higher order multiphase image segmentation and registration
TRANSCRIPT
![Page 1: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/1.jpg)
Higher Order Multiphase ImageSegmentation and Registration
Stephen KeelingInstitute for Mathematics and Scientific Computing
Karl Franzens University of Graz, Austria
in cooperation withStefan Furtinger and Renier Mendoza
Mathematical Image Processing Section, GAMM 2012, Darmstadt
March 28, 2012
![Page 2: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/2.jpg)
Higher Order Models: Total Generalized VariationGoal: Overcome the essentially piecewise constant model ofTV regularization.
In the classical approach, minimize:
J(I) =
∫Ω|I − I|2 + TVα(I) where
α
∫Ω|DI| = TVα(I) = sup
∫Ω
I divψ : ‖ψ‖∞≤α,ψ∈C10(Ω,Rn)
Noisy and TV-reconstructed images:
![Page 3: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/3.jpg)
Higher Order Models: Total Generalized VariationGoal: Overcome the essentially piecewise constant model ofTV regularization. In the classical approach, minimize:
J(I) =
∫Ω|I − I|2 + TVα(I)
where
α
∫Ω|DI| = TVα(I) = sup
∫Ω
I divψ : ‖ψ‖∞≤α,ψ∈C10(Ω,Rn)
Noisy and TV-reconstructed images:
![Page 4: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/4.jpg)
Higher Order Models: Total Generalized VariationGoal: Overcome the essentially piecewise constant model ofTV regularization. In the classical approach, minimize:
J(I) =
∫Ω|I − I|2 + TVα(I) where
α
∫Ω|DI| = TVα(I) = sup
∫Ω
I divψ : ‖ψ‖∞≤α,ψ∈C10(Ω,Rn)
Noisy and TV-reconstructed images:
![Page 5: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/5.jpg)
Higher Order Models: Total Generalized VariationGoal: Overcome the essentially piecewise constant model ofTV regularization. In the classical approach, minimize:
J(I) =
∫Ω|I − I|2 + TVα(I) where
α
∫Ω|DI| = TVα(I) = sup
∫Ω
I divψ : ‖ψ‖∞≤α,ψ∈C10(Ω,Rn)
Noisy and TV-reconstructed images:
![Page 6: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/6.jpg)
Higher Order Models: Total Generalized Variation
Goal: Develop a functional with a kernel which is richer thanpiecewise constants.
In the generalized approach, minimize:
J(I) =
∫Ω|I − I|2 + TGVk
α(I) where
TGVkα(I)=sup
∫ΩI divkψ : ‖divlψ‖∞≤αl︸ ︷︷ ︸
l=0,...,k−1
,ψ∈Ck0 (Ω,Symk (Rn))
Noisy and TGV2α-reconstructed images: [Bredies, Kunisch, Pock]
![Page 7: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/7.jpg)
Higher Order Models: Total Generalized Variation
Goal: Develop a functional with a kernel which is richer thanpiecewise constants. In the generalized approach, minimize:
J(I) =
∫Ω|I − I|2 + TGVk
α(I)
where
TGVkα(I)=sup
∫ΩI divkψ : ‖divlψ‖∞≤αl︸ ︷︷ ︸
l=0,...,k−1
,ψ∈Ck0 (Ω,Symk (Rn))
Noisy and TGV2α-reconstructed images: [Bredies, Kunisch, Pock]
![Page 8: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/8.jpg)
Higher Order Models: Total Generalized Variation
Goal: Develop a functional with a kernel which is richer thanpiecewise constants. In the generalized approach, minimize:
J(I) =
∫Ω|I − I|2 + TGVk
α(I) where
TGVkα(I)=sup
∫ΩI divkψ : ‖divlψ‖∞≤αl︸ ︷︷ ︸
l=0,...,k−1
,ψ∈Ck0 (Ω,Symk (Rn))
Noisy and TGV2α-reconstructed images: [Bredies, Kunisch, Pock]
![Page 9: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/9.jpg)
Higher Order Models: Total Generalized Variation
Goal: Develop a functional with a kernel which is richer thanpiecewise constants. In the generalized approach, minimize:
J(I) =
∫Ω|I − I|2 + TGVk
α(I) where
TGVkα(I)=sup
∫ΩI divkψ : ‖divlψ‖∞≤αl︸ ︷︷ ︸
l=0,...,k−1
,ψ∈Ck0 (Ω,Symk (Rn))
Noisy and TGV2α-reconstructed images: [Bredies, Kunisch, Pock]
![Page 10: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/10.jpg)
Higher Order Models: Total Generalized VariationNote: For example, TGV2
α reformulated with duality as
TGV2α(I) = min
G
∫Ω
α1|DI −G|+ 1
2α1|∇GT +∇G|
Locally:I DI smooth⇒ G = ∇I ≈ optimal⇒ TGV2
α(I) ≈ α0∫
loc |∇2I|.
