Lec10: Medical Image Segmentation as an Energy Minimization Problem

MEDICAL IMAGE COMPUTING (CAP 5937)

LECTURE 10: Medical Image Segmentation as an Energy Minimization Problem

Dr. Ulas BagciHEC 221, Center for Research in Computer Vision (CRCV), University of Central Florida (UCF), Orlando, FL [email protected] or [email protected]

1SPRING 2017

Outline• Energy functional

– Data and Smoothness terms• Graph Cut

– Min cut– Max Flow

• Applications in Radiology Images

2

Motivation 3

Manual annotation through expert raters. Shown are image patches with the tumor structures that are annotated in the different modalities (top left) and the final labels for the whole dataset (right). Image patches show from left to right: the whole tumor visible in FLAIR (A), the tumor core visible in T2 (B), the enhancing tumor structures visible in T1c (blue), surrounding the cystic/necrotic components of the core (green) (C). Segmentations are combined to generate the final labels of the tumor structures (D): edema (yellow), non-enhancing solid core (red), necrotic/cystic core (green), enhancing core(blue). Credit: BRATS paper/TMI

Labeling & Segmentation• Labeling is a common way for modeling various computer

vision problems (e.g. optical flow, image segmentation, stereo matching, etc)

4



• The set of labels can be discrete (as in image segmentation)

5

L = {l1, . . . , lm} with L = m



• The set of labels can be discrete (as in image segmentation)

• Or continuous (tracking, etc.)

6

L = {l1, . . . , lm} with L = m

L ⇢ Rnfor n � 1

Labeling is a function• Labels are assigned to sites (pixel locations)

7

Labeling is a function• Labels are assigned to sites (pixel locations)• For a given image, we have

8

|⌦| = Ncols

.Nrows

Labeling is a function• Labels are assigned to sites (pixel locations)• For a given image, we have• Identifying a labeling function (with segmentation) is

9

|⌦| = Ncols

.Nrows

f : ⌦ ! L


We aim at calculating a labeling function that minimizes a given (total) error or energy

10

|⌦| = Ncols

.Nrows

f : ⌦ ! L


We aim at calculating a labeling function that minimizes a given (total) error or energy

11

|⌦| = Ncols

.Nrows

f : ⌦ ! L

E(f) =X

p2⌦

[Edata

(p, fp

) +X

q2A(p)

Esmooth

(fp

, fq

)]

*A is an adjacency relation between pixel locations

Example of Energy Minimization Methods

12

Credit: Zhao et al., IJNMBE 15

Energy Function?• Penalizing results which are not compatible with the

observed images/volumes

13


observed images/volumes

14

Unary (data) cost (inverted)


observed images/volumes• Enforcing spatial coherence.

15

Pairwise (boundary) cost (inverted)


observed images/volumes• Enforcing spatial coherence.

16

Pairwise (boundary) cost (inverted)

p q

Segmentation as an Energy Minimization Problem• Edata assigns non-negative penalties to a pixel location p

when assigning a label to this location.

10/1/15

17


when assigning a label to this location.• Esmooth assigns non-negative penalties by comparing the

assigned labels fp and fq at adjacent positions p and q.

18




19

Node/vertexEdge

pixel

p q

Image as a graphrepresentation




20

This optimization model is characterized by local interactions along edges between adjacent pixels, and often called MRF (Markov Random Field) model.

Energy Function-Details

21

E(f) =X

p2⌦

[Edata

(p, fp

) +X

q2A(p)

Esmooth

(fp

, fq

)]


22

E(f) =X

p2⌦

[Edata

(p, fp

) +X

q2A(p)

Esmooth

(fp

, fq

)]

Edata(p, fp) = (p)

(p = 0) = �logP (p 2 BG)

(p = 1) = �logP (p 2 FG)

Example Data Term:


23

E(f) =X

p2⌦

[Edata

(p, fp

) +X

q2A(p)

Esmooth

(fp

, fq

)]

Edata(p, fp) = (p)

(p = 0) = �logP (p 2 BG)

(p = 1) = �logP (p 2 FG)

Example Data Term:

(p, q) = Kpq�(p 6= q) where

Kpq =exp(��(Ip � Iq)2/(2�2))

||p, q||

Example Smoothness Term:


24

E(f) =X

p2⌦

[Edata

(p, fp

) +X

q2A(p)

Esmooth

(fp

, fq

)]

E(p) =X

p2⌦

p(p) +X

p2⌦

X

q2A(p)

pq(p, q)

p⇤ = argminp2L

E(p)


25

E(f) =X

p2⌦

[Edata

(p, fp

) +X

q2A(p)

Esmooth

(fp

, fq

)]

E(p) =X

p2⌦

p(p) +X

p2⌦

X

q2A(p)

pq(p, q)

p⇤ = argminp2L

E(p)

To solve this problem, transform the energy functional into min-cut/max-flow problem and solve it!

