Rationale and Objectives
The creation of two-dimensional (2D) ultrasound mosaics is becoming a common clinical practice with a high clinical value. The next step coming along with the increasing availability of 2D array transducers is the creation of three-dimensional mosaics. The correct alignment of multiple ultrasound images is, however, a complex task. In the literature of ultrasound registration, the alignment of two images has been often addressed, but not the alignment of multiple images. Therefore, we propose registration strategies for multiple image alignment and ultrasound specific similarity measures, which are able to cope with problems when aligning ultrasound images.
Materials and Methods
In this study, we investigate the following strategies for multiple image alignment: pairwise registration with a successive Lie group normalization and simultaneous registration, which urges the usage of multivariate similarity measures. We propose alternative multivariate extensions for similarity measures based on a maximum likelihood framework. Moreover, we take the inherent contamination of ultrasound images by speckle patterns into consideration by using alternative noise models based on multiplicative Rayleigh distributed noise. This leads us to ultrasound-specific similarity measures.
Results
We compare the performances of pairwise and simultaneous registration approaches for the mosaicing scenario. Bivariate similarity measures are highly overlap-dependent, so that they rather favor the total overlap of the images than their correct alignment. Using ultrasound-specific bivariate measures leads to better results; however, a local optimum at the total overlap remains. In contrast, the derived multivariate similarity measures have a clear and single optimum at the correct alignment of the volumes.
Conclusion
The experiments indicate that standard, pairwise registration techniques have problems by aligning multiple ultrasound images with partial overlap. We demonstrate that the usage of an ultrasound specific similarity measure leads to better results for pairwise registration. The highest robustness, however, can be achieved by using simultaneous registration with the developed multivariate similarity measures.
At the moment, a paradigm shift is taking place in ultrasound (US) imaging, moving from two-dimensional (2D) to three-dimensional (3D) image acquisition. The next generation of 2D array US transducers with capacitive micromachined US transducer technology could accelerate this shift by offering superior and efficient volumetric imaging at a lower cost.
From a current perspective, the only drawbacks that remain are the limited field of view of the acquired images and the reflectance of the beam from structures with high acoustic impedance causing occlusion. The idea of mosaicing addresses these issues by combining the information of several images taken from different poses. The focus can rest on quality improvement by imaging the same scene from different directions, or the extension of the field of view by stitching together consecutively taken images. Whatever we are interested in, the first step is to calculate the correct global alignment, for which we propose solutions in this report.
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Clinical value of US mosaicing
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Problems Statement
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Mosaicing Strategies
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Pairwise Registration with Lie Normalization
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τi,j=T−1i⋅Tj⋅Ti,j. τ
i
,
j
=
T
i
−
1
·
T
j
·
T
i
,
j
.
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μ2τ(τi,j)=logId(τi,j)T⋅∑−1ττ⋅logId(τi,j). μ
τ
2
(
τ
i
,
j
)
=
log
Id
(
τ
i
,
j
)
T
⋅
∑
τ
τ
−
1
⋅
log
Id
(
τ
i
,
j
)
.
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[Tˆ1,…,Tˆn]=argmin[T1,…,Tn]12∑(i,j)ωi,j⋅μ2τ(τi,j). [
T
1
,
…
,
T
n
]
=
arg
min
[
T
1
,
…
,
T
n
]
1
2
∑
(
i
,
j
)
ω
i
,
j
·
μ
τ
2
(
τ
i
,
j
)
.
with the quality weights ω i,j . These weights model the quality of each pairwise registration. Because we are interested in an automated registration, we use the amount of overlap as an indicator of the registration quality. The final algorithm using the Lie group normalization is stated in Table 1 . The registration is accepted if the total error τt=∑(i,j)ωi,j⋅μ2τ(τi,j) τ
t
=
∑
(
i
,
j
)
ω
i
,
j
·
μ
τ
2
(
τ
i
,
j
) is below a scenario dependent threshold δ. An alternative for using an acceptance criterion based on the absolute error τ t would be to calculate the relative error between two iterations τitert−τiter−1t τ
t
iter
−
τ
t
iter
−
1 .