I I jumps⇒ G = 0 ≈ optimal⇒ TGV2α(I) ≈ α1
∫loc |∇I|.
Generally:I So computing TGV2
α can be seen as solving aminimization problem,
I in which terms of first and second order are optimallybalanced out,
I and the vector field G represents the smooth part of themeasure DI.
![Page 11: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/11.jpg)
Higher Order Models: Total Generalized VariationNote: For example, TGV2
α reformulated with duality as
TGV2α(I) = min
G
∫Ω
α1|DI −G|+ 1
2α1|∇GT +∇G|
Locally:I DI smooth⇒ G = ∇I ≈ optimal⇒ TGV2
α(I) ≈ α0∫
loc |∇2I|.
I I jumps⇒ G = 0 ≈ optimal⇒ TGV2α(I) ≈ α1
∫loc |∇I|.
Generally:I So computing TGV2
α can be seen as solving aminimization problem,
I in which terms of first and second order are optimallybalanced out,
I and the vector field G represents the smooth part of themeasure DI.
![Page 12: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/12.jpg)
Higher Order Models: Total Generalized VariationNote: For example, TGV2
α reformulated with duality as
TGV2α(I) = min
G
∫Ω
α1|DI −G|+ 1
2α1|∇GT +∇G|
Locally:I DI smooth⇒ G = ∇I ≈ optimal⇒ TGV2
α(I) ≈ α0∫
loc |∇2I|.
I I jumps⇒ G = 0 ≈ optimal⇒ TGV2α(I) ≈ α1
∫loc |∇I|.
Generally:I So computing TGV2
α can be seen as solving aminimization problem,
I in which terms of first and second order are optimallybalanced out,
I and the vector field G represents the smooth part of themeasure DI.
![Page 13: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/13.jpg)
Higer Order Models for Segmentation and Registration
Example:
Objective: Remove the motion in a DCE-MRI sequence so thatindividual tissue points can be investigated.
Challenges: Contrast changes with time, and the images arefar from piecewise constant.
Plan: Segment the images, transform the edge sets to diffusesurfaces using blurring, register the diffuse surfaces withprogressively less blurring.
![Page 14: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/14.jpg)
Higer Order Models for Segmentation and Registration
Example:
Objective: Remove the motion in a DCE-MRI sequence so thatindividual tissue points can be investigated.
Challenges: Contrast changes with time, and the images arefar from piecewise constant.
Plan: Segment the images, transform the edge sets to diffusesurfaces using blurring, register the diffuse surfaces withprogressively less blurring.
![Page 15: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/15.jpg)
Higer Order Models for Segmentation and Registration
Example:
Objective: Remove the motion in a DCE-MRI sequence so thatindividual tissue points can be investigated.
Challenges: Contrast changes with time, and the images arefar from piecewise constant.
Plan: Segment the images, transform the edge sets to diffusesurfaces using blurring, register the diffuse surfaces withprogressively less blurring.
