[PPT] - Segmentation by discrete watersheds Part 1: Watershed cuts Jean PowerPoint Presentation

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Segmentation by discrete watersheds Part 1: Watershed cuts

Jean Cousty Four-Day Course

n

Mathematical Morphology in image analysis

Bangalore 19-22 October 2010

J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology

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An applicative introduction to segmentation in medicine

Magnetic Resonance Imagery (MRI) is more and more used for cardiac diagnosis

J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology

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An applicative introduction to segmentation in medicine

A cardiac MRI examination includes three steps: Spatio-temporal acquisition (cin´ e MRI) im

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An applicative introduction to segmentation in medicine

A cardiac MRI examination includes three steps: Spatio-temporal acquisition (cin´ e MRI) Spatio-temporal acquisition during contrast agent injection (Perfusion) im

J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology

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An applicative introduction to segmentation in medicine

A cardiac MRI examination includes three steps: Spatio-temporal acquisition (cin´ e MRI) Spatio-temporal acquisition during contrast agent injection (Perfusion) Volumic acquisition after the evacuation of the contrast agent (delayed enhanced MRI) im

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Medical problem #1

Problem Visualizing objects of interests in 3D or 4D images rendu2

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Medical problem #2

Problem Determining measures useful for cardiac diagnosis

Infarcted volumes, ventricular volumes, ejection fraction, myocardial mass, movement . . .

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Technical problem

Problem Segmentation of object of interest

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Technical problem

Problem Segmentation of object of interest A morphological solution Watershed

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Watershed: introduction

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Watershed: introduction

For topographic purposes, the watershed has been studied since the 19th century (Maxwell, Jordan, . . . )

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Watershed: introduction

One hundred years later (1978), it was introduced by Digabel and Lantu´ ejoul for image segmentation

J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology

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Watershed: introduction

One hundred years later (1978), it was introduced by Digabel and Lantu´ ejoul for image segmentation

J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology

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SLIDE 14

Watershed: introduction

One hundred years later (1978), it was introduced by Digabel and Lantu´ ejoul for image segmentation

J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology

6/36

SLIDE 15

Watershed: introduction

One hundred years later (1978), it was introduced by Digabel and Lantu´ ejoul for image segmentation

J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology

6/36

SLIDE 16

Watershed: introduction

One hundred years later (1978), it was introduced by Digabel and Lantu´ ejoul for image segmentation

J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology

6/36

SLIDE 17

Watershed: introduction

One hundred years later (1978), it was introduced by Digabel and Lantu´ ejoul for image segmentation

J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology

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Watershed: problem #1

Problem How to define the watershed of digital image?

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Watershed: problem #1

Problem How to define the watershed of digital image? Which mathematical framework(s)?

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Watershed: problem #1

Problem How to define the watershed of digital image? Which mathematical framework(s)? Which properties?

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Watershed: problem #1

Problem How to define the watershed of digital image? Which mathematical framework(s)? Which properties? Which algorithms ?

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Watershed: problem #2

Problem In practice: over-segmentation

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Over-segmentation and region merging

Solution 1 Region merging methods consist of improving an initial segmentation by progressively merging pairs of neighboring regions

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Over-segmentation and region merging

Solution 1 Region merging methods consist of improving an initial segmentation by progressively merging pairs of neighboring regions Example : delayed enhanced cardiac MRI [DOUBLIER03]

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Over-segmentation and region merging

Solution 1 Region merging methods consist of improving an initial segmentation by progressively merging pairs of neighboring regions Example : delayed enhanced cardiac MRI [DOUBLIER03]

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Over-segmentation and region merging

Solution 1 Region merging methods consist of improving an initial segmentation by progressively merging pairs of neighboring regions Example : delayed enhanced cardiac MRI [DOUBLIER03]

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Over-segmentation and region merging

Solution 1 Region merging methods consist of improving an initial segmentation by progressively merging pairs of neighboring regions

infarctus myocarde cavité sanguine

Example : delayed enhanced cardiac MRI [DOUBLIER03]

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Over-segmentation

Solution 2 Seeded watershed (or marker based watershed)

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Over-segmentation

Solution 2 Seeded watershed (or marker based watershed) Methodology proposed by Beucher and Meyer (1993)

1 Recognition 2 Delineation (generally done by watershed) 3 Smoothing

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Over-segmentation

Solution 2 Seeded watershed (or marker based watershed) Methodology proposed by Beucher and Meyer (1993)

1 Recognition 2 Delineation (generally done by watershed) 3 Smoothing

Semantic information taken into account at steps 1 and 3 To kow more about this framework, wait for the second lecture of today

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Outline

1 Defining discrete watersheds is difficult

Grayscale image as vertex weighted graphs Region merging problems

2 Watershed in edge-weighted graphs

Watershed cuts: definition and consistency Minimum spanning forests: watershed optimality

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Defining discrete watersheds is difficult

Can we draw a watershed of this image?

