Connected Component Tree

Transcription

1 onnected omponent Tree Jean ousty OUR-DAY OURSE on Mathematical Morphology in image analysis Bangalore 19- October 010 Motivation onnectivity plays important role for image processing Morphological connected filters Morphological filters acting on components of a set Morphological filters acting on components of the flat zones of a function Graphs are adapted for handling digital connectivity Problem How can connected filters be efficiently implemented? J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 1 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 1

2 Outline Structuring a discrete set onnected filters on (weighted) graphs onnected components omponent tree Example of application Algorithms for connected filters on (weighted) graphs Union of disjoint set onnected component labeling omponent tree extraction Let V be a set (eg, the set of all image pixels) Let X be a subset of V (eg, a binary object to be processed) J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 3 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 4

3 Discrete set Discrete set V (white & black dots) and X (black dots) J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 5 V (white & black dots) and X (black dots) J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 6 3

4 Graph Graph Neighborhood (in blue) of a point (in red) J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 7 Vertex set (dots) and edge set E (line segments) of a graph (V,E) J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 8 4

5 onnected sets Example: Discrete set that is not connected Let G = (V,E) be a (undirected) graph Let X be a subset of V Definition X is connected if there exists a path between any two points of X J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 9 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 10 5

6 Example: onnected discrete set onnected components Let G = (V,E) be a (undirected) graph. Let X be a subset of V Let Y be a subset of X Definition Y is a (connected) component of X if Y is connected no proper superset of Y included in X is connected Property Y is a component of X iff for any x in Y Y is the union of all connected subsets of X that conains x J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 11 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 1 6

7 Example : onnected components partition onnected opening a discrete set According to a given criterion (e.g., the size), keep the most important components of X J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 13 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 14 7

8 Example: area filtering a discrete set Example: area filtering a discrete set J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 15 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 16 8

9 rom maps to sets rom maps to sets k k k = {x E, (x) k}, i.e., points above k J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 17 k = {x E, (x) < k}, i.e., points below k J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 18 9

10 Structuring a discrete map: level sets decomposition Structuring a discrete map: (upper) omponent tree Definition Let be a map from V to N and let k be an integer The (upper) level-set of at level k is the set made of the points of V whose value is greater than or equal to k k = {x E, (x) k} Any connected component of any (upper) level set of is called a (upper) component of Remark Any two components of are either nested or disjoint omponents + inclusion relation tree structure J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 19 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 0 10

11 Structuring a discrete map: (upper) omponent tree Level-sets, components: the D case J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 1 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 11

12 Level-sets, components: the D case Level-sets, components: the D case 1 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 3 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 4 1

13 Level-sets, components: the D case Level-sets, components: the D case 3 4 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 5 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 6 13

14 ross-sections, components: the D case omponent tree: the D case Level 1 1 Level Level 3 Level omponent tree omponent mapping J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology

15 omponent tree: the D case omponent tree: the D case Level Level Level 3 8 Level Level 3 Level Level 3 Level omponent tree omponent mapping omponent tree omponent mapping

16 omponent tree: the D case omponent tree: the D case Level 1 1 Level Level Level 3 Level 3 Level Level 3 Level omponent tree omponent mapping omponent tree omponent mapping

17 iltering a discrete map : flooding looding: intuitive example step 1 A leaf of the component tree T of is: a regional minimum of, if T is a lower component tree a regional maximum of, if T is an upper component tree According to a given criterion, we can iteratively remove the leaf components of Such a removal is called a flooding J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 33 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 34 17

18 looding: intuitive example step 1 looding: intuitive example step J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 35 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 36 18

19 looding: intuitive example step 3 looding: formal definition Definition Let x in V The elementary flooding of at x is defined by: η x (y) = (y) + 1 if x and y belong to the same minimum η x (y) = (y) otherwise Any map obtained from by composition of elementary floodings is called a flooding of We remove one branch of T We flood one lobe of J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 37 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 38 19

20 iltering a grayscale map Attributes: height What criterion (attribute) to select the non-significant minima to be flooded? h J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 39 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 40 0

21 Attributes: area Attributes: volume a v J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 41 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 4 1

22 iltering by volume iltering by volume Regional minima J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 43 Regional minima J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 44

23 Application: keep the 10 most volumic lobes Algorithms requested for achieving this result omponent tree algorithm Original image (top) and its many minima (bottom) iltered image (top) and its 10 minima (bottom) looding the minima with attribute below a given threshold Or alternatively, keeping only the k most significant minima J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 45 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 46 3

