medical image processing with orientation scores erik franken*,...

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1 1 Medical Image Processing with Orientation Scores Erik Franken*, Remco Duits, Markus van Almsick, Bart ter Haar Romeny *E-mail: [email protected] Eindhoven University of Technology Department of Biomedical Engineering Mathematica Technology Conference 2006 October 13th 2006, Champaign

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Page 1: Medical Image Processing with Orientation Scores Erik Franken*, …library.wolfram.com/infocenter/Conferences/6473/Image... · 1 Medical Image Processing with Orientation Scores Erik

11

Med

ical Im

ag

e P

rocessin

g

wit

h O

rien

tati

on

Sco

res

Erik

Fra

nken

*, R

emco

Dui

ts,

Mar

kus

van

Alm

sick

,B

art t

erH

aar

Rom

eny

*E-m

ail:

e.m

.fran

ken@

tue.

nl

Ein

dhov

en U

nive

rsity

of T

echn

olog

yD

epar

tmen

t of B

iom

edic

al E

ngin

eerin

g

Mat

hem

atic

a T

echn

olog

yC

onfe

renc

e 20

06O

ctob

er13

th 2

006,

Cha

mpa

ign

Page 2: Medical Image Processing with Orientation Scores Erik Franken*, …library.wolfram.com/infocenter/Conferences/6473/Image... · 1 Medical Image Processing with Orientation Scores Erik

22

Ou

tlin

e

•A

bo

ut

ou

rR

es

ea

rch

Gro

up

•O

rient

atio

nS

core

s•

Diff

usio

nin

Orie

ntat

ion

Sco

res

•S

toch

astic

Com

plet

ion

Fie

lds

•U

sing

Mat

hem

atic

a

Page 3: Medical Image Processing with Orientation Scores Erik Franken*, …library.wolfram.com/infocenter/Conferences/6473/Image... · 1 Medical Image Processing with Orientation Scores Erik

33

Th

e B

iom

ed

icalIm

ag

e A

naly

sis

gro

up

Page 4: Medical Image Processing with Orientation Scores Erik Franken*, …library.wolfram.com/infocenter/Conferences/6473/Image... · 1 Medical Image Processing with Orientation Scores Erik

44

Sta

rrin

g:

Pet

ered

a(P

ilsen

, Cze

chR

epub

lic)

Tim

Pee

ters

(E

cht,

The

Net

herla

nds)

Math

em

ati

ca in

th

e B

ioM

IMla

b

Page 5: Medical Image Processing with Orientation Scores Erik Franken*, …library.wolfram.com/infocenter/Conferences/6473/Image... · 1 Medical Image Processing with Orientation Scores Erik

55

Ou

rM

ath

em

ati

ca In

frastr

uctu

re

•F

ullc

ampu

s lic

ense

for

Mat

hem

atic

a•

The

nee

dfo

r“b

igm

ath”

kern

else

rver

s–

Big

mat

h1: T

yan

TX

46, 4

x O

pter

on2.

2Ghz

, 32G

B–

Big

mat

h2: T

yan

VX

50, 4

x D

ualc

ore

Opt

eron

2.2G

hz, 6

4GB

–+

3 o

lder

serv

ers

•U

seof

Par

alle

lMat

hem

atic

a

Page 6: Medical Image Processing with Orientation Scores Erik Franken*, …library.wolfram.com/infocenter/Conferences/6473/Image... · 1 Medical Image Processing with Orientation Scores Erik

66

Ou

rM

ath

em

ati

ca In

frastr

uctu

re

•F

ullc

ampu

s lic

ense

for

Mat

hem

atic

a•

The

nee

dfo

r“b

igm

ath”

kern

else

rver

s–

Big

mat

h1: T

yan

TX

46, 4

x O

pter

on2.

2Ghz

, 32G

B–

Big

mat

h2: T

yan

VX

50, 4

x D

ualc

ore

Opt

eron

2.2G

hz, 6

4GB

–+

3 o

lder

serv

ers

•U

seof

Par

alle

lMat

hem

atic

a

Page 7: Medical Image Processing with Orientation Scores Erik Franken*, …library.wolfram.com/infocenter/Conferences/6473/Image... · 1 Medical Image Processing with Orientation Scores Erik

77

Math

Vis

ion

To

ols

Com

pute

r V

isio

nLi

brar

yfo

rM

athe

mat

ica:

•G

auss

ian

deriv

ativ

es•

Geo

met

rydr

iven

diffu

sion

•O

rient

atio

nsc

ore

func

tions

•Im

age

tran

sfor

mat

ions

•D

ICO

M im

port

/exp

ort

ww

w.m

ath

vis

ion

too

ls.n

et

Page 8: Medical Image Processing with Orientation Scores Erik Franken*, …library.wolfram.com/infocenter/Conferences/6473/Image... · 1 Medical Image Processing with Orientation Scores Erik

88

1

2

3

4

5

6

7

8

9

10

11

Math

em

ati

ca in

Bach

elo

r E

du

cati

on

Imag

e A

naly

sis

for

Pat

holo

gy.