Table 1
Algorithm for Pairwise Registration with Lie Group Normalization
1. Start with initial global transformations T = {T 1 , … , T n } 2. Do 2.1 Deduce initial pairwise transformations T i,j from T T using T i,j = T j −1 ·T i 2.2 Compute all pairwise registrations T i,j with intensity-based rigid registration 2.3 Estimate new T from calculated T i,j with Lie group normalization in Equation 3 3. While (τ t > δ) 4. Return T T
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Simultaneous Registration
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Table 2
Algorithm for Semi-simultaneous Registration
1. For number of cycles 1.1 For each i in {1, … , n } 1.1.1 Simultaneously register image u__i to { u__1 , … , u__i–1 , u__i+1 , … u__n } for k optimization steps, changing matrix T__i 1.2 END 2. END 3. Return T
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Multivariate Similarity Measures
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logL(T,ε,f)=logP(u|v,T,ε,f)=logP(ε=u(x)−f(v(T(x)))) log
L
(
T
,
ε
,
f
)
=
log
P
(
u
|
v
,
T
,
ε
,
f
)
=
log
P
(
ε
=
u
(
x
)
−
f
(
v
(
T
(
x
)
)
)
)
with P the probability density function. In the work of Viola ( ) and Roche et al ( ), the deduction of the four measures based on this equation is shown by varying the assumptions for the intensity mapping. We are extending this approach to multiple images under the assumption of conditional independent images. The extended MLE denoting the transformed images u↓i=ui(Ti(⋅)) u
i
↓
=
u
i
(
T
i
(
·
)
)
logL(T,ε→,f→)=logP(u↓1u↓2,…,u↓n,ε→,f→) log
L
(
T
,
ε
→
,
f
→
)
=
log
P
(
u
1
↓
u
2
↓
,
…
,
u
n
↓
,
ε
→
,
f
→
)
=logP(ε2=u↓1−f2(u↓2),…,εn=u↓1−fn(u↓n))
log
P
(
ε
2
=
u
1
↓
−
f
2
(
u
2
↓
)
,
…
,
ε
n
=
u
1
↓
−
f
n
(
u
n
↓
)
)
=∑ni=2logP(εi=u↓1−fi(u↓i))
∑
i
=
2
n
log
P
(
ε
i
=
u
1
↓
−
f
i
(
u
i
↓
)
)
with intensity mappings f→=(f2,…,fn) f
→
=
(
f
2
,
…
,
f
n
) and Gaussian noise ε→=(ε2,…,εn) ε
→
=
(
ε
2
,
…
,
ε
n
) . Each summand corresponds to the bivariate formula in Equation 4 , and the deduction of the four similarity measures can therefore be done analogously, as elsewhere ( ). This shows that we directly obtain multivariate extensions of that form by summing up the bivariate measures. In this type of extension, we pick one reference image, in the formulas above u 1 , which works well for the semi-simultaneous registration approach. Setting up a similarity matrix M with the entries M__i,j = SM ( u__i__, u__j ), this corresponds to summing up its first row. An adaptation of this approach to the full-simultaneous registration is obtained by summing up the whole similarity matrix, which can often be limited to the upper triangular part because of the symmetry of the measures. Additionally, the pairs are weighted by an overlap-dependent factor ω i,j emphasizing pairs with high overlap. The final similarity measures are shown in Table 3 .