![Page 16: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/16.jpg)
Established Approaches to Segmentation
Method of kmeans:
minpk ,χk
K∑
k=1
∫Ω|pkχk − I|2 : pk ∈ P0, χk : Ω→ 0,1
Minimizing the Mumford-Shah functional:
minI,Γ
∫Ω|I − I|2 + δ−1
∫Ω\Γ|∇I|2 + β|Γ|
or the Ambrosio-Tortorelli phase function approximation:
minI,χ
∫Ω
[|I − I|2 + δ−1|∇I|2χ2 + ε|∇χ|2 + ε−1|1− χ|2
]
![Page 17: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/17.jpg)
Established Approaches to Segmentation
Method of kmeans:
minpk ,χk
K∑
k=1
∫Ω|pkχk − I|2 : pk ∈ P0, χk : Ω→ 0,1
Minimizing the Mumford-Shah functional:
minI,Γ
∫Ω|I − I|2 + δ−1
∫Ω\Γ|∇I|2 + β|Γ|
or the Ambrosio-Tortorelli phase function approximation:
minI,χ
∫Ω
[|I − I|2 + δ−1|∇I|2χ2 + ε|∇χ|2 + ε−1|1− χ|2
]
![Page 18: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/18.jpg)
Established Approaches to Segmentation
Method of kmeans:
minpk ,χk
K∑
k=1
∫Ω|pkχk − I|2 : pk ∈ P0, χk : Ω→ 0,1
Minimizing the Mumford-Shah functional:
minI,Γ
∫Ω|I − I|2 + δ−1
∫Ω\Γ|∇I|2 + β|Γ|
or the Ambrosio-Tortorelli phase function approximation:
minI,χ
∫Ω
[|I − I|2 + δ−1|∇I|2χ2 + ε|∇χ|2 + ε−1|1− χ|2
]
![Page 19: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/19.jpg)
Higher Order Counterparts
Method of kmeans:
minpk ,χk
M∑
m=1
∫Ω|pkχk − I|2 : pk ∈ Pm−1, χk : Ω→ 0,1
Minimizing the Mumford-Shah functional:
minI,Γ
∫Ω|I − I|2 + δ−1
∫Ω\Γ|∇mI|2 + β|Γ|
or the Ambrosio-Tortorelli phase function approximation:
minI,χ
∫Ω
[|I − I|2 + δ−1|∇mI|2χ2 + ε|∇χ|2 + ε−1|1− χ|2
]
![Page 20: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/20.jpg)
Representative Problems with These Methodskmeans leads to staircasing and disconnectedness:
Ambrosio-Tortorelli gives a fuzzy edge function:
![Page 21: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/21.jpg)
Representative Problems with These Methodskmeans leads to staircasing and disconnectedness:
Ambrosio-Tortorelli gives a fuzzy edge function:
![Page 22: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/22.jpg)
Higher Order on Connected Components of Segments
Initial Final
Con
stan
tLi
near
Qua
drat
ic
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Higher Order on Connected Components of SegmentsInitial Final
Con
stan
tLi
near
Qua
drat
ic
![Page 24: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/24.jpg)
Proposed Multiphase Segmentation Approach
Use multiple phase functions χk and model functions Ik.
Estimate I ≈∑K
k=1 Ikχk
through minimizing:
minIk,χk
K∑
k=1
∫Ω
[|Ik − I|2χk
2 + (ε+ ε−1χk2)|∇mIk |2
+δ|∇χk |2 + δ−1|χk (χk − 1)|2]
+ δ−1∫
Ω
[ K∑l=1
χl − 1]2
Combines elements of kmeans and Ambrosio Tortorelli.
![Page 25: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/25.jpg)
Proposed Multiphase Segmentation Approach
Use multiple phase functions χk and model functions Ik.
Estimate I ≈∑K
k=1 Ikχk through minimizing:
minIk,χk
K∑
k=1
∫Ω
[|Ik − I|2χk
2 + (ε+ ε−1χk2)|∇mIk |2
+δ|∇χk |2 + δ−1|χk (χk − 1)|2]
+ δ−1∫
Ω
[ K∑l=1
χl − 1]2
Combines elements of kmeans and Ambrosio Tortorelli.
![Page 26: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/26.jpg)
Proposed Multiphase Segmentation Approach
Use multiple phase functions χk and model functions Ik.
Estimate I ≈∑K
k=1 Ikχk through minimizing:
minIk,χk
K∑
k=1
∫Ω
[|Ik − I|2χk
2 + (ε+ ε−1χk2)|∇mIk |2
+δ|∇χk |2 + δ−1|χk (χk − 1)|2]
+ δ−1∫
Ω
[ K∑l=1
χl − 1]2
Combines elements of kmeans and Ambrosio Tortorelli.
![Page 27: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/27.jpg)
Proposed Multiphase Segmentation Approach
Use multiple phase functions χk and model functions Ik.
Estimate I ≈∑K
k=1 Ikχk through minimizing:
minIk,χk
K∑
k=1
∫Ω
[|Ik − I|2χk
2 + (ε+ ε−1χk2)|∇mIk |2
+δ|∇χk |2 + δ−1|χk (χk − 1)|2]
+ δ−1∫
Ω
[ K∑l=1
χl − 1]2
Combines elements of kmeans and Ambrosio Tortorelli.
