Applications of parametric max-flow problem
in computer vision
A parametric max-flow problem: minimize for all l’s
E (x)   Du (l ) xu  Vuv ( xu , xv )
l
u
( u ,v )
l
F (l )  min E (x)
ES algorithm [Eisner & Severence’76], [Gusfield’80]
x
General case:
has finite number
of breakpoints:
F (l )
x2
Du(l) – linear functions
Vuv(xu , xv) – submodular
- 2K calls to maxflow (K is # of breakpoints)
- K may be exponential!
x1
Monotonic case:
xk
Du(l) are all non-decreasing/all non-increasing
- Nestedness property
much more efficient implementation of ES
x3
-
l1
l4
l3
l5 l2
- K=O(n)
- Worst-case complexity can be improved to that of a
single max-flow [Gallo, Grigoriadis, Tarjan’89]
Cosegmentation
[Rother,Kolmogorov,Minka,Blake’CVPR06]
Applications
Ratio minimization (*)
PDE cuts (*)
Co-segmentation (*)
Learning
Training
…..
PDE cuts
[BKCD’ECCV06]
Goal: compute “gradient flow” of contour C
- gradient descent of some functional F(C)
set of contours within
small distance e from C
C
C
C’
- Key subproblem: 1 image + target histogram
- Minimize
C’
- best contour in
the neighbourhood
Each step – parametric max-flow problem:
E (C )  F (C )  l  dist (C , C0 )
2
controls time step
- In general: non-monotonic case
- This work: smallest detectable move can be
computed via monotonic max-flow algorithm
(much faster!)
segmentation
E (x)  EMRF (x)  Ehist (x)
Trust region graph cuts – TRGC [RKMB’06]
- Approximate Ehist(z) as a linear function z, y  const
~
- Given current x, choose new approximation z, y  const
l
~
~
- Interpolate between current y and new y : y  ly  (1  l )y
- Compute minimum
space of all contours
l
- Given 2 images, find segmentations so that histograms match
l
x
of EMRF (x)  x, y
- Choose l that minimizes
l
for l  [0,1]
l
E(x )
Searching for l:
A: as in [RKMB’06] (binary search)
B: Essentially, first breakpoint in [0,1]
C: compute all solutions in [0,1]
input image
strategy A
target histogram
strategy B
strategy C