Table 3
Summary of Bivariate and Multivariate Similarity Measures in Shortened Notation
Pairwise Semi-simultaneous Full-simultaneous Voxel-wise Sum of squared differencesE[(u−v↓)2] E
[
(
u
−
v
↓
)
2
] ∑ni=2ω1,iE[(u1−u↓i)2] ∑
i
=
2
n
ω
1
,
i
E
[
(
u
1
−
u
i
↓
)
2
] ∑i<jωi,jE[(u↓i−u↓j)2] ∑
i
<
j
ω
i
,
j
E
[
(
u
i
↓
−
u
j
↓
)
2
] ∑xk∈ΩωkEi[(μk−u↓i(xk))2] ∑
x
k
∈
Ω
ω
k
E
i
[
(
μ
k
−
u
i
↓
(
x
k
)
)
2
] Normalized cross-correlationE[u˜1⋅v˜↓] E
[
u
˜
1
⋅
v
˜
↓
] ∑ni=2ω1,iE[u˜1⋅u˜↓i] ∑
i
=
2
n
ω
1
,
i
E
[
u
˜
1
⋅
u
˜
i
↓
] ∑i<jωi,jE[u˜↓1⋅u˜↓j] ∑
i
<
j
ω
i
,
j
E
[
u
˜
1
↓
⋅
u
˜
j
↓
] ∑xk∈ΩωkE[u˜↓1⋅u˜↓2…u˜↓3] ∑
x
k
∈
Ω
ω
k
E
[
u
˜
1
↓
⋅
u
˜
2
↓
…
u
˜
3
↓
] Correlation-ratioVar[E(u∣∣v↓)]Var(u) Var
[
E
(
u
|
v
↓
)
]
Var
(
u
) ∑ni=2ω1,iVar[E(u1∣∣u↓i)]Var(u1) ∑
i
=
2
n
ω
1
,
i
Var
[
E
(
u
1
|
u
i
↓
)
]
Var
(
u
1
) ∑i≠jωi,jVar[E(u↓i∣∣uj)]Var(u↓i) ∑
i
≠
j
ω
i
,
j
Var
[
E
(
u
i
↓
|
u
j
)
]
Var
(
u
i
↓
) — Mutual information (MI) MI( u , v ↓ )∑ni=2ωi,jMI(ui,u↓i) ∑
i
=
2
n
ω
i
,
j
MI
(
u
i
,
u
i
↓
) ∑i<jωi,jMI(u↓i,u↓j) ∑
i
<
j
ω
i
,
j
M
I
(
u
i
↓
,
u
j
↓
) ∑xk∈ΩωkH(Pk) ∑
x
k
∈
Ω
ω
k
H
(
P
k
)
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logL(T)=logP(u↓1,u↓2,…,u↓n) log
L
(
T
)
=
log
P
(
u
1
↓
,
u
2
↓
,
…
,
u
n
↓
)
=1|Ω|log∏xk∈ΩPk(u↓1(xk),…,u↓n(xk))
1
|
Ω
|
log
∏
x
k
∈
Ω
P
k
(
u
1
↓
(
x
k
)
,
…
,
u
n
↓
(
x
k
)
)
≈1|Ω|∑xk∈Ωlog∏ni=1Pk(u↓i(xk)) ≈
1
|
Ω
|
∑
x
k
∈
Ω
log
∏
i
=
1
n
P
k
(
u
i
↓
(
x
k
)
)
with the grid Ω. By further assuming a Gaussian distribution of values at each location with mean μ__k and variance σ2k σ
k
2 , the log-likelihood function is
logL⎛⎝⎜T⎞⎠⎟=1|Ω|∑xk∈Ω∑ni=1log⎛⎝⎜12π√σe−12(u↓i(xk)−μk)2σ2k⎞⎠⎟ log
L
(
T
)
=
1
|
Ω
|
∑
x
k
∈
Ω
∑
i
=
1
n
log
(
1
2
π
σ
e
−
1
2
(
u
i
↓
(
x
k
)
−
μ
k
)
2
σ
k
2
)
≈−1|Ω|∑xk∈Ω1σ2k∑ni=1(u↓i(xk)−μk)2⋅ ≈
−
1
|
Ω
|
∑
x
k
∈
Ω
1
σ
k
2
∑
i
=
1
n
(
u
i
↓
(
x
k
)
−
μ
k
)
2
·
We consider this criterion as a voxel-wise extension of SSD because similar assumptions for its pairwise deduction shown elsewhere ( ) hae been used. When not taking the assumption of a Gaussian distribution, Equation 6 can be further developed as was done for the congealing by Zollei et al ( ):
logL(T)=1N∑xk∈Ω∑ni=1logPk(u↓i(xk)) log
L
(
T
)
=
1
N
∑
x
k
∈
Ω
∑
i
=
1
n
log
P
k
(
u
i
↓
(
x
k
)
)
≈∑xk∈ΩH(Pk) ≈
∑
x
k
∈
Ω
H
(
P
k
)
with the sample entropy H. We added the congealing criterion ( ) as an extension of MI to Table 3 , because they are both based on the estimation of the entropy H, although they have different properties. We also use a voxel-wise criterion for NCC that, in our opinion, captures the basic idea of it by multiplying the values at each voxel location of the normalized images u˜i u
˜
i . This is obviously not a rigorous deduction, but rather based on analogies. For all, we added the weighting factor ω k emphasizing locations with a higher number of overlapping images. The usual extensions based on higher dimensional probability density functions are not applicable to mosaicing because they are not flexible enough to allow for varying numbers of overlapping images.
