Active Contour without Edges - University of...
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Active Contour without Edges Tony Chan and Luminita Vese Tony Chan and Luminita Vese
Transcript of Active Contour without Edges - University of...
Active Contour without Edges
Tony Chan and Luminita VeseTony Chan and Luminita Vese
Basic idea in classical active contoursCurve evolution and deformation (internal forces):
Min Length(C)+Area(inside(C))Boundary detection: stopping edge-function (external forces)
0)(lim0 tggg
Example:
0)(lim , ,0
tgggt
pGug
||11|)(| 0
puG ||1 00
Snake model (Kass, Witkin, Terzopoulos 88)
111
0
20
00
2 |))((||)(''||)('|)(inf dssCudssCdssCCFC
1
Geodesic model (Caselles, Kimmel, Sapiro 95)
1
0
|)))(((||)('|2)(inf dssCIgsCCFC
Basic idea in classical active contours
Curve evolution and deformation (internal forces):
Min Length(C)+Area(inside(C))Boundary detection: stopping edge-function (external forces)
0)(lim , ,0
tgggt
1Example: puGug
||11|)(|
00
fihiG 00 ofversion smoother a is uuG
edgeson theonly0|)(| ug edges on theonly 0|)(| 0 ugedges fromfar 0constant|)(| 0 ug
Classical Snakes and Active Contour Models
)(Models in variational formulation
* Snake model (Kass, Witkin, Terzopoulos 88)
)(sCs a parameterization of C
11
* Geodesic model (Caselles, Kimmel, Sapiro 95)
00
2 |)))(((||)('|)(inf dssCIgdssCCFC
1
0
|)))(((||)('|2)(inf dssCIgsCCFC
* C ll C tt C ll Dib 93
Other related models:
* Caselles, Catte, Coll, Dibos 93
* Malladi, Sethian, Vemuri 93
* Caselles Kimmel Sapiro 95 Caselles, Kimmel, Sapiro 95
* Kichenassamy, Kumar, Olver, Tannenbaum, Yezzi ‘95
Some active contours with edge-functionModels in variational formulation
* Snake model: Kass, Witkin, Terzopoulos ’88, , p
* Geodesic model: Caselles, Kimmel, Sapiro ’95
M d l i l l t f l ti
* Kichenassamy, Kumar, Olver, Tannenbaum, Yezzi ‘95Models in level set formulation:
* Geometric model: Caselles, Catte, Coll, Dibos ‘93
M ll di S hi V i ‘93Malladi, Sethian, Vemuri ‘93
* Geodesic model: Caselles, Kimmel, Sapiro ‘95
A fitting term “without edges”
2 )(
220
2
)(10 ||||
CoutsideCinside
dxdycudxdycu
whereCuaveragec
Cuaveragecoutside)(
inside )( 01
Cuaveragec outside)( 02
Fit > 0 Fit > 0 Fit > 0 Fit ~ 0
Minimize: (Fitting +Regularization)
Fitting not depending on gradient detects “contours without gradient”
An active contour model “without edges”
Fitting + Regularization terms (length, area)(C. + Vese 98)
))((||)(inf CinsideAreaCCccF
Fitting Regularization terms (length, area)
22
21,,
||||
))((||),,(inf21 Ccc
d dd d
CinsideAreaCCccF
)( )(
220
210 ||||
Cinside Coutside
dxdycudxdycu
C = boundary of an open and bounded domain
|C| = the length of the boundary-curve C
Advantages- detects objects without sharp edges
- detects cognitive contours
- robust to noise; no need for pre-smoothing
Examplesp
Relation with the Mumford-Shah segmentation model* The Mumford-Shah functional ‘87
C
MS
CudxdyudxdyuuCCuF 22
0,||||||),(inf
- u is an optimal approximation of the initial image- u is an optimal approximation of the initial image
- C is the set of jumps or edges of u
i th t id th d C* The model is a particular case of the Mumford-Shah functional
- u is smooth outside the edges C
Cuc inside)(average
Cuc
Cucu
outside )(average inside )(average
02
01
u is the best approximation of taking only two values and1c 2cu- u is the best approximation of taking only two values and , and with one edge C
- the snake C is the boundary between the sets and
1c 2c
cu cu
0u
the snake C is the boundary between the sets and
- we do not need regularization in u, because are constant
1cu 2cu
21,cc
Level Set Representation (S. Osher - J. Sethian ‘87)n 0),(|),( yxyxC
Inside C
O d C
Outside C0 0
),(|),( yy
Outside C0
0C
nn n
C= boundary of an open domain
||
,||
Normal
divKCurvaturen
Example: mean curvature motion* All t ti t l h i d b ki* Allows automatic topology changes, cusps, merging and breaking.