2 2 2 2 2 40 30 30 30 40 40 20 20 20 40 40 40 20 40 40 1 5 20 5 1

Image equipped with the 4-adjacency

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Defining discrete watersheds is difficult

Can we draw a watershed of this image?

A A A A A 40 30 30 30 40 40 20 20 20 40 40 40 20 40 40 B 5 20 5 C

Image equipped with the 4-adjacency Label the pixels according to catchment basins letters A,B and C

J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology

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Defining discrete watersheds is difficult

Possible drawings

A A A A A 40 30 30 30 40 40 20 20 20 40 40 40 20 40 40 B 5 20 5 C A A A A A A A A A A 40 20 20 20 40 B B 20 C C B B 20 C C

Topographical watershed

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Defining discrete watersheds is difficult

Possible drawings

A A A A A 40 30 30 30 40 40 20 20 20 40 40 40 20 40 40 B 5 20 5 C A A A A A A A A A A C C C C C B B C C C B B C C C

Flooding from the minima

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Defining discrete watersheds is difficult

Possible drawings

A A A A A 40 30 30 30 40 40 20 20 20 40 40 40 20 40 40 B 5 20 5 C A A A A A A A A A A 40 A A A 40 B 40 A 40 C B B 20 C C

Flooding with divide

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Defining discrete watersheds is difficult

Possible drawings

A A A A A 40 30 30 30 40 40 20 20 20 40 40 40 20 40 40 B 5 20 5 C A A A A A 40 30 30 30 40 B B 20 C C B B 20 C C B B 20 C C

Topological watershed

J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology

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Defining discrete watersheds is difficult

Possible drawings

A A A A A 40 30 30 30 40 40 20 20 20 40 40 40 20 40 40 B 5 20 5 C A A A A A 40 30 30 30 40 B B 20 C C B B 20 C C B B 20 C C

Topological watershed Conclusion Not easy to define watersheds of digital images

J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology

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Defining discrete watersheds is difficult

Region merging: Problem #1

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Defining discrete watersheds is difficult

Region merging: Problem #1

J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology

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Defining discrete watersheds is difficult

Region merging: Problem #1

A B C D

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Defining discrete watersheds is difficult

Region merging: Problem #1

A B C D

x

y

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Defining discrete watersheds is difficult

Region merging: Problem #1

A B C D

Problem : “When 3 regions meet”, [PAVLIDIS-77]

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Defining discrete watersheds is difficult

Region merging: Problem #1

adjacence directe adjacence indirecte ?

Problem : “When 3 regions meet”, [PAVLIDIS-77] Is there some adjacency relations (graphs) for which any pair of neighboring regions can always be merged, while preserving all

ther regions?
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Defining discrete watersheds is difficult

Region merging: Problem # 2

A cleft is a set of vertices from which a point cannot be removed while leaving unchanged the number of connected components of its complement

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Defining discrete watersheds is difficult

Region merging: Problem # 2

A cleft is a set of vertices from which a point cannot be removed while leaving unchanged the number of connected components of its complement A cleft is thin if all its vertices are adjacent to its complement

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Defining discrete watersheds is difficult

Region merging: Problem # 2

A cleft is a set of vertices from which a point cannot be removed while leaving unchanged the number of connected components of its complement A cleft is thin if all its vertices are adjacent to its complement

Problem : Thick cleft (or binary watershed)

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Defining discrete watersheds is difficult

Region merging: Problem # 2

A cleft is a set of vertices from which a point cannot be removed while leaving unchanged the number of connected components of its complement A cleft is thin if all its vertices are adjacent to its complement

Problem : Thick cleft (or binary watershed) Is there some graphs in which any cleft is thin?

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Defining discrete watersheds is difficult

The familly of watersheds

Topological watersheds Only watersheds that preserve the altitudes of the passes On pixels Fusion graphs Link between thinness, region merging, and watersheds On edges Watershed cuts Optimality, drop of water principle Power watersheds Framework for seeded image segmentation (graph cuts, random walker, . . .) Energy minimization q = 2 = ⇒ uniqueness Ultrametric watersheds Hierarchical segmentation On complexes Simplicial stacks Link between collapse, watersheds and optimal spanning forests

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Watershed in edge-weighted graphs

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Watershed in edge-weighted graphs

Let G = (V , E) be a graph. Let F be a map from E to R

5 5 5 5 5 2 4 4 3 2 4 5 2 5 3 4 4 3

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Watershed in edge-weighted graphs

Image and edge-weighted graph

For applications to image analysis V is the set of pixels E corresponds to an adjacency relation on V , (e.g., 4- or 8-adjacency in 2D) F is a “gradient” of I: The altitude of u, an edge between two pixels x and y, represents the dissimilarity between x and y

F(u) = |I(x) − I(y)|.