24 Algorithms for the component tree There exist several algorithms for the component tree Breen & Jones 1996 Salembier, Oliveras & Garido 1998 Mattes & Demongeot 000 ouprie & Najman 006 Tarjan s union find onnected components labeling Quasi-linear time component tree extraction J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 47 Disjoint set problem Maintaining a collection of disjoint subsets of a set V under the operation of union Each subset X is represented by a unique element of X called the canonical element of X Three operations allow the collection to be managed: MakeSet(x): add the set {x} to the collection ind(x): return the canonical element of the set containing x Link(x,y): Remove the two sets whose canonical elements are x and y Add their union J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 48 4

25 Tarjan s union-find algorithm onnected components labeling with union-find Quasi-linear algorithm: O(m X α(m, V )) for m operations α : inverse of Ackerman function (in practice never greater than 4) J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 49 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 50 5

26 onnected components labeling with union-find onnected components labeling with union-find J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 51 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 5 6

27 onnected components labeling with union-find onnected components labeling with union-find J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 53 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 54 7

28 onnected components labeling with union-find onnected components labeling with union-find J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 55 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 56 8

29 onnected components labeling with union-find onnected components labeling with union-find J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 57 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 58 9

30 onnected components labeling with union-find onnected components labeling with union-find J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 59 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 60 30

31 onnected components labeling with union-find omponent tree algorithm General case Two collections of disjoint subsets of V Particular case: all values are distinct One collection of disjoint subsets of V J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 61 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 6 31

32 omponent tree algorithm: all values distinct foreach point x in V in decreasing order of altitude do foreach neighbor y of x do If ((y) > (x)) and (ind(x)!= ind(y)) do adjnode := lowestnode[ind(y)] nodes[x].addhild(nodes[adjnode]) curnode := Link(x,adjNode) lowestnode[curnode] := nodes[x] omponent tree algorithm: all values distinct foreach point x in V in decreasing order of altitude do foreach neighbor y of x do If ((y) > (x)) and (ind(x)!= ind(y)) do adjnode := lowestnode[ind(y)] nodes[x].addhild(nodes[adjnode]) curnode := Link(x,adjNode) lowestnode[curnode] := nodes[x] (init) Image Lowest Node nodes Disjoint sets A:7B:1:9 D:6E:4 : G:8H:5 I:3 A B D E G H I E B I H A D G J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 63 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 64 3

33 omponent tree algorithm: all values distinct (step 1) foreach point x in V in decreasing order of altitude do A:7B:1:9 D:6E:4 : G:8H:5 I:3 foreach neighbor y of x do If ((y) > (x)) and (ind(x)!= ind(y)) do adjnode := lowestnode[ind(y)] nodes[x].addhild(nodes[adjnode]) curnode := Link(x,adjNode) lowestnode[curnode] := nodes[x] A B D E G H I E B x = Image Lowest Node nodes Disjoint sets I H A D G omponent tree algorithm: all values distinct (step ) foreach point x in V in decreasing order of altitude do A:7B:1:9 D:6E:4 : G:8H:5 I:3 foreach neighbor y of x do If ((y) > (x)) and (ind(x)!= ind(y)) do adjnode := lowestnode[ind(y)] nodes[x].addhild(nodes[adjnode]) curnode := Link(x,adjNode) lowestnode[curnode] := nodes[x] A B D E G H I E B x = G Image Lowest Node nodes Disjoint sets G I H A D G J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 65 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 66 33

34 omponent tree algorithm: all values distinct (step 3) foreach point x in V in decreasing order of altitude do A:7B:1:9 D:6E:4 : G:8H:5 I:3 foreach neighbor y of x do If ((y) > (x)) and (ind(x)!= ind(y)) do adjnode := lowestnode[ind(y)] nodes[x].addhild(nodes[adjnode]) curnode := Link(x,adjNode) lowestnode[curnode] := nodes[x] A B D E G H I E B x = A Image Lowest Node nodes Disjoint sets G A I H A D G omponent tree algorithm: all values distinct (step 4) foreach point x in V in decreasing order of altitude do A:7B:1:9 D:6E:4 : G:8H:5 I:3 foreach neighbor y of x do If ((y) > (x)) and (ind(x)!= ind(y)) do adjnode := lowestnode[ind(y)] nodes[x].addhild(nodes[adjnode]) curnode := Link(x,adjNode) lowestnode[curnode] := nodes[x] A B D E G H I E B x = D Image Lowest Node nodes Disjoint sets G A I H A D G J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 67 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 68 34