•Originally developed for tracking fluid interfaces.
Variational Formulations and Level Sets(Following Zhao, Chan, Merriman and Osher ’96)
h i id f i
0),( :),( yxyxC
HC |)(|||L thThe Heaviside function
0if00 if ,1
)(
H
dxdyHCinsideArea
HC
)())((
|)(|||Length
0 if ,0
dxdyHCinsideArea )())((
))),((1()),((),( 21 yxHcyxHcyxu The level set formulation of the active contour model
ccF ),,(inf 21
))),((()),((),( 21 yyy
dxdyHHccF
cc
)(|)(|),,(
),,(
21
21,, 21
dxdyHcyxudxdyHcyxu ))(1(|),(|)(|),(| 220
210
HHFF )())(())()((
Existence of minimizers among characteristic functions of sets with finite perimeter (in any dim. N)
dxxF
HHFccF
)(||)(
)( )),(()),(),(( }0{21
dxxcudxxcu ))(1(|)(|)(|)(| 220
210
2010
z. Lipschit withbounded open, be Let : RTheorem N
.,.1,0)(),( ),(inf problem onminimizati following the then),( If 0
eadxxBVFLu
solution.a has
,,)(),(),(
i ib d dff if)(BV BVin scompactnes and TV of :s variationof calculus in standard :Proof
variationboundedoffunctions of space)(lsc
BV
The Euler-Lagrange equations)(i f U i h i i f),,(inf 21, ,21
ccFcc
dfE ti
Using smooth approximations for the Heaviside and Delta functions
dxdyHudxdyHu ))(1()( 00 2 1 andfor Equations cc
dxdyHc
dxdyHc
))(1()( ,
)()( 21
),,(for Equation yxt
)()(||
)( 220
210 cucudiv
t
),(),,0(
||
0 yxyx
t
Two approximations of the Heaviside and Delta functions
xxH arctan21
21)(
||ifsin111 - xif,0
if ,1)(
xxx
xxH
,-support compact small has -
of curves level allon acts -everywhere 0 -
||if,sin12
x
computed are minima localonly - of curve
level-zerothearround localy,only acts - lyautomatica contoursinterior detects -
minima global compute to tendency thehas -
AdvantagesExperimental Results
CofEvolution )(Averages cc AdvantagesAutomatically detects i t i t !
C of Evolution ),( Averages 21 cc
interior contours!
Works very well for concave objects
Robust w.r.t. noise
Detects blurred contours
Th i i i l bThe initial curve can be placed anywhere!
