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

Cédric Allène, Jean-Yves Audibert, Michel Couprie, Jean Cousty & Renaud Keriven

ENPC – CERTIS

ESIEE – A²SI

Université Paris-Est - France

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

1) Motivation

Outline

2) Usual algorithms in graph a) Min-cuts

b) Watersheds c) Shortest-path spanning forests cuts (SPSF cuts) d) Comparisons of results

3) Links

a) Link between watersheds and SPSF cuts b) Link between min-cuts and watersheds

4) Conclusion

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

- 1 Motivation: graph segmentation

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

1 – Motivation: graph segmentation

Image segmentation

An image

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

1 – Motivation: graph segmentation

Image segmentation

A set of markers on the image

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

1 – Motivation: graph segmentation

Image segmentation

How to find a good segmentation?

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

1 – Motivation: graph segmentation

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

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

1 – Motivation: graph segmentation

What about graphs? A graph G is composed of: ƒ a set of nodes, denoted by V,

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

1 – Motivation: graph segmentation

What about graphs? A graph G is composed of: ƒ a set of nodes, denoted by V, ƒ

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

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

1 – Motivation: graph segmentation

What about graphs? A graph G is composed of: ƒ a set of nodes, denoted by V, ƒ

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

Edge weighted graph: ƒ Application

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

1 – Motivation: graph segmentation

Image to graph

Building of the graph: • Each pixel → node • Pixels s and t are neighbours → edge e={s,t} of weight

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

1 – Motivation: graph segmentation

Image to graph

Edge-weighted graph

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

1 – Motivation: graph segmentation

Markers

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

1 – Motivation: graph segmentation

Result: frontier

Link to energy: minimization of the sum of the dashed edges

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

1 – Motivation: graph segmentation

Result: partition

Link to energy: maximization of the sum of the bold edges

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

- 2 Usual algorithms in graph segmentation

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

1 – Graph segmentation

Vocabulary on graphs

Marker M subgraph of G

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

1 – Graph segmentation

Vocabulary on graphs

Marker M

Extension relative to M subgraph of G for which each connected component contains exactly one connected component of M

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

1 – Graph segmentation

Vocabulary on graphs

Marker M

Extension

Spanning extension relative to M extension relative to M which contains all the nodes of G

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

1 – Graph segmentation

Vocabulary on graphs

Marker M

Extension

Spanning extension

Maximal extension relative to M spanning extension relative to M which can’t be strictly contained by another extension relative to M

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

1 – Graph segmentation

Vocabulary on graphs

Marker M

Extension

Spanning extension

Maximal extension

Spanning forest relative to M spanning extension relative to M from which you can’t remove any edge without loosing the spanning extension property

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

Vocabulary on graphs 1 – Graph segmentation

Maximal extension

Marker M

Extension

Spanning extension Spanning forest

Cut relative to M set of edges linking two different connected components of a spanning extension relative to M

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

Vocabulary on graphs 1 – Graph segmentation

Maximal extension

Marker M

Extension

Spanning extension

Cut Spanning forest

Correspondance with image

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

2 – Usual algorithms in graph segmentation a - Min-cuts

Min-cuts ƒ

ƒ

ƒ ƒ

Minimizing the cut’s weight = Maximizing the maximal extension’s 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)

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

2 – Usual algorithms in graph segmentation b - Watersheds

Watersheds ƒ ƒ

ƒ ƒ

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: doesn’t take into account all the edges of the maximal extension…

2 – Usual algorithms in graph segmentation c – SPSF cuts

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

Shortest-path spanning forest cuts (SPSF cuts) Let M be a subgraph of G, x be a node of G\M and π be a path in G from x to M. We define: ƒ Length of a path: ƒ 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 π in F from x to M such that P(x)=P(π).

2 – Usual algorithms in graph segmentation c – SPSF cuts

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

Shortest-path spanning forest cuts (SPSF cuts) ƒ

ƒ

ƒ

Origin: J.K. Udupa & al., 2002; A.X. Falcao & al., 2004; R. Audigier & R.A. Lotufo, 2006 Computing complexity: quasi-linear Drawback: doesn’t take into account all the edges of the maximal extension…

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

2 – Usual algorithms in graph segmentation b - Watersheds

Remark about comparison ƒ

ƒ

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 segmentation d - Comparisons of results

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

Comparisons of results

Min-cut

Watershed / SPSF cut

Similar results between watershed and SPSF cut, but differences with min-cut.