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Watershed in edge-weighted graphs

Regional minima

00 1 2 2 2 3 3 4 4 5 5 5 7 7 7 6 7 2

Definition A subgraph X of G is a minimum of F (at altitude k) if: X is connected; and k is the altitude of any edge of X; and the altitude of any edge adjacent to X is strictly greater than k

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Watershed in edge-weighted graphs

Extension

a subgraph X Definition (from Def. 12, (Ber05)) Let X and Y be two non-empty subgraphs of G We say that Y is an extension of X (in G) if X ⊆ Y and if any component of Y contains exactly one component of X.

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Watershed in edge-weighted graphs

Extension

a subgraph X an extension Y of X Definition (from Def. 12, (Ber05)) Let X and Y be two non-empty subgraphs of G We say that Y is an extension of X (in G) if X ⊆ Y and if any component of Y contains exactly one component of X.

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Watershed in edge-weighted graphs

Extension

a subgraph X an extension Y of X Definition (from Def. 12, (Ber05)) Let X and Y be two non-empty subgraphs of G We say that Y is an extension of X (in G) if X ⊆ Y and if any component of Y contains exactly one component of X.

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Watershed in edge-weighted graphs

Extension

a subgraph X an extension Y of X Definition (from Def. 12, (Ber05)) Let X and Y be two non-empty subgraphs of G We say that Y is an extension of X (in G) if X ⊆ Y and if any component of Y contains exactly one component of X.

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Watershed in edge-weighted graphs

Extension

a subgraph X an extension Y of X Definition (from Def. 12, (Ber05)) Let X and Y be two non-empty subgraphs of G We say that Y is an extension of X (in G) if X ⊆ Y and if any component of Y contains exactly one component of X.

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Watershed in edge-weighted graphs

Extension

a subgraph X an extension Y of X Definition (from Def. 12, (Ber05)) Let X and Y be two non-empty subgraphs of G We say that Y is an extension of X (in G) if X ⊆ Y and if any component of Y contains exactly one component of X.

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Watershed in edge-weighted graphs

Extension

a subgraph X an extension Y of X Definition (from Def. 12, (Ber05)) Let X and Y be two non-empty subgraphs of G We say that Y is an extension of X (in G) if X ⊆ Y and if any component of Y contains exactly one component of X.

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Watershed in edge-weighted graphs

Extension

a subgraph X an extension Y of X Definition (from Def. 12, (Ber05)) Let X and Y be two non-empty subgraphs of G We say that Y is an extension of X (in G) if X ⊆ Y and if any component of Y contains exactly one component of X.

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Watershed in edge-weighted graphs

Graph cut

a subgraph X a (graph) cut S for X Definition (Graph cut) Let X be a subgraph of G and S ⊆ E, a set of edges. We say that S is a (graph) cut for X if S is an extension of X and if S is minimal for this property

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Watershed in edge-weighted graphs

Watershed cut

The church of Sorbier (a topographic intuition) Definition (drop of water principle) The set S ⊆ E is a watershed cut of F if S is an extension of M(F) and if for any u = {x0, y0} ∈ S, there exist x0, . . . , xn and y0, . . . , ym, two descending paths in S such that:

1 xn and ym are vertices of two distinct minima of F; and 2 F(u) ≥ F({x0, x1}) if n > 0 and F(u) ≥ F({y0, y1}) if m > 0

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Watershed in edge-weighted graphs

Watershed cut: example

2 6 6 2 6 5 5 7 3 6 5 3 1 2 5 6 7 5 7 3 4 5 3 8

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Watershed in edge-weighted graphs

Watershed cut: example

2 6 6 2 6 5 5 7 3 6 5 3 1 2 8 5 6 7 5 7 3 4 5 3

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Watershed in edge-weighted graphs

Watershed cut: example

2 6 6 2 6 5 5 7 3 6 5 3 1 2 8 5 6 7 5 7 3 4 5 3

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Watershed in edge-weighted graphs

Watershed cut: example

2 6 6 2 6 5 5 7 3 6 5 3 1 2 8 5 6 7 5 7 3 4 5 3

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Watershed in edge-weighted graphs

Watershed cut: example

2 6 6 2 6 5 5 7 3 6 5 3 1 2 8 5 6 7 5 7 3 4 5 3

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Watershed in edge-weighted graphs

Watershed cut: example

2 6 6 2 6 5 5 7 3 6 5 3 1 2 8 5 6 7 5 7 3 4 5 3

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Watershed in edge-weighted graphs