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US-specific Similarity Measures
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SK 1 : Multiplicative Rayleigh Noise
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u(x)=v(T(x))⋅ε u
(
x
)
=
v
(
T
(
x
)
)
·
ε
with the Rayleigh distribution
P(y)=π⋅y2⋅exp(−π⋅y24),y>0 P
(
y
)
=
π
·
y
2
·
exp
(
−
π
·
y
2
4
)
,
y
0
having the variance 2π 2
π . Setting it into the MLE framework, Equation 4 , leads to:
logL(T,ε)=log∏xk⊂Ω1v(T(xk))P(u(xk)v(T(xk))) log
L
(
T
,
ε
)
=
log
∏
x
k
⊂
Ω
1
v
(
T
(
x
k
)
)
P
(
u
(
x
k
)
v
(
T
(
x
k
)
)
)
≈∑xk∈Ωlog(u(xk)v(T(xk))2)−π4u(xk)2v(T(xk))2 ≈
∑
x
k
∈
Ω
log
(
u
(
x
k
)
v
(
T
(
x
k
)
)
2
)
−
π
4
u
(
x
k
)
2
v
(
T
(
x
k
)
)
2
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SK 2 : Signal-dependent Gaussian Noise
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u(x)=v(T(x))+v(T(x))−−−−−−√⋅ε⋅ u
(
x
)
=
v
(
T
(
x
)
)
+
v
(
T
(
x
)
)
·
ε
·
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logL(T,ε)=log∏xk∈Ω1v(T(xk))√exp(−[u(xk)−v(T(xk))]22⋅σ2⋅v(T(xk)) log
L
(
T
,
ε
)
=
log
∏
x
k
∈
Ω
1
v
(
T
(
x
k
)
)
exp
(
−
[
u
(
x
k
)
−
v
(
T
(
x
k
)
)
]
2
2
·
σ
2
·
v
(
T
(
x
k
)
)
=∑xk∈Ω−log[v(T(xk))]−[u(xk)−v(T(xk))]22⋅σ2⋅v(T(xk))
∑
x
k
∈
Ω
−
log
[
v
(
T
(
x
k
)
)
]
−
[
u
(
x
k
)
−
v
(
T
(
x
k
)
)
]
2
2
⋅
σ
2
·
v
(
T
(
x
k
)
)
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CD 1 : Division of Rayleigh Noises
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u(x)=v(T(x))⋅εwithε=ε1ε2 u
(
x
)
=
v
(
T
(
x
)
)
·
ε
with
ε
=
ε
1
ε
2
and the probability density function for a division of Rayleigh noises is:
P(y)=2⋅y(y2+1)2,y>0. P
(
y
)
=
2
⋅
y
(
y
2
+
1
)
2
,
y
0.
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logL(T,ε)=log∏xk∈Ω1v(T(xk))P(u(xk)v(T(xk))) logL
(
T
,
ε
)
=
log
∏
x
k
∈
Ω
1
v
(
T
(
x
k
)
)
P
(
u
(
x
k
)
v
(
T
(
x
k
)
)
)
=log∏xk∈Ω1v(T(xk))2⋅u(xk)v(T(xk))((u(xk)v(T(xk)))2+1)2
log
∏
x
k
∈
Ω
1
v
(
T
(
x
k
)
)
2
⋅
u
(
x
k
)
v
(
T
(
x
k
)
)
(
(
u
(
x
k
)
v
(
T
(
x
k
)
)
)
2
+
1
)
2
=∑xk∈Ωlog2⋅u(xk)v(T(xk))2−2⋅log[(u(xk)v(T(xk)))2+1]
∑
x
k
∈
Ω
log
2
⋅
u
(
x
k
)
v
(
T
(
x
k
)
)
2
−
2
⋅
log
[
(
u
(
x
k
)
v
(
T
(
x
k
)
)
)
2
+
1
]
≈∑xk∈Ωlogu(xk)−logv(T(xk))−log[(u(xk)v(T(xk)))2+1]. ≈
∑
x
k
∈
Ω
log
u
(
x
k
)
−
log
v
(
T
(
x
k
)
)
−
log
[
(
u
(
x
k
)
v
(
T
(
x
k
)
)
)
2
+
1
]
.
