Medical Image Segmentation: Beyond Level Sets (Ismail’s part) - PowerPoint Presentation

Slide1

Medical Image Segmentation: Beyond Level Sets (Ismail’s part)

1Slide2

Basics of Level Sets (Ismail)

14Slide3

Active Curves

1Slide4

Active Curves

1

S

SSlide5

Gradient Descent (1)

2

Functional

derivative

Regional terms of the form <

S

,

f

> Slide6

Gradient Descent (2)

3



E



S

-

S =



t



E



SSlide7

Gradient Descent (2)

3



E



S

-



E



S

-

S =



t



E



SSlide8

Gradient Descent (2)

3



E



S

-



E



S

-

S =



t



E



SSlide9

Standard boundary terms:

Geodesic Active

Contours

4

e.g.,

Caselles

et al., 97Slide10

5

Standard boundary terms:

General derivative with

E-L equations Slide11

5

1

Boundary length

Standard boundary terms:

General derivative with

E-L equations Slide12

5

Standard boundary terms:

General derivative with

E-L equations

Depends on image gradient

Attracts curve to strong edges Slide13

5

Standard boundary terms:

General derivative with

E-L equations

Depends on image gradient

Attracts curve to strong edges Slide14

6

Standard region terms

:

P

iecewise constant case

e.g., Chan and et

Vese

, 01Slide15

6

Standard region terms

:

P

iecewise constant case

e.g., Chan and et

Vese

, 01

Alternate minimization

(1) Fix parameters and evolve the curve

(2) Fix curve, optimize w.r.t parameters

Slide16

7

Standard region terms:

Log-Likelihood

Slide17

7

Standard region terms:

Log-Likelihood

Distributions fixed by prior learning

e.g.,

Paragios

and

Dercihe

, 02Slide18

7

Standard region terms:

Log-Likelihood

Distributions updated iteratively

e.g., Gaussian

:

Rousson

and

Deriche

02

Gamma: Ben Ayed et al., 05Slide19

Functional derivatives for region terms

(E-L equations and Green’s theorem)

8

See Zhu and

Yuille

, 96

Mitiche

and Ben Ayed, 11Slide20

8

Curve flow in the log-likelihood case

S

Functional derivatives for region terms

(E-L equations and Green’s theorem)

See Zhu and

Yuille

, 96

Mitiche

and Ben Ayed, 11Slide21

8

>0

Curve flow in the log-likelihood case

S

Functional derivatives for region terms

(E-L equations and Green’s theorem)Slide22

8

<

0

Curve flow in the log-likelihood case

S

Functional derivatives for region terms

(E-L equations and Green’s theorem)Slide23

Level set representation of the curve

9Slide24

Level set representation of the curve

9

We can replace everything Slide25

Level set representation of the curve

9

Easy to show from

the facts that on the curve:

See

Mitiche

and Ben Ayed, 11Slide26

Alternatively, we can embed the level set function in the energy directly

10

Region terms:

e.g., Chan and

Vese

, 01

Li et al., 2005Slide27

Alternatively, we can embed the level set function in the energy directly

10

L

ength term:

e.g., Chan and

Vese

, 01

Li et al., 2005Slide28

Alternatively, we can embed the level set function in the energy directly

10

e.g., Chan and

Vese

, 01

Li et al., 2005

Region terms:

L

ength term:

Compute E-L equations directly w.r.t the level set

function Slide29

Pros of level

sets (1)

11

Applicable to any differentiable functional: Slide30

Pros of level

sets (1)

11

Applicable to any differentiable functional: Slide31

Pros of level

sets (2)

12

Direct extension to higher dimensionsSlide32

Small moves + Fixed and small time step

+ Can be slow in practice:

Cons of level

sets (1)

13

S =



t



E



S

Courant-

Friedrichs

-

Lewy

(

CFL)

conditions for

evolution

stability



t

<

cst

See, for example,

Estellers

et al.,

IEEE TIP 12Slide33

Sometimes very weak local optima

Cons of level

sets (2)

14

= 0



E



SSlide34

Dependence on the choice of an

approximate numerical scheme (for stable evolution)

Cons of level

sets (3)

15

e.g., Complex upwind schemes for PDE discretization

See, for example,

Sethian

99

See, for example,

S. Osher and R. Fedkiw 2002

Keep a distance function by

ad hoc re-initialization procedures