Applications of parametric maxflow problem

1
Applications of parametric max-flow problem in
computer vision
A parametric max-flow problem minimize for all
ls
Du(l) linear functions Vuv(xu , xv)
submodular

• ES algorithm Eisner Severence76,
  Gusfield80

General case
has finite number of breakpoints
- 2K calls to maxflow (K is of breakpoints) - K
may be exponential!
Monotonic case Du(l) are all
non-decreasing/all non-increasing

• - Nestedness property
• much more efficient implementation
  of ES
• - KO(n)
• - Worst-case complexity can be improved to that
  of a single max-flow Gallo, Grigoriadis,
  Tarjan89

Applications
Cosegmentation Rother,Kolmogorov,Minka,BlakeCVPR
06
Ratio minimization () PDE cuts
() Co-segmentation () Learning Training ..
- Given 2 images, find segmentations so that
histograms match - Key subproblem
1 image target histogram segmentation
- Minimize
PDE cuts BKCDECCV06

• Trust region graph cuts TRGC RKMB06

• - Approximate Ehist(z) as a linear function
• - Given current x, choose new approximation
• - Interpolate between current and new
• - Compute minimum xl of
  for
• - Choose l that minimizes E(xl)

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
- best contour in the neighbourhood
C
space of all contours
Searching for l
Each step parametric max-flow problem
A as in RKMB06 (binary search) B
Essentially, first breakpoint in 0,1 C compute
all solutions in 0,1
target histogram
input image
controls time step
- In general non-monotonic case - This work
smallest detectable move can be computed via
monotonic max-flow algorithm (much faster!)
strategy B
strategy A
strategy C