Allows for automatical change of topolgy
Contours without gradient (cognitive contours)
A spiral from an art picture(the initial curve is the border of the image)
Europe nightlightsTh d l d t t l t f i t ( iti t )The model detects clusters of points (cognitive contours)
A plane in a noisy environment
A galaxyg y
From Active Contours (2D) to Active Surfaces (3D)
MRI DATA FROM LONI-UCLA
Potential Application to Data Mining:pp g
(Data from V. Kumar et al: Chameleon, IEEE Computer 2000)
Extension to Vector-Valued ImagesTony F. CHAN & Luminita VESE & B. Yezrielev SANDBERGApplications
* Color images (RGB)
* M l i l (PET MRI CT)* Multi-spectral (PET, MRI, CT)
channels ),,...,,( ,02,01,00 Nuuuu N
Vector-Valued image:
( f h h l i idectorsconstant v are , 21 cc
The model
(averages of each channel inside and outside the curve)
),,(inf 21 CccF
21 ))(()(),,( CinsideAreaCLengthCccF
),,( 21,, 21 Ccc
)( )(
2,2,0
2,1,0 ||||
Cinside Coutsideiiii
idxdycudxdycu
* The model can be solved using level sets, similar with the scalar case
Extension to Vector-Valued ImagesChannel 1 with occlusion Channel 2 with noise
Recovered object and averages
Color (RGB) picture Intensity (gray-level) pictureColor Images
R G B
Recovered objects and contours in RGB modeRecovered objects and contours in RGB mode
45 transforms of image with Gabor functionActive Contour for Texture
F F120 120
120
150
120
150
180 180
F
120
User Picked Gabor transforms yield final contour
150
180
Texture Segmentation with User picked Gabor transforms
Gabor Transforms
Original Image F=120
F=180
F=150
Initial Contour Final Contour Final Contour on the original image
U i d T S i S h iUnsupervised Texture Segmentation SyntheticAutomatically select small subset of Gabor frames for AC model
Final Contour on the original imageOriginal Image Final Contour
Original Image Final Contour on the original imageFinal Contour
PlanIntroduction and Motivations:
Active contours “without edges” Chan-Vese, Active Contours without Edges, SS ’99, IEEE IP
Generalization to the Mumford-Shah model:Chan-Sandberg-Vese, Active Contours without Edges for Vector-Valued Images, JVCI
Generalization to the Mumford Shah model:
The piecewise-constant case
The piecewise-smooth caseChan Vese I t ti i l l t d th i i t t M f dChan-Vese, Image segmentation using level sets and the piecewise-constant Mumford-Shah model, A level set algorithm for minimizing the Mumford-Shah functional in image processing, preprints 2000, to appear in Special Issue of IJCV, for VLSM/SS’01
PlanIntroduction and Motivations:
Active contours “without edges”
Generalization to the Mumford-Shah model:Generalization to the Mumford Shah model:
The piecewise-constant case
The piecewise-smooth case
Variational Segmentation ModelsM f d Sh h D l M M l S li i i Sh h Ch b llMumford-Shah, Dal Maso-Morel-Solimini, Shah, Chambolle, Gobino, March, Koepfler-Lopez-Morel, Ambrosio-Tortorelli, Morel-Solimini BraidesMorel Solimini, Braides,
Mumford-Nitzberg-Shiota, Zhu-Lee-Yuille,
Chambolle-Dal Maso,
Chambolle-Bourdin, Cortesani-Toader,Chambolle Bourdin, Cortesani Toader,
Malladi-Sarti-Sethian, …
Works Related to Ours:
Samson-Feraud-Aubert-Zerubia, Sifakis-Garcia-Tziritas,, ,
Paragios-Deriche, Yezzi-Tsai-Willsky
Mumford-Shah Segmentation 89
dSuuu ))(||(min 22
S S
SudSuuu ))(||(min
, 0
S=“edges”
Algorithms for Minimizing Mumford-Shah energy
Difficult problem in practice: non-convex; lower-dim. unknown C
Solutions proposed:
* Region growing and merging: use MS energy in greedy algorithm (Dal Maso - Morel - Solimini, Koepfler-Lopez-Morel)
* Elliptic approximations: embed C in 2D phase-field function (Ambrosio - Tortorelli, Chambolle, Braides, March)
* Statistical framework: (Zhu-Lee-Yuille)
Our work:
* provides an efficient level set formulation for the M-S modelp
* inherits the advantages of level sets
Relation C-Vese AC & Mumford-Shah segmentation models
