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Lena Gorelick Joint work with Frank Schmidt and Yuri Boykov Rochester Institute of Technology, Center of Imaging Science January 2013. Fast Trust Region for Segmentation. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A A. - PowerPoint PPT Presentation
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Fast Trust Region for SegmentationLena Gorelick
Joint work with Frank Schmidt and Yuri Boykov
Rochester Institute of Technology, Center of Imaging Science January 2013
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Sf,E(S) B(S)Sf,E(S)
Image segmentation Basics
I
Fg)|Pr(I Bg)|Pr(I
Sx
f(x)s(x)E(S)
bg)|Pr(I(x)fg)|Pr(I(x)lnf(x)
S
3
Standard Segmentation Energy
Fg
Bg
Intensity
ProbabilityDistribution
Target AppearanceResulting Appearance
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Minimize Distance to Target Appearance Model
KL( R(S) Bha( R(S)
)||
),
2L||-|| R(S)
Sp p
p
bg)|Pr(Ifg)|Pr(I
ln
Non-linear harder to optimizeregional term
TS
TS
TS
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Non-linear Energies with High- Order Terms
complex appearance models shape
non-linear regional
term
B(S)R(S)E(S)
S
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Related Work
Can be optimized with gradient descent First order (linear) approximation models
We use more accurate non-linear approximation models based on trust region
Ben Ayed et al. Image Processing 2008,Foulonneau et al., PAMI 2006Foulonneau et al., IJCV 2009
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Our contributions
General class of non-linear regional functionals
Optimization algorithm based on trust region framework – Fast Trust Region
)S,f,,S,fF(R(S) k1
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Outline
Non-linear Regional Functionals Overview of Trust Region
Framework Trust region sub-problem
Lagrangian Formulation for the sub-problem
Fast Trust Region method Results
Regional FunctionalExamples
Volume Constraint
1f(x) ibin for (x)fi
S,fiS1,|S|
20 )VS1,(R(S)
9
10
Regional FunctionalExamples
Bin Count Constraint
1f(x) ibin for (x)fi
2ii
k
1i)VS,f(ΣR(S)
S1,|S| S,fi
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Regional FunctionalExamples Histogram Constraint
1f(x) ibin for 1(x)fi
S1,S,f
(S)P ii
S1,|S| S,fi
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Regional FunctionalExamples Histogram Constraint
VS)PΣR(S) 2ii
k
1i
(
2L||-|| R(S) TS
S1,S,f
(S)P ii
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Regional FunctionalExamples Histogram Constraint
V(S)P(S)logPΣR(S)
i
ii
k
1i
KL( R(S) )|| TS
S1,S,f
(S)P ii
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Regional FunctionalExamples Histogram Constraint
ii
k
1iV(S)PΣlogR(S)
Bha( R(S) ), TS
S1,S,f
(S)P ii
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Shape Prior
Volume Constraint is a very crude shape prior
Can be encoded using a set of shape moments mpq
p+q is the order
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Volume Constraint is a very crude shape prior
Shape Prior
01pq yxy)(x,f 22pq yxy)(x,f 32pq yxy)(x,f
qppq
pqpq
yxy)(x,f
Sf(S)m
,
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Shape Prior using Shape Moments mpq
Volumem00
...RatioAspect
nOrientatio Principalmmmm
0211
1120
Mass OfCenter )m,(m 0110
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Shape Prior Constraint
Shape Prior using Shape moments
Dist( ),R(S)
kqp
2pqpq (T))m(S)(mR(S)
qppq
pqpq
yxy)(x,f
f(S)m
S,
S T
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Optimization of Energies with Regional Functional
B(S)R(S)E(S)
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Gradient Descent with Level Sets
Gradient Descent First Order Taylor Approximation for
R(S) First Order approximation for B(S)
(“curvature flow”) Only robust with tiny steps
Slow Sensitive to initialization
B(S)R(S)E(S)
http://en.wikipedia.org/wiki/File:Level_set_method.jpg
Ben Ayed et al. CVPR 2010,Freedman et al. tPAMI 2004
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Energy Specific vs. General
Speedup via energy- specific methods Bhattacharyya Distance Volume Constraint
In contrast: Fast optimization algorithm
for general high-order energies
Based on more accurate non-linear approximation models
Ben Ayed et al. CVPR 2010,Werner, CVPR2008Woodford, ICCV2009
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B(S)(S)U(S)E 0d||SS|| 0
~min
General Trust Region ApproachAn overview
The goal is to optimize
Trust regio
nTrust
Region Sub-
Problem
B(S)R(S)E(S)
B(S)(S)U(S)E 0 ~
• First Order Taylor for R(S)
• Keep quadratic B(S)
0Sd
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General Trust Region ApproachAn overview
The goal is to optimize B(S)R(S)E(S)
B(S)(S)U(S)E 0 ~
Trust Region Sub-
Problem
B(S)(S)U(S)E 0d||SS|| 0
~min
0Sd
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||SS||λB(S)(S)U(S)L 00λ
How to Solve Trust Region Sub-Problem Constrained optimization
minimize
Unconstrained Lagrangian Formulationminimize
Can be optimized globally using graph-cut
d||SS||s.t.B(S)(S)U(S)E
0
0
~
||SS||λ(S)E(S)L 0λ ~
Can be approximated with
unary termsBoykov et al. ECCV
2006
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Spectrum of Solutions for different λ or d
• Newton step• “Gradient Descent”• Exact Line Search
(ECCV12)
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General Trust Region
Repeat Solve Trust Region Sub-problem
around S0 with radius d Update solution S0 Update Trust Region Size d
Until Convergence
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Fast Trust Region
General Trust Region Control of
the distance constraint d
Lagrangian Formulation Control of
the Lagrange multiplier λ λd1λ
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Comparison Simulated Gradient Descent Exact Line-Search (ECCV 12) Newton step Fast Trust Region (CVPR 13)
Volume Constraint for Vertebrae segmentation
Log-Lik. + length + volumeFast Trust Region
InitializationsLog-Lik. + length
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maxVminV0
)(SR
|| S
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Shape Prior with Geometric moments for liver segmentation
Fast Trust Region
Log-LikelihoodsNo Shape Prior
Second order geometric moments computed for the user provided initial ellipse
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Appearance model with KL Divergence Constraint
Init
Fast Trust Region
“Gradient Descent”
Exact Line Search
Appearance model is obtained from the ground truth
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Appearance Model with Bhattacharyya Distance Constraint
“ “
Fast Trust Region
“Gradient Descent”
Exact Line Search
Appearance model is obtained from the ground truth
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Shape prior with Tchebyshev moments for spine segmentation
Log-Lik. + length + Shape PriorFast Trust Region
Second order Tchebyshev moments computed for the user scribble
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Appearance Model with Bhattacharrya Distance Constraint
BHA. + length Fast Trust Region
Ground Truth
Appearance model is obtained from the ground truth
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Future Directions
Multi-label Fast Trust Region Binary shape prior:
affine-invariant Legendre/Tchebyshev moments
Learning class specific distribution of moments
Multi-label shape prior moments of multi-label atlas map
Experimental evaluation and comparison between level-sets and FTR.
42
Thank you
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