![Page 28: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/28.jpg)
Proposed Multiphase Segmentation Approach
Simplification:
minIk
K∑k=1
∫Ω
[|Ik − I|2χk + (ε+ ε−1χk )|∇mIk |2
]with each χk binary and depending upon Il:
χk (x) =
1, |Ik (x)− I(x)| < |Il(x)− I(x)|, ∀l 6= k0, otherwise.
Effects:I ε−1χk |∇mIk |2 ⇒ Ik nearly in Pm−1 on each connected
component of (χk = 1).I ε|∇mIk |2 ⇒ Ik extended naturally outside (χk = 1).I |Ik − I|2χk ⇒ Ik ≈ I on (χk = 1).
![Page 29: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/29.jpg)
Proposed Multiphase Segmentation Approach
Simplification:
minIk
K∑k=1
∫Ω
[|Ik − I|2χk + (ε+ ε−1χk )|∇mIk |2
]with each χk binary and depending upon Il:
χk (x) =
1, |Ik (x)− I(x)| < |Il(x)− I(x)|, ∀l 6= k0, otherwise.
Effects:I ε−1χk |∇mIk |2 ⇒ Ik nearly in Pm−1 on each connected
component of (χk = 1).
I ε|∇mIk |2 ⇒ Ik extended naturally outside (χk = 1).I |Ik − I|2χk ⇒ Ik ≈ I on (χk = 1).
![Page 30: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/30.jpg)
Proposed Multiphase Segmentation Approach
Simplification:
minIk
K∑k=1
∫Ω
[|Ik − I|2χk + (ε+ ε−1χk )|∇mIk |2
]with each χk binary and depending upon Il:
χk (x) =
1, |Ik (x)− I(x)| < |Il(x)− I(x)|, ∀l 6= k0, otherwise.
Effects:I ε−1χk |∇mIk |2 ⇒ Ik nearly in Pm−1 on each connected
component of (χk = 1).I ε|∇mIk |2 ⇒ Ik extended naturally outside (χk = 1).
I |Ik − I|2χk ⇒ Ik ≈ I on (χk = 1).
![Page 31: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/31.jpg)
Proposed Multiphase Segmentation Approach
Simplification:
minIk
K∑k=1
∫Ω
[|Ik − I|2χk + (ε+ ε−1χk )|∇mIk |2
]with each χk binary and depending upon Il:
χk (x) =
1, |Ik (x)− I(x)| < |Il(x)− I(x)|, ∀l 6= k0, otherwise.
Effects:I ε−1χk |∇mIk |2 ⇒ Ik nearly in Pm−1 on each connected
component of (χk = 1).I ε|∇mIk |2 ⇒ Ik extended naturally outside (χk = 1).I |Ik − I|2χk ⇒ Ik ≈ I on (χk = 1).
![Page 32: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/32.jpg)
Computational Investigation of the ApproachExample: K = 2, m = 2, χk & Ik by splitting, χ = χ1.
Given:
Since |I1 − I| < |I2 − I| on and just outside (χ = 1), next curves:
![Page 33: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/33.jpg)
Computational Investigation of the ApproachExample: K = 2, m = 2, χk & Ik by splitting, χ = χ1.
Given:
Since |I1 − I| < |I2 − I| on and just outside (χ = 1), next curves:
![Page 34: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/34.jpg)
Computational Investigation of the Approach(χ = 1) has grown to include (I > 0), but also some (x < δ),
Since |I1 − I| < |I2 − I| in (x < δ), converged result:
![Page 35: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/35.jpg)
Computational Investigation of the Approach(χ = 1) has grown to include (I > 0), but also some (x < δ),
Since |I1 − I| < |I2 − I| in (x < δ), converged result:
![Page 36: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/36.jpg)
Computational Investigation of the Approach
Converged result:
Effects: (K = 2, m = 2, χ = χ1)I ε−1χk |∇mIk |2 ⇒ Ik nearly in Pm−1 on each connected
component of (χk = 1).I ε|∇mIk |2 ⇒ Ik extended naturally outside (χk = 1).I |Ik − I|2χk ⇒ Ik ≈ I on (χk = 1).
![Page 37: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/37.jpg)
Computational Investigation of the Approach
Converged result:
Effects: (K = 2, m = 2, χ = χ1)I ε−1χk |∇mIk |2 ⇒ Ik nearly in Pm−1 on each connected
component of (χk = 1).I ε|∇mIk |2 ⇒ Ik extended naturally outside (χk = 1).I |Ik − I|2χk ⇒ Ik ≈ I on (χk = 1).