* Our AC model is a particular case of the Mumford-Shah functional
CIc
u inside )(average1
CIc
u outside )(average2
- u is the best approximation of I taking only two values and , and with one edge C
1c 2c
- the contour C is the boundary between the sets and
- we do not need regularization in u, because are constant
1cu 2cu
21,cc
Relation C-Vese AC & Mumford-Shah segmentation models* The Mumford-Shah functional 87
C
MS
CudxdyudxdyIuCLengthCuF 22
,||||)(),(inf
- u is an optimal approximation of Iu is an optimal approximation of I
- C is the set of jumps or edges of u
- u is smooth outside the edges C
* Our AC model is a particular case of the Mumford-Shah functional
CIc inside)(average1
CIcCIc
u outside )(average
inside)(average
2
1
i th b t i ti f I t ki l t l dc c- u is the best approximation of I taking only two values and , and with one edge C
- the snake C is the boundary between the sets and
1c 2c
1cu 2cu
- we do not need regularization in u, because are constant21,cc
Generalizing CV-AC to MS
• From 2 segments (1 level set function) to multiple segments and level set functionsp g
• From p.w. constant to p.w. smoothapproximationsapproximations
Generalization to Multiple (>2) SegmentsThe Mumford Shah piecewise constant segmentation modelThe Mumford-Shah piecewise-constant segmentation model
N d 1 l l f i 2 i l j i
ii
MS CdxdycuCuF || ||),( 20
Need > 1 level set function to represent > 2 segments, triple junctions, …
Classical methods for Multiphase Level Set Representation:
i
(Zhao-Chan-Merriman-Osher):
- associate one level set function to each phase, allows for triple junctions
hi h t ti l t- high computational cost
- needs additional constraints to prevent vacuum and overlap
New Multiphase Level Set Representation:segmentsor phases 2represent can wefunctions,set level using - nn
- reduced computational cost
New Multiphase Level Set Representation:
p
- by definition of the partition, no vacuum and no overlap
- allows for triple junctions and other complex topologies
Multiphase level set representations and partitionsallows for triple junctions, with no vacuum and no overlap of phases
phases2)( n 4-phase segmentation2 l l f i:Curves
phases2 ),...,( 1 n
2 level set functions
}0{:Curves
}0{}0{:Curves
21
0
2-phase segmentation
1 l l t f ti
00
1
00
1
00
2
1
1 level set function
02 02
0
0
0
00
2
1
Example: two level set functions and four phases:segmentsorphases4)(functionssetlevel2
functionssetlevelThe)( 0,0 ,0,0 ,0,0 ,0,0
:segmentsor phases4 ),(functionsset level 2
21212121
21
vectorConstant ),,,( functionsset level The ),(
00011011
21 ccccc
))(1))((1()())(1( ))(1)(()()(
21002101
21102111
HHcHHcHHcHHcu
))(1)((||)()(||),(
Energy ))())((()())((
212
100212
110
21002101
dxdyHHcudxdyHHcucFInf
))(1))((1(||)())(1(||
))()((||)()(||),(
212
000212
010
2110021110),(
dxdyHHcudxdyHHcu
yyfc
|)(||)(| 21 HH
Existence of minimizers f
Theorem:
Existence of minimizers among characteristic functions of sets with finite perimeterp
Similar with the previous case
Remarks:
Non-convex minimization problem
No uniqueness of minimizers
The Euler-Lagrange equations
00in)( constants The
umeanc
Using smooth approximations for the Heaviside and Delta functions
00i)(0,0in )( 0,0in )( 0,0in )(
21001
21010
21011
umeancumeancumeanc
),(in sPDE' The
0,0in )(
21
21000
umeanc
))(1(||||)(||||
||)( 2
2000
21002
2010
2110
1
11
1
HcucuHcucudivt
))(1(||||)(||||
||)( 1
2000
21001
2010
2110
2
22
2
HcucuHcucudivt
),...,,(for Similarly 21 n
Results with 2 level set functions (4 phases)* t diff t i iti li ti* two different initializations
* interior contours are automatically detected
* model robust to noise
Faster
Energy Decrease With Iterations for 3 initial conditions(here, only with (a) and (c) it converge to the global minimum)y g g
( )(a)
(b)
(c)
Non-convex problem: different (I.C.) may converge to local/global min.