•G

roup

sof

8 2

nd y

ear

stud

ents

•“I

nven

t”im

age

anal

ysis

algo

rithm

sin

M

ath

em

atica

•C

ompe

titiv

eel

emen

t•

6 w

eeks

pro

ject

Page 9: Medical Image Processing with Orientation Scores Erik Franken*, …library.wolfram.com/infocenter/Conferences/6473/Image... · 1 Medical Image Processing with Orientation Scores Erik

99

1stor

der

(edg

es)

2ndor

der

(rid

ges)

3rdor

der

(T-ju

nctio

ns)

For

exa

mpl

eR

otat

ion

inva

riant

T-ju

nctio

n de

tect

ion:

Exam

ple

: D

iffe

ren

tialin

vari

an

ts

Page 10: Medical Image Processing with Orientation Scores Erik Franken*, …library.wolfram.com/infocenter/Conferences/6473/Image... · 1 Medical Image Processing with Orientation Scores Erik

1010

Ou

tlin

e

•A

bout

our

Res

earc

h gr

oup

•O

rie

nta

tio

nS

co

res

•D

iffus

ion

in O

rient

atio

nS

core

s•

Sto

chas

ticC

ompl

etio

nF

ield

s•

Usi

ngM

athe

mat

ica

Page 11: Medical Image Processing with Orientation Scores Erik Franken*, …library.wolfram.com/infocenter/Conferences/6473/Image... · 1 Medical Image Processing with Orientation Scores Erik

1111

1.T

he r

etin

a co

ntai

nsre

cept

ive

field

sof

var

ying

size

s�

multi-sca

lesa

mpl

ing

devi

ce2.

Prim

ary

visu

alco

rtex

is m

ulti-ori

enta

tion

Bio

log

ical In

sp

irati

on

•C

ells

in th

e pr

imar

y vi

sual

cor

tex

are

orie

ntat

ion-

spec

ific

•S

tron

g co

nnec

tivity

be

twee

n ce

lls th

at

resp

ond

to (

near

ly)

the

sam

e or

ient

atio

n

Mea

sure

men

t in

Prim

ary

Vis

ual C

orte

x

Bos

king

et a

l., J

. Neu

rosc

ienc

e 17

:211

2-21

27, 1

997

Page 12: Medical Image Processing with Orientation Scores Erik Franken*, …library.wolfram.com/infocenter/Conferences/6473/Image... · 1 Medical Image Processing with Orientation Scores Erik

1212

Ori

en

tati

on

Sco

res

Fro

m2D

imag

e f(

x,y

) to

orie

ntat

ion

scor

e U

f(x,y

,θ)

with

pos

ition

(x,y

) an

d or

ient

atio

n θ

An

orie

ntat

ion

scor

e is

a fu

nctio

non

the

Eu

clid

ea

nm

otio

ng

rou

p

Page 13: Medical Image Processing with Orientation Scores Erik Franken*, …library.wolfram.com/infocenter/Conferences/6473/Image... · 1 Medical Image Processing with Orientation Scores Erik

1313

Ou

rap

pro

ach

: Im

ag

e P

rocessin

g v

ia

Ori

en

tati

on

Sco

res

“Enh

ance

men

t”op

erat

ion

Initi

alim

age

“Enh

ance

d”im

age

Orie

ntat

ion

scor

e tr

ansf

orm

atio

n

Inve

rse

orie

ntat

ion

scor

e tr

ansf

orm

atio

n

Seg

men

ted

stru

ctur

es

Seg

men

t str

uctu

res

of

inte

rest

Page 14: Medical Image Processing with Orientation Scores Erik Franken*, …library.wolfram.com/infocenter/Conferences/6473/Image... · 1 Medical Image Processing with Orientation Scores Erik

1414

Invert

ible

Ori

en

tati

on

Sco

re T

ran

sfo

rmati

on

Des

ign

cons

ider

atio

ns: r

econ

stru

ctio

n,

dire

ctio

nal,

spat

iall

ocal

izat

ion,

qua

drat

ure

Page 15: Medical Image Processing with Orientation Scores Erik Franken*, …library.wolfram.com/infocenter/Conferences/6473/Image... · 1 Medical Image Processing with Orientation Scores Erik