Graph Cuts for Optimal Boundary Detection (Boykov ICCV 2001)

26

26

F

B

F

B

F

F F

F B

B

B

• Binary label: foreground vs. background• User labels some pixels • Exploit

– Statistics of known Fg & Bg– Smoothness of label

• Turn into discrete graph optimization– Graph cut (min cut / max flow)

Graph Cut27

Each pixel is connected to its neighbors in an undirected graphGoal: split nodes into two sets A and B based on pixel values, and try to classify Neighbors in the same way too!

Graph Cut28

Each pixel is connected to its neighbors in an undirected graphGoal: split nodes into two sets A and B based on pixel values, and try to classify Neighbors in the same way too!

Graph-Cut29

• Each pixel = node• Add two nodes F & B• Labeling: link each pixel to either F or B

F

B

F

B

F

F F

F B

B

B

Desired result

Cost Function: Data term

30

• Put one edge between each pixel and both F & G• Weight of edge = minus data term

B

F

Cost Function: Smoothness term31

• Add an edge between each neighbor pair• Weight = smoothness term

B

F

Min-Cut32

• Energy optimization equivalent to graph min cut• Cut: remove edges to disconnect F from B• Minimum: minimize sum of cut edge weight

B

Fcut

Min-Cut33

Source

Sink

v1 v2

2

5

9

42

1Graph (V, E, C)

Vertices V = {v1, v2 ... vn}Edges E = {(v1, v2) ....}Costs C = {c(1, 2) ....}

Min-Cut34

Source

Sink

v1 v2

2

5

9

42

1

What is a st-cut?

An st-cut (S,T) divides the nodes between source and sink.

What is the cost of a st-cut?

Sum of cost of all edges going from S to T

5 + 2 + 9 = 16

Min-Cut35

What is a st-cut?

An st-cut (S,T) divides the nodes between source and sink.

What is the cost of a st-cut?

Sum of cost of all edges going from S to T

Source

Sink

v1 v2

2

5

9

42

1

2 + 1 + 4 = 7

What is the st-mincut?

st-cut with the minimum cost

How to compute min-cut?36

Source

Sink

v1 v2

2

5

9

42

1

Solve the dual maximum flow problem

In every network, the maximum flow equals the cost of the st-mincut

Min-cut\Max-flow Theorem

Compute the maximum flow between Source and Sink

Constraints

Edges: Flow < Capacity

Nodes: Flow in & Flow out

Max-Flow Algorithms

37

Augmenting Path Based Algorithms

1. Find path from source to sink with positive capacity

2. Push maximum possible flow through this path

3. Repeat until no path can be found

Source

Sink

v1 v2

2

5

9

42

1

Algorithms assume non-negative capacity

Flow=0

Max-Flow Algorithms38






Source

Sink

v1 v2

2

5

9

42

1

Flow=0

Max-Flow Algorithms

39






Source

Sink

v1 v2

2-2

5-2

9

42

1

Flow=0+2

Max-Flow Algorithms

40






Source

Sink

v1 v2

0

3

9

42

1

Flow=2

Max-Flow Algorithms

41






Source

Sink

v1 v2

0

3

9

42

1

Flow=2

Max-Flow Algorithms

42






Source

Sink

v1 v2

0

3

9

42

1

Flow=2

Max-Flow Algorithms

43






Source

Sink

v1 v2

0

3

5

02

1

Flow=2+4

Max-Flow Algorithms

44






Source

Sink

v1 v2

0

3

5

02

1

Flow=6

Max-Flow Algorithms

45






Source

Sink

v1 v2

0

3

5

02

1

Flow=6

Max-Flow Algorithms

46






Source

Sink

v1 v2

0

2

4

02+1

1-1

Flow=6+1

Max-Flow Algorithms

47






Source

Sink

v1 v2

0

2

4

03

0

Flow=7

Max-Flow Algorithms

48






Source

Sink

v1 v2

0

2

4

03

0

Flow=7

Another Example-Max Flow

49

source

sink

9

5

6

8

42

2

2

5

3

5

5

311

6

2

3

Ford & Fulkerson algorithm (1956)

Find the path from source to sink

While (path exists)

flow += maximum capacity in the path

Build the residual graph (“subtract” the flow)

Find the path in the residual graph

End


50

source

sink

9

5

6

8

42

2

2

5

3

5

5

311

6

2

3



While (path exists)