Detection of contours without gradient -cognitive contours
Triple junctions can be detected and represented using only two level set functions, without vacuum and without overlap (new).
0 0 21
An MRI brain image
Phase 11 Phase 10 Phase 01 Phase 00
mean(11)=45 mean(10)=159 mean(01)=9 mean(00)=103mean(11)=45 mean(10)=159 mean(01)=9 mean(00)=103
Numerical results on an MRI brain picture
Multiphase segmentation model using two level set functions and four phases
A real outdoor picture
Phase 11 Phase 10 Phase 01 Phase 00
mean(11)=159 mean(10)=205 mean(01)=23 mean(00)=97mean(11)=159 mean(10)=205 mean(01)=23 mean(00)=97
Three level set functions representing up to eight phasesSix phases are detected, together with the triple junctions.
T
Evolution of the 3 level sets Segmented image
T
Evolution of the 3 individual level sets.
Phase 11 Phase 10 Phase 01 Phase 00
Original gene chip data
(Terry Speed)
Detected contours
Segmented images for three different scale parameters
Original gene data (G) Detected contours Segmented image
1 Segment(11)=180 2 Segment(10)=93 3 Segment(01)=56
Gene Microarray4 Segment(00)=6 Gene Microarray
Expression Data
Narrow band
It is useful to compute the level set function not on the whole image domain but in a narrow band near to the contour. Abs()<d
Decreasing the computational complexity.
Narrow Band AlgorithmNarrow Band Algorithm
Narrow Band AlgorithmNarrow Band Algorithm
• Tag alive points in narrow bandTag alive points in narrow band
• Build land mines to indicate near edge
i i li f i id h• Initialize far away points outside the narrow band with large positive (negative) values if l id (i id ) h f i lfvalues are outside(inside) the front itself.
• Solve level set equation until land mine hit
• Rebuild and loop.
• Bandwidth can be set as 12Bandwidth can be set as 12
PlanIntroduction and Motivations:
Active contours “without edges” Chan-Vese, Active Contours without Edges, SS ’99, and IEEE IP
Generalization to the Mumford-Shah model:Chan-Sandberg-Vese, Active Contours without Edges for Vector-Valued Images, JVCI
Generalization to the Mumford Shah model:
The piecewise-constant case
The piecewise-smooth caseChan Vese I t ti i l l t d th i i t t M f dChan-Vese, Image segmentation using level sets and the piecewise-constant Mumford-Shah model, A level set algorithm for minimizing the Mumford-Shah functional in image processing, preprints 2000, to appear in Special Issue of IJCV for VLSM/SS’01
PlanIntroduction and Motivations:
Active contours “without edges”
Generalization to the Mumford-Shah model:Generalization to the Mumford Shah model:
The piecewise-constant case
The piecewise-smooth case
Generalization to Piecewise-Smooth Approximations by the Mumford-Shah segmentation modely f gThe Mumford-Shah segmentation model ‘89
),(inf CuF MS
MS CdxdyudxdyuuCuF ||||||),( 220
2
),(inf,
CuFCu
C
A level set algorithm for the M-S model
( fi h C i h b d f d i )
0if),,(
)(0)(|)(yxu
C
(assume first that C is the boundary of an open domain)
0 if ),,(0),,(
),( ,0),( |),(
yxuyxu
yxuyxyxC
))(1)(,()(),(),( HyxuHyxuyxu
),,(inf,,
uuFuu
0 0
20
220
2 ||||),,(
dxdyuudxdyuuuuF
0
2
0
2 ||||||
dxdyudxdyuC
Using the Heaviside function:
2222
|)(|))(1(||)(||
))(1(||)(||),,(
22
20
220
2
HdxdyHudxdyHu
dxdyHuudxdyHuuuuF
|)(|))(1(||)(|| HdxdyHudxdyHu
The model is a common framework for activeThe model is a common framework for active contours, denoising, segmentation and edge detection.