1515

Ou

tlin

e

•A

bout

our

Res

earc

h gr

oup

•O

rient

atio

nS

core

s•

Dif

fus

ion

in O

rie

nta

tio

nS

co

res

•S

toch

astic

Com

plet

ion

Fie

lds

•U

sing

Mat

hem

atic

a

Page 16: Medical Image Processing with Orientation Scores Erik Franken*, …library.wolfram.com/infocenter/Conferences/6473/Image... · 1 Medical Image Processing with Orientation Scores Erik

1616

Th

e D

iffu

sio

nE

qu

ati

on

on

Imag

es

f=

imag

eu

= s

cale

spac

eof

imag

eD

= d

iffus

ion

tens

or

Line

ardi

ffusi

onP

eron

a&M

alik

Coh

eren

ce-e

nhan

cing

diff.

t=

0

Page 17: Medical Image Processing with Orientation Scores Erik Franken*, …library.wolfram.com/infocenter/Conferences/6473/Image... · 1 Medical Image Processing with Orientation Scores Erik

1717

Th

e D

iffu

sio

nE

qu

ati

on

on

Imag

es

f=

imag

eu

= s

cale

spac

eof

imag

eD

= d

iffus

ion

tens

or

Line

ardi

ffusi

onP

eron

a&M

alik

Coh

eren

ce-e

nhan

cing

diff.

t=

10

Page 18: Medical Image Processing with Orientation Scores Erik Franken*, …library.wolfram.com/infocenter/Conferences/6473/Image... · 1 Medical Image Processing with Orientation Scores Erik

1818

Dif

fusio

nin

ori

en

tati

on

sco

res

curv

atur

e

Diff

usio

nor

thog

onal

toor

ient

edst

ruct

ures

Diff

usio

nta

ngen

t to

orie

nted

stru

ctur

esD

iffus

ion

inor

ient

atio

n

Evo

lvin

gor

ient

atio

nsc

ore

Rot

atin

gta

ngen

t spa

ceco

ordi

nate

basi

s

Left-

inva

riant

deriv

ativ

es

are

left-

inva

riant

deriv

ativ

eson

Euc

lidea

nm

otio

ngr

oup,

i.e.

∂ξ

∂η

∂θ

x

θ

y

∂ξ

∂η

∂θ

∂ξ,∂η,and∂θ

Lg∂iU=∂iLgU,i∈{ξ,η,θ}

Page 19: Medical Image Processing with Orientation Scores Erik Franken*, …library.wolfram.com/infocenter/Conferences/6473/Image... · 1 Medical Image Processing with Orientation Scores Erik

1919

Exam

ple

dif

fusio

nkern

els

Page 20: Medical Image Processing with Orientation Scores Erik Franken*, …library.wolfram.com/infocenter/Conferences/6473/Image... · 1 Medical Image Processing with Orientation Scores Erik

2020

•O

rient

edre

gion

s: D

’ 11

and

D33

smal

l, D

22la

rge

and κ

acco

rdin

gto

estim

ate

•N

on-o

rient

edre

gion

s: D

’ 11

larg

e,

D22

=D

33la

rge,

κ=

0∂ξ

∂η

∂θ

x

θ

y

∂ξ

∂η

∂θ

Ho

wto

Ch

oo

se

Co

nd

ucti

vit

yC

oeff

icie

nts

Page 21: Medical Image Processing with Orientation Scores Erik Franken*, …library.wolfram.com/infocenter/Conferences/6473/Image... · 1 Medical Image Processing with Orientation Scores Erik

2121

Diff

usio

nin

orie

ntat

ion

scor

eC

oher

ence

enha

ncin

gdi

ffusi

onR

esu

lts

Siz

e: 1

28 x

128

x 6

4

t=

0

Page 22: Medical Image Processing with Orientation Scores Erik Franken*, …library.wolfram.com/infocenter/Conferences/6473/Image... · 1 Medical Image Processing with Orientation Scores Erik

2222

Diff

usio

nin

orie

ntat

ion

scor

eC

oher

ence

enha

ncin

gdi

ffusi

onR

esu

lts

Siz

e: 1

28 x

128

x 6

4

t=

10

Page 23: Medical Image Processing with Orientation Scores Erik Franken*, …library.wolfram.com/infocenter/Conferences/6473/Image... · 1 Medical Image Processing with Orientation Scores Erik

2323

Co

llag

en

imag

e

Diff

usio

nin

orie

ntat

ion

scor

eC

oher

ence

enha

ncin

gdi

ffusi

on

Siz

e: 2

00 x

200

x 6

4

t=

0

Page 24: Medical Image Processing with Orientation Scores Erik Franken*, …library.wolfram.com/infocenter/Conferences/6473/Image... · 1 Medical Image Processing with Orientation Scores Erik