35 omponent tree algorithm: all values distinct (step 4) foreach point x in V in decreasing order of altitude do A:7B:1:9 D:6E:4 : G:8H:5 I:3 foreach neighbor y of x do If ((y) > (x)) and (ind(x)!= ind(y)) do adjnode := lowestnode[ind(y)] nodes[x].addhild(nodes[adjnode]) curnode := Link(x,adjNode) lowestnode[curnode] := nodes[x] D B D E G H I E B x = D y = A Image Lowest Node nodes Disjoint sets G A D I H A D G omponent tree algorithm: all values distinct (step 4) foreach point x in V in decreasing order of altitude do A:7B:1:9 D:6E:4 : G:8H:5 I:3 foreach neighbor y of x do If ((y) > (x)) and (ind(x)!= ind(y)) do adjnode := lowestnode[ind(y)] nodes[x].addhild(nodes[adjnode]) curnode := Link(x,adjNode) lowestnode[curnode] := nodes[x] D B D E G H I E B x = D y = G Image Lowest Node nodes Disjoint sets G A D I H A D G J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 69 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 70 35

36 omponent tree algorithm: all values distinct (step 5) foreach point x in V in decreasing order of altitude do A:7B:1:9 D:6E:4 : G:8H:5 I:3 foreach neighbor y of x do If ((y) > (x)) and (ind(x)!= ind(y)) do adjnode := lowestnode[ind(y)] nodes[x].addhild(nodes[adjnode]) curnode := Link(x,adjNode) lowestnode[curnode] := nodes[x] H B D E G H I E B x = H y = G Image Lowest Node nodes Disjoint sets G A D H I H A D G omponent tree algorithm: all values distinct (step 6) foreach point x in V in decreasing order of altitude do A:7B:1:9 D:6E:4 : G:8H:5 I:3 foreach neighbor y of x do If ((y) > (x)) and (ind(x)!= ind(y)) do adjnode := lowestnode[ind(y)] nodes[x].addhild(nodes[adjnode]) curnode := Link(x,adjNode) lowestnode[curnode] := nodes[x] E B D E G H I E B x = E y = D Image Lowest Node nodes Disjoint sets G A D H E I H A D G J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 71 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 7 36

37 omponent tree algorithm: all values distinct (step 6) foreach point x in V in decreasing order of altitude do A:7B:1:9 D:6E:4 : G:8H:5 I:3 foreach neighbor y of x do If ((y) > (x)) and (ind(x)!= ind(y)) do adjnode := lowestnode[ind(y)] nodes[x].addhild(nodes[adjnode]) curnode := Link(x,adjNode) lowestnode[curnode] := nodes[x] E B D E G H I E B x = E y = H Image Lowest Node nodes Disjoint sets G A D H E I H A D G omponent tree algorithm: all values distinct (step 7) foreach point x in V in decreasing order of altitude do A:7B:1:9 D:6E:4 : G:8H:5 I:3 foreach neighbor y of x do If ((y) > (x)) and (ind(x)!= ind(y)) do adjnode := lowestnode[ind(y)] nodes[x].addhild(nodes[adjnode]) curnode := Link(x,adjNode) lowestnode[curnode] := nodes[x] I B D E G H I E B x = I y = H Image Lowest Node nodes Disjoint sets G A D H E I I H A D G J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 73 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 74 37

38 omponent tree algorithm: all values distinct (step 8) foreach point x in V in decreasing order of altitude do A:7B:1:9 D:6E:4 : G:8H:5 I:3 foreach neighbor y of x do If ((y) > (x)) and (ind(x)!= ind(y)) do adjnode := lowestnode[ind(y)] nodes[x].addhild(nodes[adjnode]) curnode := Link(x,adjNode) lowestnode[curnode] := nodes[x] I B D E G H I E B x = y = Image Lowest Node nodes Disjoint sets G A D H E I I H A D G omponent tree algorithm: all values distinct (step 7) foreach point x in V in decreasing order of altitude do A:7B:1:9 D:6E:4 : G:8H:5 I:3 foreach neighbor y of x do If ((y) > (x)) and (ind(x)!= ind(y)) do adjnode := lowestnode[ind(y)] nodes[x].addhild(nodes[adjnode]) curnode := Link(x,adjNode) lowestnode[curnode] := nodes[x] I B D E G H I E B x = y = E Image Lowest Node nodes Disjoint sets G A D H E I I H A D G J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 75 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 76 38