The Euler-Lagrange equations
u 0on 00,on )( 02
unuuuu
0on 0 0,on )( 02
nuuuu
2220
220
2 ||||||||)( uuuuuudiv
t
00 ||||||||
||)(
The model is a common framework for active
Curvature Jump of energy terms in u
The model is a common framework for active contours, denoising, segmentation and edge detection.
functions by 0 on and 0 on extend tohave We :Remark
1Cuu
Sbert) Morel,(Caselles, functions of ionInterpolat :direction) normal thein diffusion(by solutions Several
y
Aslam) (Fedkiw, method fluidGhost Cao) Crandall, (Jensen, extensions Lipschitzmizing Mini
),( ,p
)( ,
||
,||
),( Normal
yxyx nnN
:Examples||||
O
on 0 0,on 2 22
yyyxyyxxxxt nuununnunu
,...0in 0 :Or
yyxxt ununu
Combining p.w. smooth approx. with multiple LS’sWe consider (again) two level set functions with four phases
Remark: Based on Four Color Thm., 2 LS’s should suffice.
))(1))((1()())(1(
))(1)(()()( 2121
HHuHHu
HHuHHuu
))(1))((1()())(1( 2121 HHuHHu
)( ),( 21
The energy and the Euler-Lagrange equation can be written and solvedThe energy and the Euler Lagrange equation can be written and solved in a similar manner
To handle triple junctions and more general casesWe consider (again) two level set functions with four phases
Remark: Based on the Four Color Thm., it should suffice.
))(1))((1()())(1(
))(1)(()()( 2121
HHuHHu
HHuHHuu
))(1))((1()())(1( 2121 HHuHHu
)( ),( 21
The energy and the Euler-Lagrange equation can be written and solvedThe energy and the Euler Lagrange equation can be written and solved in a similar manner
Existence of minimizers in the space SBV
(Special functions of Bounded Variation)
LBVSBV NN )()()( 1/ Proof: standard in calculus of i i
BVu )(
variations
JuCudxxuDu )( Compactness and l.s.c. results on SBV by L. Ambrosio
JudxxuDuSBVu
)()( )(
Signal denoising - segmentation - jump detection(numerical results in 1D; note jumps well located and preserved)
Remark: Related works by Anne Gelb and Eitan Tadmor usingRemark: Related works by Anne Gelb and Eitan Tadmor using spectral methods
Piecewise-smooth approximationsActive Contours+Denoising+Segmentation g g
M S Th th bj t d t t d ith th t l !M-S: The three objects are detected with the correct values! Before (A-C): all objects were detected, but with the same mean
2 level sets (four phases) in the piecewise-smooth case
Initial Noisy Image Initial curves
Evolutions to the steady state (denoised - segmented image)
Future worksConsider other surface energies depending on the normal- Consider other surface energies, depending on the normal
or on the curvature (beyond image processing applications)
Th lti h b d f l i (P P )- The multiphases can be used for occlusion (P. Perona)
- Higher order regularizations, to detect jumps in the derivatives
- Study the convergence of the algorithm, when adding some y g g gregularizing/elliptic terms
- Use the method for related problems arising in fractureUse the method for related problems arising in fracture mechanics and optimal design
Viscosity solutions for these geometric curve evolution PDEs- Viscosity solutions for these geometric curve evolution PDEs
- ...
Features & Advantages of PDE Imaging ModelsConclusion
Features & Advantages of PDE Imaging Models
* Use PDE & Geometry concepts: gradients diffusion curvature level Use PDE & Geometry concepts: gradients, diffusion, curvature, level sets
* Exploit sophisticated PDE and CFD (e.g. shock capturing) techniquesp p ( g p g) q
Restoration:
sharper edges less edge artifacts often morphological- sharper edges, less edge artifacts, often morphological
Segmentation (using level set techniques):
l d i i b d ll d l i f b d i- scale adaptivity, geometry-based, controlled regularity of boundaries, segments can have complex topologies
Newer, less well developed/accepted.