![Page 38: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/38.jpg)
Computational Investigation of the ApproachAbove I was piecewise linear, now piecewise quadratic:
Converged result with an unnatural edge in left piece of (I > 0):
![Page 39: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/39.jpg)
Computational Investigation of the ApproachAbove I was piecewise linear, now piecewise quadratic:
Converged result with an unnatural edge in left piece of (I > 0):
![Page 40: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/40.jpg)
Computational Investigation of the Approach
This result motivates changing ε−1χk to αχk where α ε−1
as ε→ 0 (small, i.e., ε need not be tuned).
New simplified approach:
minIk
K∑k=1
∫Ω
[|Ik − I|2χk + (ε+ αχk )|∇mIk |2
]again with each χk binary and depending upon Il:
χk (x) =
1, |Ik (x)− I(x)| < |Il(x)− I(x)|, ∀l 6= k0, otherwise.
(Alternative to choosing α: Increase the order m.)
![Page 41: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/41.jpg)
Computational Investigation of the Approach
This result motivates changing ε−1χk to αχk where α ε−1
as ε→ 0 (small, i.e., ε need not be tuned).
New simplified approach:
minIk
K∑k=1
∫Ω
[|Ik − I|2χk + (ε+ αχk )|∇mIk |2
]again with each χk binary and depending upon Il:
χk (x) =
1, |Ik (x)− I(x)| < |Il(x)− I(x)|, ∀l 6= k0, otherwise.
(Alternative to choosing α: Increase the order m.)
![Page 42: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/42.jpg)
Computational Investigation of the Approach|I1 − I| small near (χ = 1) and |I2 − I| large near (χ = 0):
α < ε−1 ⇒ |I1 − I| < |I2 − I| always near (χ = 1). Finally:
![Page 43: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/43.jpg)
Computational Investigation of the Approach|I1 − I| small near (χ = 1) and |I2 − I| large near (χ = 0):
α < ε−1 ⇒ |I1 − I| < |I2 − I| always near (χ = 1). Finally:
![Page 44: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/44.jpg)
Computational Investigation of the ApproachBut the method can still get stuck:
I I is simply piecewise linear.I 0 ≈ |I1 − I| < |I2 − I| on (χ = 1).I 0 ≈ |I2 − I| < |I1 − I| on (χ = 0).I Result is converged.I Such cases are more likely with K > 2.
![Page 45: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/45.jpg)
Computational Investigation of the ApproachBut the method can still get stuck:
I I is simply piecewise linear.I 0 ≈ |I1 − I| < |I2 − I| on (χ = 1).I 0 ≈ |I2 − I| < |I1 − I| on (χ = 0).
I Result is converged.I Such cases are more likely with K > 2.
![Page 46: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/46.jpg)
Computational Investigation of the ApproachBut the method can still get stuck:
I I is simply piecewise linear.I 0 ≈ |I1 − I| < |I2 − I| on (χ = 1).I 0 ≈ |I2 − I| < |I1 − I| on (χ = 0).I Result is converged.
I Such cases are more likely with K > 2.
![Page 47: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/47.jpg)
Computational Investigation of the ApproachBut the method can still get stuck:
I I is simply piecewise linear.I 0 ≈ |I1 − I| < |I2 − I| on (χ = 1).I 0 ≈ |I2 − I| < |I1 − I| on (χ = 0).I Result is converged.I Such cases are more likely with K > 2.
![Page 48: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/48.jpg)
Determining EdgesExamples motivate starting with χk which respect edges.
Determining non-fuzzy edge set (χ = 0) for χ : Ω→ 0,1:
minχ
∫Ω|I(χ)−I|2 where I(χ) = arg min
I
∫Ω
[|I − I|2χ+ (ε+ αχ)|∇mI|2
]
Here the edge set (χ = 0) = (|x | < δ) can be determinedexplicitly by minimizing with respect to δ. In general?...
![Page 49: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/49.jpg)
Determining EdgesExamples motivate starting with χk which respect edges.