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CD 2 : Logarithm of Division of Rayleigh Noises
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logu(x)=log[v(T(x))⋅ε] log
u
(
x
)
=
log
[
v
(
T
(
x
)
)
·
ε
]
=logv(T(x))+logε.
log
v
(
T
(
x
)
)
+
log
ε
.
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ε(x)=exp(u˜(x)+v˜(x)) ε
(
x
)
=
exp
(
u
˜
(
x
)
+
v
˜
(
x
)
)
leading to the log-likelihood function:
logL(T,ε)=log∏xk∈Ωexp(u˜(xk))exp(v˜(xk))⋅P(exp(u˜(xk)−u˜(xk))) log
L
(
T
,
ε
)
=
log
∏
x
k
∈
Ω
exp
(
u
˜
(
x
k
)
)
exp
(
v
˜
(
x
k
)
)
⋅
P
(
exp
(
u
˜
(
x
k
)
−
u
˜
(
x
k
)
)
)
=log∏xk∈Ωexp(u˜(xk))exp(v˜(xk))⋅2⋅exp(u˜(xk)−u˜(xk))[exp(u˜(xk)−u˜(xk))2+1]2
log
∏
x
k
∈
Ω
exp
(
u
˜
(
x
k
)
)
exp
(
v
˜
(
x
k
)
)
·
2
⋅
exp
(
u
˜
(
x
k
)
−
u
˜
(
x
k
)
)
[
exp
(
u
˜
(
x
k
)
−
u
˜
(
x
k
)
)
2
+
1
]
2
=log∏xk∈Ω2⋅exp(2(u˜(xk)−u˜(xk)))[exp(2(u˜(xk)−u˜(xk)))+1]2
log
∏
x
k
∈
Ω
2
⋅
exp
(
2
(
u
˜
(
x
k
)
−
u
˜
(
x
k
)
)
)
[
exp
(
2
(
u
˜
(
x
k
)
−
u
˜
(
x
k
)
)
)
+
1
]
2
≈∑xk∈Ωu˜(xk)−v˜(xk)−log[exp(2(u˜(xk)−v˜(xk)))+1]. ≈
∑
x
k
∈
Ω
u
˜
(
x
k
)
−
v
˜
(
x
k
)
−
log
[
exp
(
2
(
u
˜
(
x
k
)
−
v
˜
(
x
k
)
)
)
+
1
]
.
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Multivariate Extension
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Table 4
Summary of Multivariate Ultrasound-specific Similarity Measures
SK 1 SK 2 ∑i≠jωi,j⋅E[log(uiu2j)−π4u2iu2j] ∑
i
≠
j
ω
i
,
j
⋅
E
[
log
(
u
i
u
j
2
)
−
π
4
u
i
2
u
j
2
] ∑i≠jωi,j⋅E[loguj+(ui−uj)2uj] ∑
i
≠
j
ω
i
,
j
⋅
E
[
log
u
j
+
(
u
i
−
u
j
)
2
u
j
] CD 1 CD 2 ∑i≠jωi,j⋅E[loguiu2j((uiuj)2+1)−2] ∑
i
≠
j
ω
i
,
j
⋅
E
[
log
u
i
u
j
2
(
(
u
i
u
j
)
2
+
1
)
−
2
] ∑i≠jωi,j⋅E[u˜i−u˜j−log(e2(u˜i−u˜j)+1)] ∑
i
≠
j
ω
i
,
j
⋅
E
[
u
˜
i
−
u
˜
j
−
log
(
e
2
(
u
˜
i
−
u
˜
j
)
+
1
)
]
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Results
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Conclusion
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