OUTLINE
• Review of Variational Level Set Segmentation• Fast AlgorithmFast Algorithm
A Fast Algorithm for Level Set Based Optimization(Bing Song and Tony Chan, 2003)
Available at www math ucla edu/applied/cam/index htmlwww.math.ucla.edu/applied/cam/index.html
CAM 02-68(Related work by Gibou & Fedkiw)(Related work by Gibou & Fedkiw)
Outline
• Motivation• General AlgorithmGeneral Algorithm• Application to the Chan-Vese segmentation
modelmodel• Convergence of algorithm• Examples• Closely related work by Gibou and Fedkiwy y
Motivation
• Minimize level set based functional
))((min HF
• Examples:
))((min
HF
– Image segmentation– Inverse problem– Shape analysis and optimizationShape analysis and optimization– Data Clustering– …
How to solve it?
))(( HFt
• Usually use gradient descent flow
• Drawback
))(( HFt
Drawback– Slow (CFL)– F need to be differentiable
An Insight
• Segmentation only needs sign of but not its value
Di t h ki f bj ti f ti d• Direct search making use of objective function does not require derivatives of functional
New Algorithm
1. Initialization. Partition the domain into and
0
0
2. Advance. For each point x in the domain, if the energy F lower when we change )(xthe energy F lower when we change to , then update this point.
3 Repeat step 2 until the energy F remains
)(x
)(x
3. Repeat step 2 until the energy F remains unchanged
An example: Chan-Vese model
)(|||)(|(),),(( 2
10121 HcuHccHF
))(1(|| 2202
0
Hcu
Core Step of AlgorithmChange in F if is changed to at a pixel:0 0
0 01
~ 111
muccc
g g p
0 0
)(~1
~
2
222
mcuFF
nuccc
0
1)(~
1)(
2222
111
nncuFF
mcuFF
n
lengthinchangemcuncuF 1
)(1
)( 21
2212 gg
mn 1)(
1)( 1212
How to calculate length term?2
,11,2
,,1 ))()(())()((|)(| jijijijiij HHHHH
Change in length term can be updated locally.
Only possible value for length term locally is 0 1 sqrt(2)Only possible value for length term locally is 0, 1, sqrt(2) depending the 3 points in the length term belong to the same region or not.
A 2-phase example
(a), (b), (c), (d) are four different initial condition.All of them converge in one sweep!All of them converge in one sweep!
Example with Noise
Converged in 4 steps.
(G di D E l L k 400 )(Gradient Descent on Euler-Lagrange took > 400 steps.)
Convergence of the algorithm
Theorem: For a two phase image, the algorithm will converge in one sweep, independent of sweeping order.
Proof considers the following 2 cases:Green part of inside of contour will flip sign. Only one of the 2 black parts can change sign
00
0
0
Why is 1-step convergence possible?
• Problem is global: usually cannot have finite step convergence based on local p gupdates only
• But, in our case, we can exactly calculate the global energy change via local updatethe global energy change via local update (can update global average locally)
Application to piecewise linear CV modelApplication to piecewise linear CV model(Vese 2002)
))(1(||
)(|||)(|(),),((2
22100121
Hybxbbu
HyaxaauHccHF
))(1(|| 21002 Hybxbbu
Original P.W.ConstantConverged in 4
steps
P.W. LinearConverged in 6
steps
Application to multiphase CV modelApplication to multiphase CV model(C.- Vese 2000)
• Using n level set to represent 2n phases.
Ii
IIIn HdxdycucF
m 121
20 |)(|||),(
mInI m 121
Multiphase CV model (cont’d)
Converged in 1 step.