Determining non-fuzzy edge set (χ = 0) for χ : Ω→ 0,1:
minχ
∫Ω|I(χ)−I|2 where I(χ) = arg min
I
∫Ω
[|I − I|2χ+ (ε+ αχ)|∇mI|2
]
Here the edge set (χ = 0) = (|x | < δ) can be determinedexplicitly by minimizing with respect to δ. In general?...
![Page 50: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/50.jpg)
Determining EdgesExamples motivate starting with χk which respect edges.
Determining non-fuzzy edge set (χ = 0) for χ : Ω→ 0,1:
minχ
∫Ω|I(χ)−I|2 where I(χ) = arg min
I
∫Ω
[|I − I|2χ+ (ε+ αχ)|∇mI|2
]
Here the edge set (χ = 0) = (|x | < δ) can be determinedexplicitly by minimizing with respect to δ.
In general?...
![Page 51: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/51.jpg)
Determining EdgesExamples motivate starting with χk which respect edges.
Determining non-fuzzy edge set (χ = 0) for χ : Ω→ 0,1:
minχ
∫Ω|I(χ)−I|2 where I(χ) = arg min
I
∫Ω
[|I − I|2χ+ (ε+ αχ)|∇mI|2
]
Here the edge set (χ = 0) = (|x | < δ) can be determinedexplicitly by minimizing with respect to δ. In general?...
![Page 52: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/52.jpg)
Edge Determination Approach
Edge set is (χ = 0) for χ : Ω→ 0,1,
χ(x) =
1, |Ib(x)− E(x)| < θ|If(x)− E(x)|0, otherwise.
Fuzzy edge function E = |∇Is|,
Is = arg minI
∫Ω
[|I − I|2χ+ (ε+ αχ)|∇mI|2
]Ib and If are background and foreground estimations of E ,
Ib = arg minI
∫Ω
[|I − E |2χ+ (ε+ αχ)|∇I|2
]If = arg min
I
∫Ω
[|I − E |2(1− χ) + (ε+ α(1− χ))|∇I|2
]
![Page 53: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/53.jpg)
Edge Determination Approach
Example:
Computed by splitting, starting with χ = 1, then
· · · → χ→ Is → E → If, Ib, χ → χ→ · · ·
Theorem: There exists a fixed point for this mapping.[Furtinger & Keeling]
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Edge Determination Approach
Example:
Computed by splitting, starting with χ = 1, then
· · · → χ→ Is → E → If, Ib, χ → χ→ · · ·
Theorem: There exists a fixed point for this mapping.[Furtinger & Keeling]
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Segmentation Regularization
Segments are regularized by smoothing χl according to
ψl = arg minψ
∫Ω
[|ψ − χl |2 + δ|∇ψ|2
], l = 1, . . . ,L
and updatingφ(x) = l , ∀x : χl(x) = 1
for redefined
χl(x) =
1, ψl(x) > ψk (x), ∀k 6= l0, otherwise
Resulting segments are smoother with increasing δ.
![Page 56: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/56.jpg)
Segmentation Regularization
Segments are regularized by smoothing χl according to
ψl = arg minψ
∫Ω
[|ψ − χl |2 + δ|∇ψ|2
], l = 1, . . . ,L
and updatingφ(x) = l , ∀x : χl(x) = 1
for redefined
χl(x) =
1, ψl(x) > ψk (x), ∀k 6= l0, otherwise
Resulting segments are smoother with increasing δ.
![Page 57: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/57.jpg)
Computational Investigation of the Approach2D Examples:
![Page 58: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/58.jpg)
Computational Investigation of the Approach2D Examples:
![Page 59: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/59.jpg)
Computational Investigation of the Approach2D Examples:
![Page 60: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/60.jpg)
Computational Investigation of the Approach2D Examples:
![Page 61: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/61.jpg)
Obtaining a SegmentationWith χ in hand, the multiphase approach can be well initialized.For the above examples:
![Page 62: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/62.jpg)
Obtaining a SegmentationWith χ in hand, the multiphase approach can be well initialized.For the above examples:
![Page 63: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/63.jpg)
Defining a SegmentationWith χ in hand, the multiphase approach can be well initialized.For the above examples:
![Page 64: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/64.jpg)
Defining a SegmentationWith χ in hand, the multiphase approach can be well initialized.For the above examples:
![Page 65: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/65.jpg)
Application to Measured Images
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Application to Measured Images
![Page 67: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/67.jpg)
Application to Measured Images
![Page 68: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/68.jpg)
Application to Measured Images
![Page 69: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/69.jpg)
Application to Measured ImagesComparison of the fuzzy edge function
with a higher orderAmbrosio-Tortorelli approach:
and the respective edge functions,
![Page 70: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/70.jpg)
Application to Measured ImagesComparison of the fuzzy edge function with a higher orderAmbrosio-Tortorelli approach:
and the respective edge functions,
![Page 71: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/71.jpg)
Application to Measured ImagesComparison of the fuzzy edge function with a higher orderAmbrosio-Tortorelli approach:
and the respective edge functions,
![Page 72: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/72.jpg)
Application to Measured Images
![Page 73: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/73.jpg)
Application to Measured Images
![Page 74: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/74.jpg)
Application to Measured Images
![Page 75: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/75.jpg)
Application to Measured Images
![Page 76: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/76.jpg)
Registration of Edge SetsFor mapping a Purkinje fiber network system[Furtinger & Keeling]:
Performed using 2D slices,
minu
∫Ω
|Iε0 (Id + u)− Iε1|2 + µ|∇uT +∇u|2
with diffuse images Iε0 and Iε1, providing strong registration force,then ε→ 0.