Steepest descent

2 6 6 2 6 5 5 7 3 6 5 3 1 2 8 5 6 7 5 7 3 4 5 3 Definition Let π = x0, . . . , xl be a path in G. The path π is a path with steepest descent for F if: ∀i ∈ [1, l], F({xi−1, xi}) = min{xi−1,y}∈E F({xi−1, y})

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Watershed in edge-weighted graphs

Catchment basins by a steepest descent property

Definition Let S be a cut for M(F), the minima of F We say that S is a basin cut of F if, from each point of V to M(F), there exists, in the graph induced by S, a path with steepest descent for F

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SLIDE 72

Watershed in edge-weighted graphs

Catchment basins by a steepest descent property

Theorem (consistency) An edge-set S ⊆ E is a basin cut of F if and only if S is a watershed cut of F

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SLIDE 73

Watershed in edge-weighted graphs

Illustration to grayscale image segmentation

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Watershed in edge-weighted graphs

Illustration to grayscale image segmentation

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Watershed in edge-weighted graphs

Illustration to mesh segmentation

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Watershed in edge-weighted graphs

Watershed optimality?

Problem Are watersheds optimal segmentations? Which combinatorial optimization problem do they solve?

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SLIDE 77

Watershed in edge-weighted graphs

Relative minimum spanning forest: an image intuitition

cut forest spanning regions

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SLIDE 78

Watershed in edge-weighted graphs

Relative forest: a botanical intuition

A tree (Lal Bagh)

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SLIDE 79

Watershed in edge-weighted graphs

Relative forest: a botanical intuition

Cuting the roots yield a forest of several trees

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SLIDE 80

Watershed in edge-weighted graphs

Relative forest: a botanical intuition

Roots may contain cycles

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SLIDE 81

Watershed in edge-weighted graphs

Relative forest

a subgraph X a forest Y relative to X Definition Let X and Y be two non-empty subgraphs of G. We say that Y is a forest relative to X if:

1 Y is an extension of X; and 2 any cycle of Y is also a cycle of X

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SLIDE 82

Watershed in edge-weighted graphs

Minimum spanning forest

The weight of a forest Y is the sum of its edge weights i.e.

u∈E(Y ) F(u).
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SLIDE 83

Watershed in edge-weighted graphs

Minimum spanning forest

The weight of a forest Y is the sum of its edge weights i.e.

u∈E(Y ) F(u).

Definition We say that Y is a minimum spanning forest (MSF) relative to X

if Y is a spanning forest relative to X and if the weight of Y is less than or equal to the weight of any other spanning forest relative to X

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SLIDE 84

Watershed in edge-weighted graphs

Minimum spanning forest: example

5 5 5 5 5 2 4 4 3 2 4 5 2 5 3 4 4 3

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SLIDE 85

Watershed in edge-weighted graphs

Minimum spanning forest: example

5 5 5 5 5 2 4 4 3 2 4 5 2 5 3 4 3 4

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SLIDE 86

Watershed in edge-weighted graphs

Minimum spanning forest: example

5 5 5 5 5 2 4 4 3 2 4 5 2 5 3 4 3 4

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SLIDE 87

Watershed in edge-weighted graphs

Minimum spanning forest: example

5 5 5 5 5 2 4 4 3 2 4 5 2 5 3 4 3 4

If Y is a MSF relative to X, there exists a unique cut S for Y and this cut is also a cut for X;

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SLIDE 88

Watershed in edge-weighted graphs

Minimum spanning forest: example

5 5 5 5 5 2 4 4 3 2 4 5 2 5 3 4 3 4

If Y is a MSF relative to X, there exists a unique cut S for Y and this cut is also a cut for X; In this case, we say that S is a MSF cut for X.

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SLIDE 89

Watershed in edge-weighted graphs

Watershed optimality

Theorem An edge-set S ⊆ E is a MSF cut for the minima of F if and only if S is a watershed cut of F

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SLIDE 90

Watershed in edge-weighted graphs

Minimum spanning tree

Computing a MSF ⇔ computing a minimum spanning tree

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SLIDE 91

Watershed in edge-weighted graphs

Minimum spanning tree

Computing a MSF ⇔ computing a minimum spanning tree Best algorithm [CHAZEL00]: quasi-linear time

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SLIDE 92

Watershed in edge-weighted graphs

Minimum spanning tree

Computing a MSF ⇔ computing a minimum spanning tree Best algorithm [CHAZEL00]: quasi-linear time Problem Can we reach a better complexity for computing watershed cuts?

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SLIDE 93

Watershed in edge-weighted graphs

Minimum spanning tree

Computing a MSF ⇔ computing a minimum spanning tree Best algorithm [CHAZEL00]: quasi-linear time Problem Can we reach a better complexity for computing watershed cuts? A morphological solution To know the answer, come back after the coffee break

J. Serra, J. Cousty, B.S. Daya Sagar : Course on Math. Morphology

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