Some links between mincuts, optimal spanning forests and watersheds

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Some links between min-cuts,optimal spanning
forests and watersheds
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Outline

• Motivation
• Usual algorithms in graph
• Min-cuts
• Watersheds
• Shortest-path spanning forests cuts (SPSF cuts)
• Comparisons of results
• Links
• Link between watersheds and SPSF cuts
• Link between min-cuts and watersheds
• Conclusion

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- 1 -Motivationgraph segmentation
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Image segmentation
1 Motivation graph segmentation
An image
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Image segmentation
1 Motivation graph segmentation
A set of markers on the image
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Image segmentation
1 Motivation graph segmentation
How to find a good segmentation?
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Energy to minimize

• Let I be an image. Let s and t be two pixels of
  I.
• We denote by
• the value of the pixel s
• the neighbourhood of the pixel s
• the label given to the pixel s.
• Example of energy to minimize for the
  segmentation

1 Motivation graph segmentation
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What about graphs?

• A graph G is composed of

• a set of nodes, denoted by V,

• a set of edges linking a couple of nodes, denoted
  by E.

1 Motivation graph segmentation

• Edge weighted graph
• Application

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Image to graph

• Building of the graph
• Each pixel ? node
• Pixels s and t are neighbours ? edge es,t of
  weight

1 Motivation graph segmentation
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Image to graph
1 Motivation graph segmentation
Edge-weighted graph
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Markers
1 Motivation graph segmentation
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Result frontier
1 Motivation graph segmentation
Link to energyminimization of the sum of the
dashed edges
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Result partition
1 Motivation graph segmentation
Link to energymaximization of the sum of the
bold edges
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- 2 -Usual algorithmsin graph segmentation
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Vocabulary on graphs
Marker M
Extension
Spanning extension
Maximal extension
Cut
Spanning forest
1 Graph segmentation
Correspondance with image
Spanning forest relative to Mspanning extension
relative to M from which you cant remove any
edge without loosing the spanning extension
property
Maximal extension relative to Mspanning
extension relative to M which cant be strictly
contained by another extension relative to M
Cut relative to M set of edges linking two
different connected components of a spanning
extension relative to M
Marker Msubgraph of G
Extension relative to Msubgraph of G for which
each connected component contains exactly one
connected component of M
Spanning extension relative to M extension
relative to M which contains all the nodes of G
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Min-cuts

• Minimizing the cuts weight Maximizing the
  maximal extensions weight
• Searching the maximum flow (Ford Fulkerson,
  1962) Searching the min-cut of a marker with 2
  connected components
• Computing complexity polynomial
• Drawback for more than two connected components
  in the marker, can only give an approximated
  result through successive cuts (NP-complete)

2 Usual algorithms in graph segmentationa -
Min-cuts
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Watersheds

• Origin S. Beucher C. Lantuéjoul, 1979
• Searching a watershed Searching the cut induced
  by a spanning forest of minimum weight relative
  to the minima of the graph
• Computing complexity linear (J. Cousty al.,
  2007)
• Drawback doesnt take into account all the edges
  of the maximal extension

2 Usual algorithms in graph segmentationb -
Watersheds
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Shortest-path spanning forest cuts (SPSF cuts)

• Let M be a subgraph of G, x be a node of G\M and
  p be a path in G from x to M.
• We define
• Length of a path
• Connection value of x
• Definition
• A subgraph F of G is a SPSF if and only if F is
  a spanning forest and for any node x of F there
  exists a path p in F from x to M such that
  P(x)P(p).

2 Usual algorithms in graph segmentationc
SPSF cuts
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Shortest-path spanning forest cuts (SPSF cuts)

• Origin J.K. Udupa al., 2002 A.X. Falcao
  al., 2004 R. Audigier R.A. Lotufo, 2006
• Computing complexity quasi-linear
• Drawback doesnt take into account all the edges
  of the maximal extension

2 Usual algorithms in graph segmentationc
SPSF cuts
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Remark about comparison

• Min-cuts are of low value whereas watersheds or
  SPSF cuts are of high value.
• In the objective to compare min-cuts with
  watersheds or SPSF cuts working on the same data,
  we consider a strictly decreasing function
  applied to the weights of the graph before
  computation of a watershed or a SPSF cut.