End


51

source

sink

9

5

6

8

42

2

2

5

3

5

5

311

6

2

3



While (path exists)




End

flow = 3


52

source

sink

9

5

6-3

8-3

42+3

2

2

5-3

3-3

5

5

311

6

2

3



While (path exists)




End

flow = 3

+3


53

source

sink

9

5

3

5

45

2

2

2

5

5

311

6

2

3



While (path exists)




End

flow = 3

3


54

source

sink

9

5

3

5

45

2

2

2

5

5

311

6

2

3



While (path exists)




End

flow = 3

3


55

source

sink

9

5

3

5

45

2

2

2

5

5

311

6

2

3



While (path exists)




End

flow = 6

3


56

source

sink

9-3

5

3

5-3

45

2

2

2

5

5

3-311

6

2

3-3



While (path exists)




End

flow = 6

3

+3

+3


57

source

sink

6

5

3

2

45

2

2

2

5

5

11

6

2



While (path exists)




End

flow = 6

3

3

3


58

source

sink

6

5

3

2

45

2

2

2

5

5

11

6

2



While (path exists)




End

flow = 6

3

3

3


59

source

sink

6

5

3

2

45

2

2

2

5

5

11

6

2



While (path exists)




End

flow = 11

3

3

3


60

source

sink

6-5

5-5

3

2

45

2

2

2

5-5

5-5

1+51

6

2+5



While (path exists)




End

flow = 11

3

3

3


61

source

sink

1

3

2

45

2

2

2

61

6

7



While (path exists)




End

flow = 11

3

33


62

source

sink

1

3

2

45

2

2

2

61

6

7



While (path exists)




End

flow = 11

3

33


63

source

sink

1

3

2

45

2

2

2

61

6

7



While (path exists)




End

flow = 13

3

33


64

source

sink

1

3

2-2

45

2-2

2-2

2-2

61

6

7



While (path exists)




End

flow = 13

3

33

+2

+2


65

source

sink

1

3

45

61

6

7



While (path exists)




End

flow = 13

3

33

2

2


66

source

sink

1

3

45

61

6

7



While (path exists)




End

flow = 13

3

33

2

2


67

source

sink

1

3

45

61

6

7



While (path exists)




End

flow = 15

3

33

2

2


68

source

sink

1

3-2

4-25

61

6-2

7



While (path exists)




End

flow = 15

3

33+2

2

2-2

+2


69

source

sink

1

1

5

61

4

7



While (path exists)




End

flow = 15

3

35

2

2

2


70

source

sink

1

1

5

61

4

7



While (path exists)




End

flow = 15

3

35

2

2

2


71

source

sink

1

1

5

61

4

7



While (path exists)




End

flow = 15

3

35

2

2

2


72

source

sink

1

1

5

61

4

7


Why is the solution globally optimal ?

flow = 15

3

35

2

2

2


73

source

sink

1

1

5

61

4

7



1. Let S be the set of reachable nodes in the

residual graph

flow = 15

3

35

2

2

2

S


74



1. Let S be the set of reachable nodes in the

residual graph

2. The flow from S to V - S equals to the sum

of capacities from S to V – S

source

sink

9

5

6

8

42

2

2

5

3

5

5

311

6

2

3

S


75

flow = 15



1. Let S be the set of reachable nodes in the residual graph

2. The flow from S to V - S equals to the sum of capacities from S to V – S

3. The flow from any A to V - A is upper bounded by the sum of capacities from A to V – A

4. The solution is globally optimal

source

sink

8/9

5/5

5/6

8/8

2/40/2

2/2

0/2

5/5

3/3

5/5

5/5

3/30/10/1

2/6

0/2

3/3

Individual flows obtained by summing up all paths


76

source

sink

9

5

6

8

42

2

2

5

3

5

5

311

6

2

3

S

T

cost = 18

source

sink

9

5

6

8

42

2

2

5

3

5

5

311

6

2

3

S

T


77

source

sink

9

5

6

8

42

2

2

5

3

5

5

311

6

2

3

S

T

cost = 23


78

C(x) = 5x1 + 9x2 + 4x3 + 3x3(1-x1) + 2x1(1-x3)

+ 3x3(1-x1) + 2x2(1-x3) + 5x3(1-x2) + 2x4(1-x1)

+ 1x5(1-x1) + 6x5(1-x3) + 5x6(1-x3) + 1x3(1-x6)

+ 3x6(1-x2) + 2x4(1-x5) + 3x6(1-x5) + 6(1-x4)

