Primal-dual Algorithm for Convex Markov Random Fields Vladimir Kolmogorov University College London...

Primal-dual Algorithmfor Convex Markov Random Fields

Vladimir Kolmogorov

University College London

GDR (Optimisation Discrète, Graph Cuts et Analyse d'Images) Paris, 29 November 2005

Note: these slides contain animation

Convex MRF functions

qp

pqpqp

pp xxVxDE,

)()()(x

• Functions Dp(.) , Vpq(.) are convex

• xp[0,…,K-1] (K is # of labels)

• Goal: compute global minimum of E

Example: Panoramic image stitching[Levin,Zomet,Peleg,Weiss’04]

• Main idea: gradients of output image x should match gradients of input images

),,( 1 nxx x

• Energy function:

– - output image (e.g. xp[0,…,255])

qp

pqpq xxVE,

)()(x

Vpq

xq - xp

0 255-255• Vpq(.) is convex!

Example: Panoramic image stitching[Levin,Zomet,Peleg,Weiss’04]

Algorithms for MRF minimisation

• Arbitrary Dp(.), convex Vpq(.)– “Battleship” construction [Ishikawa’03], [Ahuja et al.’04]

• Construct graph with O(nK) nodes

• Minimum cut gives global minimum

– Needs a lot of memory!

• Convex Dp(.), convex Vpq(.)– Dual algorithms (maintain dual variables - flow f)

• [Karzanov et al. ’97], [Ahuja et al. ’03].

• Best complexity is O(n m log (n2/m) log(nK))

– Primal algorithm (maintains primal variables - configuration x)• Iterative min cut [Bioucas-Dias & Valadão’05]

• Advantage: relies on maxflow algorithm

• Complexity?

• New results (convex Dp, convex Vpq):

– Establishing complexity of primal algorithm• At most 2K steps

– Extending primal algorithm to primal-dual algorithm• Maintains both primal and dual variables

• Can be speeded up using Dijkstra’s shortest path procedure

• Experimentally much faster than primal algorithm

Algorithms for MRF minimisation

Overview of primal algorithm (iterative min cut)

label K-1

label 0

label 1

Primal algorithm (iterative min cut)

Graph:

label K-1

label 0

label 1

• Start with arbitrary configuration x• Procedure UP:

nE }1,0{)(minarg bbxb



label K-1

label 0

label 1

bxx :

• Procedure UP:


label K-1

label 0

label 1

• Procedure DOWN:



label K-1

label 0

label 1

bxx :

• Procedure DOWN:



label K-1

label 0

label 1

• UP:


label K-1

label 0

label 1

• UP:• DOWN:


label K-1

label 0

label 1

• DOWN:


label K-1

label 0

label 1

• DOWN:• Done!– UP and DOWN do not decrease energy


Discussion• [Bioucas-Dias & Valadão’05]:

– Procedure yields global minimum! • No unary terms Dp, terms Vpq are convex

– Straighforward extension to convex MRF functions• Convex Dp , convex Vpq

– Non-polynomial bound on the number of steps

• [Murota ’00,’03] (steepest descent algorithm)– Procedure yields global minimum for L♮-convex functions

• Convex MRF functions are special case of L♮-convex functions– O(nK) bound on the number of steps

• New result:– Global minimum after at most 2K steps– Holds for L♮-convex functions (including convex MRF functions)

Contribution #1: Complexity of primal algorithm

Background

Two classes of functions• Consider function E(x) = E(x1,…,xn)

– xp[0,…,K-1]

• Algorithm can be applied to any such function:

UP:

DOWN:

• Question #1: When UP and DOWN can be solved efficiently?

• Question #2: When does it yield global minimum?

bxxbbxb :}1,0{)(minarg nE

bxxbbxb :}1,0{)(minarg nE

Submodular functions

L♮-convex functions

Submodular functions

)()()()( yxyxyx EEEE • E is submodular if for all configurations x, y

– “” and “” are component-wise minimum/maximum:

• Definition is extended from binary variables (K=2) to multi-valued variables (K>2)

• Can be minimised in time polynomial in K, n, m– Functions with unary, pairwise and ternary terms:

reduction to min cut/max flow [Kovtun’04]

ppp

ppp

yx

yx

, max)(

, min)(

yx

yx

L♮-convex functions

)()(22

yxyxyx

EEEE

• E is L♮-convex if for all configurations x, y

– “” and “” are component-wise round-down and round-up (floor and ceiling)

• Note: in continuous case, E is convex if for all x, y

)()(2

2 yxyx

EEE

Submodulariry and L♮-convexity• K=2: submodular functions = L♮-convex functions• K>2: submodular functions L♮-convex functions

