1. Applications of parametric max-flow problem in computer vision
A parametric max-flow problem: minimize for all l’s
Du(l) – linear functions
Vuv(xu , xv) – submodular
ES algorithm [Eisner & Severence’76], [Gusfield’80]
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
- K=O(n)
- Worst-case complexity can be improved to that of a single max-flow [Gallo, Grigoriadis, Tarjan’89]
Applications
Cosegmentation
[Rother,Kolmogorov,Minka,Blake’CVPR06]
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
[BKCD’ECCV06]
Trust region graph cuts – TRGC [RKMB’06]
- 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 C’
- best contour in
the neighbourhood C’
space of all contours
Searching for l:
Each step – parametric max-flow problem:
A: as in [RKMB’06] (binary search)
B: Essentially, first breakpoint in [0,1]
C: compute all solutions in [0,1]
input image
target histogram
controls time step
- In general: non-monotonic case
- This work: smallest detectable move can be
computed via monotonic max-flow algorithm
(much faster!)
strategy A
strategy B
strategy C