+ 8(1-x5) + 5(1-x6)

source

sink

9

5

6

8

42

2

2

5

3

5

5

311

6

2

3

x1 x2

x3

x4

x5

x6


79

C(x) = 2x1 + 9x2 + 4x3 + 2x1(1-x3)

+ 3x3(1-x1) + 2x2(1-x3) + 5x3(1-x2) + 2x4(1-x1)

+ 1x5(1-x1) + 3x5(1-x3) + 5x6(1-x3) + 1x3(1-x6)

+ 3x6(1-x2) + 2x4(1-x5) + 3x6(1-x5) + 6(1-x4)

+ 5(1-x5) + 5(1-x6)

+ 3x1 + 3x3(1-x1) + 3x5(1-x3) + 3(1-x5)

source

sink

9

5

6

8

42

2

2

5

3

5

5

311

6

2

3

x1 x2

x3

x4

x5

x6


80

C(x) = 2x1 + 9x2 + 4x3 + 2x1(1-x3)

+ 3x3(1-x1) + 2x2(1-x3) + 5x3(1-x2) + 2x4(1-x1)

+ 1x5(1-x1) + 3x5(1-x3) + 5x6(1-x3) + 1x3(1-x6)

+ 3x6(1-x2) + 2x4(1-x5) + 3x6(1-x5) + 6(1-x4)

+ 5(1-x5) + 5(1-x6)

+ 3x1 + 3x3(1-x1) + 3x5(1-x3) + 3(1-x5)

3x1 + 3x3(1-x1) + 3x5(1-x3) + 3(1-x5)

=

3 + 3x1(1-x3) + 3x3(1-x5)

source

sink

9

5

6

8

42

2

2

5

3

5

5

311

6

2

3

x1 x2

x3

x4

x5

x6


81

C(x) = 3 + 2x1 + 6x2 + 4x3 + 5x1(1-x3)

+ 3x3(1-x1) + 2x2(1-x3) + 5x3(1-x2) + 2x4(1-x1)

+ 1x5(1-x1) + 3x5(1-x3) + 5x6(1-x3) + 1x3(1-x6)

+ 2x5(1-x4) + 6(1-x4)

+ 2(1-x5) + 5(1-x6) + 3x3(1-x5)

+ 3x2 + 3x6(1-x2) + 3x5(1-x6) + 3(1-x5)

3x2 + 3x6(1-x2) + 3x5(1-x6) + 3(1-x5)

=

3 + 3x2(1-x6) + 3x6(1-x5)

source

sink

9

5

3

5

45

2

2

2

5

5

311

6

2

33

x1 x2

x3

x4

x5

x6

82

C(x) = 6 + 2x1 + 6x2 + 4x3 + 5x1(1-x3)

+ 3x3(1-x1) + 2x2(1-x3) + 5x3(1-x2) + 2x4(1-x1)

+ 1x5(1-x1) + 3x5(1-x3) + 5x6(1-x3) + 1x3(1-x6)

+ 2x5(1-x4) + 6(1-x4) + 3x2(1-x6) + 3x6(1-x5)

+ 2(1-x5) + 5(1-x6) + 3x3(1-x5)

source

sink

6

5

3

2

45

2

2

2

5

5

11

6

2

3

3

3

x1 x2

x3

x4

x5

x6


Another Example-Max Flow83

C(x) = 15 + 1x2 + 4x3 + 5x1(1-x3)

+ 3x3(1-x1) + 7x2(1-x3) + 2x1(1-x4)

+ 1x5(1-x1) + 6x3(1-x6) + 6x3(1-x5)

+ 2x5(1-x4) + 4(1-x4) + 3x2(1-x6) + 3x6(1-x5)

source

sink

1

1

5

61

4

7

3

35

2

2

2x1 x2

x3

x4

x5

x6 cost = 0

min cut

S

T

cost = 0

• All coefficients positive

• Must be global minimum

S – set of reachable nodes from s

History of Max-Flow Algorithms84

Augmenting Path and Push-Relabeln: #nodesm: #edgesU: maximum edge weight

Algorithms assume non-negative edge

weights

Software Packages for Optimization

85

Applications Used in Energy Minimization Based Segmentation Methods

86

Applications

87

Interactive Organ Segmentation (Boykov and Jolly, MICCAI 2000)

88

Segmentation of multiple objects. (a-c): Cardiac MRI. (d): Kidney CE-MR angiography

Interactive Organ Segmentation (Boykov and Jolly, MICCAI 2000)

89

Segmentation of the right lung in CT. (a): representative 2D slice of original 3D data. (b): segmentation results on the slice in (a). (c-d) 3D visualization of segmentation results.