• Example: qp

pqpqp

pp xxVxDE,

)()()(x

Dp arbitraryVpq convex

E is submodular

Dp convexVpq convex

E is L♮-convex

D1

x1

D1

x1

Contribution #1: Complexity of primal algorithm

for L♮-convex functions

Proof

• For configuration x, define +(x), (x)

– 0 +(x), (x) < K

• Prove that UP and DOWN do not increase +(x), (x)

• Prove that if +(x) > 0, then UP will decrease it – Similarly for (x) and DOWN

• Prove that +(x) = (x) = 0 implies that x is a global minimum

Overview

Property of submodular functions• Let OPT(E) be the set of global minima of E

• There exists minimal and maximal optimal configurations:

• In general, not true for non-submodular functions! (e.g. Potts interactions)

)( allfor

)(,maxmin

maxmin

EOPT

EOPT

xxxx

xx

• Step 1: Let E+ be a restriction of E to configurations y x

Defining +(x)

• Step 3: Define +(x) = || x+ - x ||= max { x+p

- xp }

• Step 2: Let x+ be the minimal optimal configuration of E+

x

x+

• Step 1: Let E ¯ be a restriction of E to configurations y x

Defining (x)

• Step 3: Define (x) = || x - x¯ ||= max { xp - x¯p }

• Step 2: Let x¯ be the maximal optimal configuration of E ¯

x

x¯

Algorithm’s behaviour

(x) = 2

(x) = 3


(x) = 1

(x) = 3


(x) = 0

(x) = 3


(x) = 0

(x) = 2


(x) = 0

(x) = 1


(x) = 0

(x) = 0

Contribution #2: Primal-dual algorithm

Primal-dual algorithm• Primal algorithm maintains only primal variables

(configuration x)– Each maxflow problem is solved independently

• Motivation: reuse flow from previous computation

• New primal-dual algorithm– Applies to convex MRF functions

– Maintains both primal variables (configuration x) and dual variables (flow f )

– Upon termination, optimal x and f

– Can be speeded up via Dijkstra’s algorithm

– Experimentally much faster than primal algorithm

Flow and reparameterisation

Dp(xp)

xp xqxq-xp

Vpq(xq-xp) Dq(xq)

Dp(xp) + f·xp

xp xqxq-xp

Vpq(xq-xp) + f·(xq-xp) Dp(xp) f·xq

Flow and reparameterisation

Dp(xp) + fp·xp

xp xq-xp

Vpq(xq-xp) + fpq·(xq-xp)

• Flow: vector f = { fp , fpq } satisfying antisymmetry and flow conservation constraints

• Any flow defines reparameterisation

Optimality conditions

Dp(xp) + fp·xp

xp xq-xp

Vpq(xq-xp) + fpq·(xq-xp)

• ( x, f ) is an optimal primal-dual pair iff

)(min)(

)(min)(

zVxxV

zDxD

pqf

zpqpq

f

pf

zpp

f

Primal-dual algorithm• UP:

– MAXFLOW-UP• Construct graph for minimising

E(x + b), b{0,1}n

• Compute maximum flow• Update x and f accordingly

– DIJKSTRA-UP• Update x

• DOWN: similar

nodes

edges

• Algorithm’s property: – Maintains optimality condition for edges

(but not necessarily for nodes)

DIJKSTRA-UP• Increase xp until Dp(xp) starts

increasing

• Maintain optimality condition for edges

• Compute maximal such labeling x– Dijkstra’s shortest path algorithm

nodes

edges

• Complexity is preserved (at most 2K steps)

Experimental results

input pair

maximaloptimal configuration

minimal optimal configuration

average

Running times

0

20

40

60

80

100

120

140

160

180

1 2 3

secs primal

primal-dual

• Primal• Primal-dual

Running times• Initialisation is important

– If x = global minimum, then terminates in 2 steps

• Two-stage process:– Solve the problem in the overlap area

• Small graph

– Use it as initialisation for the full image• Experimentally, second stage takes 3 steps

Running times• MCNF (minimum cost network flow, [Goldberg’97])• Primal-dual, 1 stage• Primal-dual, 2 stages

0

5

10

15

20

25

30

35

1 2 3

secs

MCNF

primal-dual, 1stage

primal-dual, 2 stages

Conclusions• Complexity of primal algorithm for minimising

L♮-convex functions– Tight bound on the number of steps– Improves bounds of Murota and Bioucas-Dias et al.

• New primal-dual algorithm– Applies to convex MRF functions– Experimentally much faster than primal algorithm– With good initialisation, outperforms MCNF

Primal-dual Algorithm for Convex Markov Random Fields Vladimir Kolmogorov University College London...

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