) = min [E1(x1) + θTx1] + min [E2(x2) - θTx2]

Update step: Θ’’ = Θ’ + λ (x*-x*)
1 2

Look at pixel p:

Case1: x* = x2p
1p
* then Θ’’ = Θ’

Case2: x* = 1 x2p = 0
1p
* then Θ’’ = Θ’+ λ

Case3: x* = 0 x* = 1
1p 2p then Θ’’ = Θ’- λ

Example 1: Segmentation and Connectivity
Foreground object must be connected:
E(x) = ∑ θi (xi) + ∑ θij (xi,xj) + h(x)

h(x)= { ∞ if x not 4-connected
0 otherwise

Zoom in

User input Standard MRF Standard MRF
+h
*Vicente et al ’08+

E1(x)


0 otherwise
Derive Lower bound:
min E(x) = min [ E1(x) + θTx + h(x) – θTx ]
x x
≥ min [E1(x1) + θTx1] + min [h(x2) + θTx2] = L(θ)
x1 x2

Subproblem 1: Subproblem 2:
Unary terms + Unary terms +
pairwise terms Connectivity constraint
Global minimum: Global minimum:
GraphCut Dijkstra

But: Lower bound was for no example tight.

E1(x)


0 otherwise

Derive Lower bound: x’ indicator vector of all pairwise terms

min E(x) = min [ E1(x) + θTx + θ’Tx’ + h(x) – θTx + h(x) - θ’Tx’]
x,x’ x,x’
≥ min [E1(x1) + θTx1 + θ’Tx’1] + min [h(x2) + θTx2] +
x1,x’1 x2
min [h(x3) + θ’Tx’3] =L(θ)
x3,x’3
Subproblem 1: Subproblem 2: Subproblem 3:
Unary terms + Unary terms + Pairwise terms +
pairwise terms Connectivity constraint Connectivity constraint
Global minimum: Global minimum: Lower Bound: Based on
GraphCut Dijkstra minimal paths on a dual graph

Results: Segmentation and Connectivity
Global optimum 12 out of 40 cases.

Extra
Input GlobalMin
Image Input GraphCut

Heuristic method, DijkstraGC, which is faster and gives
empirically same or better results

*Vicente et al ’08+

Example2: Dual of the LP Relaxation
(from Pawan Kumar’s part)
q*( i) = min iTx
i
xi
Wainwright et al., 2001
1
q*( 1) Va Vb Vc
Va Vb Vc
2
q*( 2) Vd Ve Vf
Vd Ve Vf
3
q*( 3) V V g V h i
Vg Vh Vi
q*( 4) q*( 5) q*( 6)
=( a0, a1,…, ab00, ab01,…) Va Vb Vc
i≥ 0
x = (xa0, xa1,…,xab00,xab01,…) Vd Ve Vf
Dual of LP
Vg Vh Vi
max i q*( i) = L({ i})
{ i} i 4 5 6
i i =
i

Example2: Dual of the LP Relaxation
“Original “i different “Lower bound”
problem” trees”
min Tx = min iTx
x
∑ ≥ ∑ min
x
iTx
i = L({ i})
x i i i

q*( i)
Subject to i =
i
Use subgradient … why?
θTx’1
q*( i) concave wrt i ; q*( i) = min iTx
i
θTx’’1
xi
θTx’’’1
Projected subgradient method:
θ
Θi = [Θi + λx *]
i Ω

Ω= {Θi| ∑ Θi = Θ }
i

Guaranteed to get the optimal lower bound !

Example 2: optimize LP of TRW
TRW-S:
• Not guaranteed to get optimal bound (DD does)
• Lower bound goes always up (DD not).
• Needs min-marginals (DD not)
• DD paralizable (every tree in DD can be optimized separately)

[Komodiakis et al ’07+

Example 3:
A global perspective on low-level vision

Global unary, er. 12.8%

Cost f

0 n ∑xi
Not NP-hard i
Add global term which enforces
(Pushmeet
Kohli’s part) a match with the marginal statistic

Cost f E(x) = ∑ θi (xi) + ∑ θij (xi,xj) + f(∑xi) “Solve with dual-
i i,jЄN i decomposition”
E1 E2
0 n ∑xi
*Woodford et al. ICCV’09+
i (see poster on Friday)

Example 3:
A global perspective on low-level vision
Image de-noising

Ground truth Noisy input Pairwise-MRF Global gradient Gradient strength
prior

Image synthesis

input [Kwatra ’03+ global color
distribution prior

Example 4: Solve GrabCut globally optimal

w Color model

E(x, w) E’(x) = min E(x, w)
w
Highly connected MRF Higher-order MRF
E(x,w): {0,1}n x {GMMs}→ R
E(x,w) = ∑ θi (xi,w) + ∑ θij (xi,xj) *Vicente et al; ICCV ’09+
(see poster on Tuesday)

E(x)= g(∑xi) + ∑ fb(∑xib) + ∑ θij (xi,xj) “Solve with dual-
i b i,jЄN decomposition”
E1 E2

g convex fb concave

0 n/2 n ∑xi 0 max ∑xib
Prefers “equal area” segmentation Each color either fore- or background

input

Globally optimal in 60% of cases, such as…

*Vicente et al; ICCV ’09+
(see poster on Tuesday)

Summary
• Dual Decomposition is a powerful technique for
challenging MRFs

• Not guaranteed to give globally optimal energy

• … but for several vision problems we get tight
bounds

Texture Restoration (table 1)
256x85 nodes; 15connec; 36% non-sub; 6.6 unary strength

Training Image Test Image GC, E=999 QPBO, QPBOP,
(0.05sec) 16.5% unlab. 0% unlab. (14sec)
(1.4sec) Global Minimum.
Sim. ann., ICM, BP,
BP+I, C+BP+I
(visually similar)

New-View Synthesis [Fitzgibbon et al ‘03]
385x385 nodes; 8con; 8% non-sub; 0.1 unary strength

Ground Truth Graph Cut E=2 (0.3sec)
Sim. Ann. E=980 (50sec)
ICM E=999 (0.2sec) (visually similar)

QPBO 3.9% unlabelled QPBOP - Global Min. (1.4sec),
BP E=18 (0.6sec)
(black) (0.7sec) P+BP+I, BP+I

Image Segmentation – region & boundary brush
321x221 nodes; 4con; 0.006% non-sub; 0 unary strength

Input Image User Input Sim. Ann. E=983 (50sec) ICM E=999 (0.07sec)

GraphCut E=873 (0.11sec) BP E=28 (0.2sec) QPBO 26.7% unlabeled QPBOP Global Min (3.8sec)
(0.08sec)

Non-truncated cost function
[HBF; Szeliski ‘06+
Fast linear system solver in
continuous domain (then discretised)

cost
|di-dj|

original input HBF

ICCV2009: MAP Inference in Discrete Models: Part 5

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ICCV2009: MAP Inference in Discrete Models: Part 5