2 – Usual algorithms in graph segmentation d - Comparisons of results

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

Comparisons of results

Markers

Min-cut

Watershed / SPSF cut

ƒ

"Best": min-cut

ƒ

Drawback of watershed / SPSF cut: leak point

2 – Usual algorithms in graph segmentation d - Comparisons of results

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

Comparisons of results

Markers ƒ

ƒ

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…

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

- 3 Links

3 – Links a - Link between watersheds and SPSF cuts

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

Link between watersheds and SPSF cuts Theorem 1: A spanning forest of minimum weight is a SPSF.

3 – Links a - Link between watersheds and SPSF cuts

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

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 – Links a - Link between watersheds and SPSF cuts

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

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. Consequence: A SPSF cut relative to the minima of the graph is a watershed. (found by us and independently by R. Audigier & al., 2007)

3 – Links a - Link between watersheds and SPSF cuts

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

Link between min-cut and watershed We denote by P[n] 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 P[n] is the cut induced by a maximum spanning forest relative to the marker M for P[n]. Consequently, if M is the maxima of the graph, this min-cut is a watershed. Remark: The value of n acts like a smoothing parameter for the cut.

3 – Links a - Link between watersheds and SPSF cuts

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

Link between min-cut and watershed We denote by P[n] 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 P[n] is the cut induced by a maximum spanning forest relative to the marker M for P[n]. Consequently, if M is the maxima of the graph, this min-cut is a watershed. Remark: The value of n acts like a smoothing parameter for the cut.

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

3 – Links b - Link between min-cuts and watersheds

Link between min-cut and watershed: Rise of the power of graph weights for min-cuts Min-cut

Watershed

Rise to square: min cut = watershed

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

3 – Links b - Link between min-cuts and watersheds

Link between min-cut and watershed: Rise of the power of graph weights for min-cuts Min-cut

Marker

Watershed

Power n = 1

Power n = 1,4

Power n = 2

Power n = 3

3 – Links b - Link between min-cuts and watersheds

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

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

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

Intuitively… Let Cn be the min-cut for the weight at power n. Weight of the cut: Looks like n-norm:

3 – Links b - Link between min-cuts and watersheds

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

Intuitively… Let Cn be the min-cut for the weight at power n. Weight of the cut: Looks like n-norm: When n is rising you converge towards infinity-norm:

3 – Links b - Link between min-cuts and watersheds

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

Intuitively… Let Cn be the min-cut for the weight at power n. Weight of the cut: Looks like n-norm: When n is rising you converge towards infinity-norm:

Consequently:

3 – Links b - Link between min-cuts and watersheds

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

Intuitively… Min-cut

Watershed

3 – Links b - Link between min-cuts and watersheds

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

Intuitively… Min-cut

Watershed

3 – Links b - Link between min-cuts and watersheds

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

Intuitively… Min-cut

Watershed

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

- 4 Conclusion

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

Conclusion

4 – Conclusion

ƒ

ƒ

ƒ

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

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

Conclusion

4 – Conclusion

ƒ

ƒ

ƒ

ƒ ƒ

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 Watersheds: fast but isn’t 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?

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

Thank you for your attention! Any question?

Marker

Min-cut (power p = 1)

Min-cut (power p = 1,4)

Min-cut Watershed, SPSF cut (power p = 2) and min-cut (power p = 3)

3 – Links b - Link between min-cuts and watersheds

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

Scheme of proof ƒ

ƒ ƒ

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’). ƒ

ƒ

ƒ

Let p1,…,pk be the different values of weight in the graph with p1>…>pk. Let nh(X) be the number of edges of weight ph in the subgraph X. k k So we have: P(F ) = ∑h =1 nh (F )× ph and P(F ') = ∑h =1 nh (F ')× ph

Which comes to proving that for any h: nh (F ) = nh (F ')

3 – Links b - Link between min-cuts and watersheds

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

Leveling (thresholding) ƒ

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

G2 (p2 = 8)

G3 (p3 = 7)

G4 (p4 = 6)

G6 (p6 = 4)

3 – Links b - Link between min-cuts and watersheds

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

Lemma 1 ƒ

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

G3 (p3 = 7)

F3 (p3 = 7)

G6 (p6 = 4)

F6 (p6 = 4)

3 – Links b - Link between min-cuts and watersheds

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

Lemma 2 ƒ

For any h, F’h is a spanning forest relative to the markers for Gh.

G3 (p3 = 7)

F’3 (p3 = 7)

G6 (p6 = 4)

F’6 (p6 = 4)

3 – Links b - Link between min-cuts and watersheds

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

Lemma 3 and epilogue ƒ

Lemma 3: Any spanning forest for a subgraph X has a constant number of edges.

ƒ

We deduce that n1(F1)=n1(F’1).

ƒ

By induction: for all h, nh(Fh)=nh(F’h).

ƒ

Consequently, P(F)=P(F’) and finally F’ is a MaxSF!