AUTOMATIC HEART ISOLATION FOR CT CORONARY VISUALIZATION USING GRAPH-CUTS (Funka-Lea et al, ISBI 2006)

• Isolating the entire heart allows the coronary vessels on the surface of the heart to be easily visualized despite the proximity of surrounding organs such as the ribs and pulmonary blood vessels.

90



• Numerous techniques have been described for segmenting the left ventricle of the heart in images from various types of medical scanners but rarely has the entire heart been segmented.

91




• Graph-cut formation:

92





93





94

(C is center)





95

If cos(.) <0, (0 otherwise.)

(C is center)


96

Top Left:A balloon is expanded within the heart. The heart wall pushes the balloon toward the heart center as the balloon grows.

Top right: volume rendering of original heart volume.

Bottom left: heart cropped based on segmentation mask.

Bottom right:volume rendering after automatic heart isolation algorithm.

Measurement of hippocampal atrophy using 4D graph-cut segmentation: Application to ADNI (Wolz, et al. NeuroImage 10)

• Proposed a method for simultaneously segmenting longitudinal magnetic resonance (MR) images

97


• Proposed a method for simultaneously segmenting longitudinal magnetic resonance (MR) images– 3D MRI + time component (longitudinal)

98



• A 4D graph is used to represent the longitudinal data: – edges are weighted based on spatial and intensity priors and connect

spatially and temporally neighboring voxels represented by vertices in the graph.

99





– Solving the min-cut/max-flow problem on this graph yields the segmentation for all time-points in a single step

100






• Time-series image can be considered as a single (4D) image, with the following dimensions: x,y,z and t

101






• Time-series image can be considered as a single (4D) image, with the following dimensions: x,y,z and t

102

6 spatial neighborsin 3DAnd 2 temporal neighbors

and


103

a. Right hippocampus segmentation (baseline), b. follow up segmentation (12 months)

Integrated Graph Cuts for Brain Image Segmentation (Song et al, MICCAI 2006)

• In addition to image intensity, tissue priors and local boundary information are integrated into the edge weight metrics in the graph.

104


• In addition to image intensity, tissue priors and local boundary information are integrated into the edge weight metrics in the graph.

105

Example of the graph with three terminals for brain MRI tissue segmentation of gray matter (GM), white matter (WM), and cerebrospinal fluid (CSF). The set of nodes V includes all voxels and terminals. The set of edges E includes all n-links and t-links.

n-links: voxel-to-voxel edgest-links: voxel-to-terminal edges


106


107

Pairwise termAtlas termData termRegularizationterm


108

Atlas termData termRegularizationterm

(T2, segmentation results)

GC + Appearance Model109

GC + Appearance Model110

GC + Shape Model

111

• Regions used for calculating Dice score, sensitivity, specificity, and robust Hausdorff score. Region T1 is the true lesion area (outline blue), T0 is the remaining normal area. P1 is the area that is predicted to be lesion by—for example—an algorithm (outlined red), and P0 is predicted to be normal. P1 has some overlap with T1 in the right lateral part of the lesion, corresponding to the area referred to as P1Λ T1 in the definition of the Dice score. (Credits: BRATS paper)

112

Summary– Data and Smoothness Terms -> Graph based segmentation methods– Additional terms can(should) be added into segmentation formulation

based observation/need and problem definition– Problems formulated as a MRF task can be solved by max-flow/min-

cut

113

Slide Credits and References• Fredo Durand• M. Tappen• R. Szelisky• http://www.csd.uwo.ca/faculty/yuri/Abstracts/eccv06-tutorial.html• J.Malcolm, Graph Cut in Tensor Scale• Interactive Graph Cuts for Optimal Boundary & Region Segmentation of

Objects in N-D images.Yuri Boykov and Marie-Pierre Jolly.In International Conference on Computer Vision, (ICCV), vol. I, 2001.http://www.csd.uwo.ca/~yuri/Abstracts/iccv01-abs.html

• http://www.cse.yorku.ca/~aaw/Wang/MaxFlowStart.htm• http://research.microsoft.com/vision/cambridge/i3l/segmentation/GrabCut.htm• http://www.cc.gatech.edu/cpl/projects/graphcuttextures/• A Comparative Study of Energy Minimization Methods for Markov Random

Fields. Rick Szeliski, Ramin Zabih, Daniel Scharstein, Olga Veksler, Vladimir Kolmogorov, Aseem Agarwala, Marshall Tappen, Carsten Rother. ECCV 2006www.cs.cornell.edu/~rdz/Papers/SZSVKATR.pdf

• P. Kumar, Oxford University.

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Lec10: Medical Image Segmentation as an Energy Minimization Problem

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