2424

Co

llag

en

imag

e

Diff

usio

nin

orie

ntat

ion

scor

eC

oher

ence

enha

ncin

gdi

ffusi

on

Siz

e: 2

00 x

200

x 6

4

t=

30

Page 25: Medical Image Processing with Orientation Scores Erik Franken*, …library.wolfram.com/infocenter/Conferences/6473/Image... · 1 Medical Image Processing with Orientation Scores Erik

2525

Ou

tlin

e

•A

bout

our

Res

earc

h gr

oup

•O

rient

atio

nS

core

s•

Diff

usio

nin

Orie

ntat

ion

Sco

res

•S

toc

ha

sti

cC

om

ple

tio

nF

ield

s

•U

sing

Mat

hem

atic

a

Page 26: Medical Image Processing with Orientation Scores Erik Franken*, …library.wolfram.com/infocenter/Conferences/6473/Image... · 1 Medical Image Processing with Orientation Scores Erik

2626

Oth

er

PD

E:

the s

toch

asti

cco

mp

leti

on

field

Res

olve

ntof

line

arP

DE

Itre

nder

spr

obab

ility

dens

ityfie

ld fo

rlin

eco

ntin

uatio

nba

sed

onra

ndom

wal

ker

prio

r

Page 27: Medical Image Processing with Orientation Scores Erik Franken*, …library.wolfram.com/infocenter/Conferences/6473/Image... · 1 Medical Image Processing with Orientation Scores Erik

2727

Co

nvo

luti

on

on

Ori

en

tati

on

Sco

res

Nor

mal

conv

olut

ion

(on

tran

slat

ion

grou

p)

G-c

onvo

lutio

n, w

here

G is

the

Euc

lidea

nm

otio

ngr

oup

An

imag

e is

a fu

nctio

non

the

tran

slat

ion

grou

p

An

orie

ntat

ion

scor

e is

a fu

nctio

non

the

Euc

lidea

nm

otio

ngr

oup

Page 28: Medical Image Processing with Orientation Scores Erik Franken*, …library.wolfram.com/infocenter/Conferences/6473/Image... · 1 Medical Image Processing with Orientation Scores Erik

2828

(Fro

mM

.Sc.

thes

is b

yR

ensk

e de

Boe

r)

Fillin

gG

ap

sin

Cu

rves

Page 29: Medical Image Processing with Orientation Scores Erik Franken*, …library.wolfram.com/infocenter/Conferences/6473/Image... · 1 Medical Image Processing with Orientation Scores Erik

2929

Sou

rce

imag

eLo

cali

nfor

mat

ion

�R

esul

t�

En

han

cin

g e

dg

es in

Med

ical im

ag

es

Page 30: Medical Image Processing with Orientation Scores Erik Franken*, …library.wolfram.com/infocenter/Conferences/6473/Image... · 1 Medical Image Processing with Orientation Scores Erik

3030

Ou

tlin

e

•A

bout

our

Res

earc

h gr

oup

•O

rient

atio

nS

core

s•

Diff

usio

nin

Orie

ntat

ion

Sco

res

•S

toch

astic

Com

plet

ion

Fie

lds

•U

sin

gM

ath

em

ati

ca

Page 31: Medical Image Processing with Orientation Scores Erik Franken*, …library.wolfram.com/infocenter/Conferences/6473/Image... · 1 Medical Image Processing with Orientation Scores Erik

3131

•M

athe

mat

ica

is h

elpf

ulin

sol

ving

the

mat

h(e

.g. n

on-c

omm

utin

gop

erat

ors)

•N

DS

olve

in m

ath

em

atica

is n

otus

able

for

our

type

of P

DE

sas

far

as I

know

•P

DE

sol

ver

is w

ritte

nin

C+

+, l

inke

dw

ithM

athl

ink

•T

ypic

alpr

oble

ms

of o

urP

DE

–H

ighl

yan

isot

ropi

c, n

otal

igne

dw

ithgr

id–

Non

-com

mut

ing

oper

ator

s–

Con

vect

ion

+ d

iffus

ion

Usin

gM

ath

em

ati

ca

Page 32: Medical Image Processing with Orientation Scores Erik Franken*, …library.wolfram.com/infocenter/Conferences/6473/Image... · 1 Medical Image Processing with Orientation Scores Erik

3232

Ackn

ow

led

gem

en

ts

•R

emco

Dui

ts•

Mar

kus

van

Alm

sick

•B

art t

er H

aar

Rom

eny

•B

art J

anss

en•

Arje

n R

icks

en•

Ren

ske

de B

oer

For

ques

tions

/ mor

e re

fere

nces

abou

tthi

sw

ork,

co

ntac

t e.m

.fran

ken@

tue.

nl