Some links between min-cuts, optimal spanning forests and watersheds Cédric Allène, Jean-Yves Audibert, Michel Couprie, Jean Cousty & Renaud Keriven ENPC – CERTISESIEE – A²SI Université Paris-Est - France

Some links between min-cuts, optimal spanning forests and watersheds Outline 1) Motivation 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 Image segmentation 1 – Motivation: graph segmentation An image

Some links between min-cuts, optimal spanning forests and watersheds Image segmentation 1 – Motivation: graph segmentation A set of markers on the image

Some links between min-cuts, optimal spanning forests and watersheds Image segmentation 1 – Motivation: graph segmentation How to find a good segmentation?

Some links between min-cuts, optimal spanning forests and watersheds 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: Energy to minimize 1 – Motivation: graph segmentation

Some links between min-cuts, optimal spanning forests and watersheds A graph G is composed of: What about graphs? 1 – Motivation: graph segmentation Edge weighted graph:  Application  a set of nodes, denoted by V,  a set of edges linking a couple of nodes, denoted by E.

Some links between min-cuts, optimal spanning forests and watersheds Image to graph 1 – Motivation: graph segmentation 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 Image to graph 1 – Motivation: graph segmentation Edge-weighted graph

Some links between min-cuts, optimal spanning forests and watersheds Markers 1 – Motivation: graph segmentation

Some links between min-cuts, optimal spanning forests and watersheds 1 – Motivation: graph segmentation Link to energy: minimization of the sum of the dashed edges Result: frontier

Some links between min-cuts, optimal spanning forests and watersheds Link to energy: maximization of the sum of the bold edges 1 – Motivation: graph segmentation Result: partition

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 Marker M subgraph of G Extension relative to M subgraph 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 Maximal extension relative to M spanning extension relative to M which can ’ t be strictly contained by another extension relative to M Spanning forest relative to M spanning extension relative to M from which you can ’ t remove any edge without loosing the spanning extension property Cut relative to M set of edges linking two different connected components of a spanning extension relative to M Vocabulary on graphs 1 – Graph segmentation Marker M ExtensionSpanning extension Maximal extension Spanning forest Cut Correspondance with image

Some links between min-cuts, optimal spanning forests and watersheds Min-cuts  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) 2 – Usual algorithms in graph segmentation a - Min-cuts

Some links between min-cuts, optimal spanning forests and watersheds 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: doesn’t take into account all the edges of the maximal extension… 2 – Usual algorithms in graph segmentation b - Watersheds

Some links between min-cuts, optimal spanning forests and watersheds Shortest-path spanning forest cuts (SPSF cuts) 2 – Usual algorithms in graph segmentation c – 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:  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 π in F from x to M such that P(x)=P(π).

Some links between min-cuts, optimal spanning forests and watersheds Shortest-path spanning forest cuts (SPSF cuts) 2 – Usual algorithms in graph segmentation c – SPSF cuts  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 Remark about comparison 2 – Usual algorithms in graph segmentation b - Watersheds  Min-cuts need are of low value whereas watersheds or SPSF cuts need cuts are of high value.  In the objective to compare min-cuts with watersheds or SPSF cuts, we consider a strictly decreasing function applied to the weights of the graph after computation of a watershed or a SPSF cut.

Some links between min-cuts, optimal spanning forests and watersheds Comparisons of results 2 – Usual algorithms in graph segmentation d - Comparisons of results Min-cut Watershed / SPSF cut Similar results between watershed and SPSF cut, but differences with min-cut.

Some links between min-cuts, optimal spanning forests and watersheds Comparisons of results 2 – Usual algorithms in graph segmentation d - Comparisons of results Markers Min-cut  "Best": min-cut  Drawback of watershed / SPSF cut: leak point Watershed / SPSF cut

Some links between min-cuts, optimal spanning forests and watersheds Comparisons of results 2 – Usual algorithms in graph segmentation d - Comparisons of results Markers Min-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… Watershed / SPSF cut

Some links between min-cuts, optimal spanning forests and watersheds - 3 – Links

Some links between min-cuts, optimal spanning forests and watersheds Link between watersheds and SPSF cuts 3 – Links a - 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)

Some links between min-cuts, optimal spanning forests and watersheds Link between min-cut and watershed 3 – Links a - Link between watersheds and SPSF cuts 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 Link between min-cut and watershed: Rise of the power of graph weights for min-cuts 3 – Links b - Link between min-cuts and watersheds Rise to square: min cut = watershed Min-cut Watershed

Some links between min-cuts, optimal spanning forests and watersheds Link between min-cut and watershed: Rise of the power of graph weights for min-cuts 3 – Links b - Link between min-cuts and watersheds Power p = 1 Power p = 1,4 Power p = 2 Power p = 3 Watershed Marker Min-cut

Some links between min-cuts, optimal spanning forests and watersheds Intuitively… 3 – Links b - Link between min-cuts and watersheds Let C n be the min-cut for the weight at power n. Weight of the cut: Looks like p-norm: When n is rising you converge towards infinity-norm: Consequently:

Some links between min-cuts, optimal spanning forests and watersheds Intuitively… 3 – Links b - Link between min-cuts and watersheds 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  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? 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

Some links between min-cuts, optimal spanning forests and watersheds  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 p 1,…,p k be the different values of weight in the graph with p 1 >…>p k.  Let n h (X) be the number of edges of weight p h in the subgraph X.  So we have: and Which comes to proving that for any h: Scheme of proof 3 – Links b - Link between min-cuts and watersheds

Some links between min-cuts, optimal spanning forests and watersheds Leveling (thresholding) 3 – Links b - Link between min-cuts and watersheds  Let X h be the subgraph of X composed with only the edges of weight greater or equal to p h. G 2 (p 2 = 8) G 3 (p 3 = 7) G 4 (p 4 = 6) G 6 (p 6 = 4)

Some links between min-cuts, optimal spanning forests and watersheds G 3 (p 3 = 7) G 6 (p 6 = 4) Lemma 1 3 – Links b - Link between min-cuts and watersheds  For any h, F h is a spanning forest relative to the markers for G h. F 3 (p 3 = 7) F 6 (p 6 = 4)

Some links between min-cuts, optimal spanning forests and watersheds G 3 (p 3 = 7) G 6 (p 6 = 4) Lemma 2 3 – Links b - Link between min-cuts and watersheds  For any h, F’ h is a spanning forest relative to the markers for G h. F’ 3 (p 3 = 7) F’ 6 (p 6 = 4)

Some links between min-cuts, optimal spanning forests and watersheds Lemma 3 and epilogue 3 – Links b - Link between min-cuts and watersheds  Lemma 3: Any spanning forest for a subgraph X has a constant number of edges.  We deduce that n 1 (F 1 )=n 1 (F’ 1 ).  By induction: for all h, n h (F h )=n h (F’ h ).  Consequently, P(F)=P(F’) and finally F’ is a MaxSF!