2 Usual algorithms in graph segmentationb -
Watersheds
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Comparisons of results
2 Usual algorithms in graph segmentationd -
Comparisons of results
Min-cut
Watershed / SPSF cut
Similar results between watershed and SPSF cut,
but differences with min-cut.
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Comparisons of results
2 Usual algorithms in graph segmentationd -
Comparisons of results
Markers
Min-cut
Watershed / SPSF cut

• "Best" min-cut
• Drawback of watershed / SPSF cut leak point

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Comparisons of results
2 Usual algorithms in graph segmentationd -
Comparisons of results
Markers
Min-cut
Watershed / SPSF cut

• "Best" watershed / SPSF cut
• Drawback of min-cut as a global minimum, the cut
  is minimum with a few edges of high value rather
  than lots of edges of low value

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- 3 -Links
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Link between watersheds and SPSF cuts
Theorem 1 A spanning forest of minimum weight
is a SPSF.
Theorem 2 A SPSF relative to the minima of the
graph is a spanning forest of minimum weight
relative to the minima of the graph.
3 Linksa - Link between watersheds and SPSF
cuts
Consequence A SPSF cut relative to the minima
of the graph is a watershed.
(found by us and independently by R. Audigier
al., 2007)
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Link between min-cut and watershed

• We denote by Pn the application of weight risen
  to power n.
• Theorem
• There exists a real number m such that for any n
  m, any min-cut relative to the marker M for
  Pn is the cut induced by a maximum spanning
  forest relative to the marker M for Pn.
• Consequently, if M is the maxima of the graph,
  this min-cut is a watershed.

3 Linksa - Link between watersheds and SPSF
cuts
Remark The value of n acts like a smoothing
parameter for the cut.
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Link between min-cut and watershed Rise of the
power of graph weights for min-cuts
Watershed
Min-cut
3 Linksb - Link between min-cuts and watersheds
Rise to square min cut watershed
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Link between min-cut and watershed Rise of the
power of graph weights for min-cuts
Min-cut
Marker
3 Linksb - Link between min-cuts and watersheds
Power n 1,4
Power n 1
Watershed
Power n 3
Power n 2
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Intuitively
Let Cn be the min-cut for the weight at power
n. Weight of the cut
3 Links b - Link between min-cuts and
watersheds
Looks like n-norm
When n is rising you converge towards
infinity-norm
Consequently
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Intuitively
Min-cut
Watershed
3 Links b - Link between min-cuts and
watersheds
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- 4 -Conclusion
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Conclusion

• Equivalence between watersheds and SPSF cuts
  relative to minima
• Link from min-cuts to watersheds
• Rise of the power for min-cuts smoothing
  parameter of the cut

4 Conclusion

• Watersheds fast but isnt global (leak points)
• Min-cuts slow but global minimum (for a marker
  with 2 connected components) and has smoothness
  parameter
• Future work link from watersheds to min-cuts?

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Thank you for your attention!Any question?
Min-cut(power p 1)
Min-cut(power p 1,4)
Min-cut(power p 2)
Watershed, SPSF cut and min-cut(power p 3)
Marker
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Scheme of proof

• F spanning forest of maximum weight (MaxSF) for
  the graph risen to power n
• C min-cut for the graph risen to power n
• F MaxSF for the graph complementary of C risen
  to power n
• We just have to prove that P(F) P(F).
• Let p1,,pk be the different values of weight in
  the graph with p1gtgtpk.
• Let nh(X) be the number of edges of weight ph in
  the subgraph X.
• So we have and
• Which comes to proving that for any h

3 Links b - Link between min-cuts and
watersheds
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Leveling (thresholding)

• Let Xh be the subgraph of X composed with only
  the edges of weight greater or equal to ph.

3 Links b - Link between min-cuts and
watersheds
G3 (p3 7)
G2 (p2 8)
G6 (p6 4)
G4 (p4 6)
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Lemma 1

• For any h, Fh is a spanning forest relative to
  the markers for Gh.

3 Links b - Link between min-cuts and
watersheds
G3 (p3 7)
F3 (p3 7)
G6 (p6 4)
F6 (p6 4)
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Lemma 2

• For any h, Fh is a spanning forest relative to
  the markers for Gh.

3 Links b - Link between min-cuts and
watersheds
G3 (p3 7)
F3 (p3 7)
G6 (p6 4)
F6 (p6 4)
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Lemma 3 and epilogue

• Lemma 3 Any spanning forest for a subgraph X has
  a constant number of edges.
• We deduce that n1(F1)n1(F1).
• By induction for all h, nh(Fh)nh(Fh).
• Consequently, P(F)P(F) and finally F is a
  MaxSF!

3 Links b - Link between min-cuts and
watersheds