![Page 77: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/77.jpg)
Registration of Edge SetsFor mapping a Purkinje fiber network system[Furtinger & Keeling]:
Performed using 2D slices,
minu
∫Ω
|Iε0 (Id + u)− Iε1|2 + µ|∇uT +∇u|2
with diffuse images Iε0 and Iε1, providing strong registration force,then ε→ 0.
![Page 78: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/78.jpg)
Registration of Edge SetsFor mapping a Purkinje fiber network system[Furtinger & Keeling]:
Performed using 2D slices,
minu
∫Ω
|Iε0 (Id + u)− Iε1|2 + µ|∇uT +∇u|2
with diffuse images Iε0 and Iε1, providing strong registration force,then ε→ 0.
![Page 79: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/79.jpg)
Registration of Edge SetsBut reducing ε→ 0⇒ argmin = 0! Landscape is
(@ left)
Theorem: There exists a minimizer uε ∈ H1(Ω) whichconverges (subsequentially) in H1(Ω) as ε→ 0.
However, with∫Ω|Iε0 (Id+u)− Iε1|2 →
∫Ω|Iε0 (Id+u)− Iε1|2
/∫Ω
[|Iε0|2 + Iε1|2]
the landscape becomes (@ right).Convergence to Hausdorf distance between edge sets to be shown.
![Page 80: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/80.jpg)
Registration of Edge SetsBut reducing ε→ 0⇒ argmin = 0! Landscape is
(@ left)
Theorem: There exists a minimizer uε ∈ H1(Ω) whichconverges (subsequentially) in H1(Ω) as ε→ 0.
However, with∫Ω|Iε0 (Id+u)− Iε1|2 →
∫Ω|Iε0 (Id+u)− Iε1|2
/∫Ω
[|Iε0|2 + Iε1|2]
the landscape becomes (@ right).Convergence to Hausdorf distance between edge sets to be shown.
![Page 81: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/81.jpg)
Registration of Edge SetsBut reducing ε→ 0⇒ argmin = 0! Landscape is (@ left)
Theorem: There exists a minimizer uε ∈ H1(Ω) whichconverges (subsequentially) in H1(Ω) as ε→ 0.
However, with∫Ω|Iε0 (Id+u)− Iε1|2 →
∫Ω|Iε0 (Id+u)− Iε1|2
/∫Ω
[|Iε0|2 + Iε1|2]
the landscape becomes (@ right).
Convergence to Hausdorf distance between edge sets to be shown.
![Page 82: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/82.jpg)
Registration of Edge SetsBut reducing ε→ 0⇒ argmin = 0! Landscape is (@ left)
Theorem: There exists a minimizer uε ∈ H1(Ω) whichconverges (subsequentially) in H1(Ω) as ε→ 0.
However, with∫Ω|Iε0 (Id+u)− Iε1|2 →
∫Ω|Iε0 (Id+u)− Iε1|2
/∫Ω
[|Iε0|2 + Iε1|2]
the landscape becomes (@ right).Convergence to Hausdorf distance between edge sets to be shown.
![Page 83: Higher Order Multiphase Image Segmentation and Registration](https://reader030.vdocuments.site/reader030/viewer/2022012422/6176d71861d1a111cc17328e/html5/thumbnails/83.jpg)
Thank You!