39 omponent tree algorithm: all values distinct (step 7) foreach point x in V in decreasing order of altitude do A:7B:1:9 D:6E:4 : G:8H:5 I:3 foreach neighbor y of x do If ((y) > (x)) and (ind(x)!= ind(y)) do adjnode := lowestnode[ind(y)] nodes[x].addhild(nodes[adjnode]) curnode := Link(x,adjNode) lowestnode[curnode] := nodes[x] I B D E G H I E B x = y = I Image Lowest Node nodes Disjoint sets G A D H E I I H A D G omponent tree algorithm: all values distinct (step 9) foreach point x in V in decreasing order of altitude do A:7B:1:9 D:6E:4 : G:8H:5 I:3 foreach neighbor y of x do If ((y) > (x)) and (ind(x)!= ind(y)) do adjnode := lowestnode[ind(y)] nodes[x].addhild(nodes[adjnode]) curnode := Link(x,adjNode) lowestnode[curnode] := nodes[x] I B B D E G H I E B x = B y = A Image Lowest Node nodes Disjoint sets G A D H E I B I H A D G J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 77 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 78 39

40 omponent tree algorithm: all values distinct (step 10) foreach point x in V in decreasing order of altitude do A:7B:1:9 D:6E:4 : G:8H:5 I:3 foreach neighbor y of x do If ((y) > (x)) and (ind(x)!= ind(y)) do adjnode := lowestnode[ind(y)] nodes[x].addhild(nodes[adjnode]) curnode := Link(x,adjNode) lowestnode[curnode] := nodes[x] I B B D E G H I E B x = B y = Image Lowest Node nodes Disjoint sets G A D H E I B I H A D G omponent tree algorithm: all values distinct foreach point x in V in decreasing order of altitude do foreach neighbor y of x do If ((y) > (x)) and (ind(x)!= ind(y)) do adjnode := lowestnode[ind(y)] nodes[x].addhild(nodes[adjnode]) curnode := Link(x,adjNode) lowestnode[curnode] := nodes[x] Time complexity : quasi-linear with respect to the number of edges and vertices Requires a sorting pre-processing step O( V ) O( V + E ) O( E.α( E, V )) O( E ) O( E ) O( E. α( E, V )) O( E ) J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 79 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 80 40

41 omponent tree algorithm: general case Build the component tree Two union-find implementations needed One for building the union of connected components over height Similar to the one in the case where all values were distinct One for building the connected components of a given altitude J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 81 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 8 41

42 Auxiliary functions Example ; 0; ; 1; 14; 13; 3; 4; 5; 8; 9; 10; 11; 6; 7 [10] [110] [100] [90] [80] [70] [50] [40] [0] J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 83 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 84 4

43 step 1 Step th1 th SubtreeRoot th1 th SubtreeRoot [1] 10 [1] 10 [0] 110 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 85 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 86 43

44 step 3 step th1 th SubtreeRoot th1 th SubtreeRoot [1] 90 [1] 10 [0] 110 [] 100 [1] 10 [0] 110 [] 100 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 87 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 88 44

45 step 5 step th1 th SubtreeRoot th1 th SubtreeRoot [1] 90 [1] 90 [13] 70 [0] 110 [] 100 [1] 10 [14] 80 [0] 110 [] 100 [1] 10 [14] 80 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 89 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 90 45

46 step 7 step th1 th SubtreeRoot th1 th SubtreeRoot [3] 50 [3] 50 [1] 90 [13] 70 [1] 90 [13] 70 [0] 110 [] 100 [1] 10 [14] 80 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 91 [0] 110 [] 100 [1] 10 [14] 80 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 9 46

47 step 9 step th1 th SubtreeRoot th1 th SubtreeRoot [3] 50 [3] 50 [1] 90 [13] 70 [1] 90 [13] 70 [0] 110 [] 100 [1] 10 [14] 80 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 93 [0] 110 [] 100 [1] 10 [14] 80 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 94 47

48 step 11 step th1 th SubtreeRoot th1 th SubtreeRoot [3] 50 [9] 50 [3] 50 [9] 50 [1] 90 [13] 70 [1] 90 [13] 70 [0] 110 [] 100 [1] 10 [14] 80 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 95 [0] 110 [] 100 [1] 10 [14] 80 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 96 48

49 step 13 step th1 th [9] 50 SubtreeRoot th1 [6] 40 th SubtreeRoot [9] 50 [1] 90 [13] 70 [1] 90 [13] 70 [0] 110 [] 100 [1] 10 [14] 80 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 97 [0] 110 [] 100 [1] 10 [14] 80 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 98 49

50 step 15 omponent tree: conclusion Another insight into connected filters (level sets stacking) Useful to compute floodings then attribute openings & closings Efficient algorithm th1 [7] 0 [6] 40 th SubtreeRoot Quasi-linear time Easy implementation Many applications in signal processing and image analysis [9] 50 Important for computing connection values [1] 90 [13] 70 There exist self-dual trees [0] 110 [] 100 [1] 10 [14] 80 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology 99 J. Serra, J. ousty, B.S. Daya Sagar: ourse on Math. Morphology