Gibou and Fedkiw’s method (02)• First discard the stiff length term in E L:• First, discard the stiff length term in E-L:
)(:)()( 222
211 xVcucu
dtd
• Take large time step enough to change sign• If , then 0)()( xVx ))(()( xsignx ,• Followed by anisotropic diffusion to handle
noisenoise• Difference (Ours wrt GF):
A ifi d f k t h dl l th t– A unified framework to handle length term– F does not need to be differentiable
Conclusion• A fast algorithm to solve the Chan Vese• A fast algorithm to solve the Chan-Vese
segmentation models.• 1-step convergence for 2 phase images (no reg )1-step convergence for 2 phase images (no reg.).• Robust wrt noise: 3-4 step convergence.• The gradient of functional is not needed• The gradient of functional is not needed.• We think it can be applied to more general level
set based optimization problemsset based optimization problems.• In general, cannot expect finite step convergence
(need global change from local update). Can get(need global change from local update). Can get stuck in local minima.
That’s all.Th k f i !
That’s all.Th k f i !Thank you for your patience!Thank you for your patience!
Logic Operators on Multi-Channel Active Contour(Chan, Sandberg 2001)
• An Example for two channel logical segmentation:• An Example for two channel logical segmentation:1A 2A
21 (a) AA 21 (b) AA 21 ~ (c) AA
Redefining the fitting term of CV Model
dxdyyyyxxxfClengthccCF nn ),..,,,,..,,()(),,(min 21,21
using the logical variables:
otherwise1Cinsidefit good 0),( (x,y)yxix
th i1C outsidefit good 0),(
otherwise 1 (x,y)yxiy
otherwise 1
2
Logical variables for CV Model:
C inside 2||)(
max
2|0|),( (x,y)ici
ouiuyx
iciuyxix
C outside 0
),( iouyx
Coutside2|0|
C inside 0 ),(
(x y)iciu
yxiy
Coutside2||),(
max0 (x,y)ici
ouiouyx
ii xx 1'And their complements:
ii yy 1'
I i f bjTruth Tables
Intersection of objects:intersection of xi inside C, and union of yi outside C.
Union of objects: (analogous)Truth Table using parameters :
Truth Table for 2 Channels
Union of objects: (analogous)
Truth Table for 2 Channels
inside C 1 1 0 0 1 1 1 1 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 1
1x 2x 1y 2y 21 AA 21 AA 21 ~ AA
outside C 0 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0
Fitting the truth table for function f:
g
2/1
2121
212/1
21
)),(),((2/)),(),((),(
2/)),(),(()),(),((),(
21
21
yxyyxyyxxyxxyxf
yxyyxyyxxyxxyxf
AA
AA
2/12121~ ))),(1)(,((2/))),(1(),((),(
21
21
yxyyxyyxxyxxyxf AA
Examples using logic operators1A 2A
1A 2A 3A
21 (a) AA
21 (b) AA 321(b) AAA 321(a) AAA 321 ~(c) AAA 321 ~(d) AAA 321( )321( ) 321( ) 321( )
Given the 3 channels 4 possible logic operations are implemented.
21 ~ (c) AA
Time evolution for original example
1A 2A
Example of an Application
Original image with contour overlapping
Contour only
Time evolution for finding the “tumor” in the first image that is not in the second.
A Fast Algorithm for Variational Level Set Segmentation
Tony F. Chan
Department of Mathematics, Univ. Calif. Los Angeles
SONAD 2003SONAD 2003
McMaster University, Ontario, Canada
May 2, 2003
Reprints: www.math.ucla.edu/~imagersp g
Supported by ONR and NSF
(J i k i h Bi S )(Joint work with Bing Song)
OUTLINE
• Review of Variational Level Set Segmentation• Fast AlgorithmFast Algorithm
Narrow band
• Initialization n=0• repeat
– n++– Determination of the narrow band– Computing c and c– Computing c1 and c2– Evolving the level-set function on the narrow band
– Re-initialization
• until the solution is stationary, or n>nor n>nmax
What is an “active contour” ?
: imagean giving 0 u
objectstheof boundarieson thestoptohas curve thein objectsdetect to curve a evolve 0
uC
Initial Curve Evolutions Detected Objects
jp
