Curvature-Based Registration for Slice Interpolation of Medical Images

Curvature-Based Registration For Slice Interpolation Of
Medical Images
Ahmadreza Baghaie, Zeyun Yu
College Of Engineering And Applied Science, University Of Wisconsin - Milwaukee
Sept 3, 2014
Ahmadreza, Zeyun (CEAS-UWM) Slice Interpolation Sept 3, 2014 1 / 27

Content
1 Introduction
2 Literature Review
3 Image Registration
4 Motivation
5 Proposed Method
6 Results and Discussion
7 Conclusion

Introduction
What is slice interpolation?
Using modern image modalities, like CT and MRI, a sequence of
images are provided from body organs that can be used in building
3D models.
The resolution of the images is not identical in all three directions.
This asymmetry in the resolution causes problems such as
step-shaped iso-surfaces and discontinuity in structures in 3D
reconstructed models.
Fig.1 A series of slices of human brain

Literature Review
What has been done before?
In general, methods of interpolation can be divided into two groups:
Scene-based interpolation:
In scene based interpolation methods, the final result of interpolation is
directly computed from the intensity values of input images.
Linear and cubic spline interpolation methods are two examples of this
group.
The major advantages of these methods are their simplicity and low
computation complexity.
They suffer from blurring effects in object boundaries, which makes
their results non-realistic and visually unpleasing.
Object-based interpolation:
In object based methods, extracted information from objects contained
in input images are used in order to guide the interpolation into a more
accurate result.

Literature Review
Registration based slice interpolation methods are based on two important
assumptions:
The consecutive slices contain similar anatomical features.
The registration method is capable of ﬁnding the appropriate
transformation map to match these similar features.
What did we do?
In the present work, we developed a novel image registration based
method for slice interpolation using curvature registration method.

Image Registration
Image Registration
Given two images, reference (Re) and template (Te), image
registration is the process of finding a valid and optimal spatial
transformation such that the transformed template image matches
the reference image.
Two different classes:
Parametric: rigid, affine, landmark-based, FFT-based, spline, etc.
Non-parametric: diffusion, fluid, curvature, elastic, etc.

Image Registration
Image registration is an ill-posed inverse problem.
The process has three components:
1 a deformation model;
2 an objective function to be optimized;
3 an optimization method.
A general objective function can be deﬁned as:
E[u] = D[Re, Te ◦ u] + αS

Image Registration
Distance measure:
Sum of Squared Diﬀerences (SSD);
Mutual Information (MI);
Cross-Correlation (CC);
Normalized Gradient Fields (NGF);
etc....
Smoothness term:
Elastic registration;
Fluid registration;
Diﬀusion registration;
Curvature registration;
etc....

Motivation
Motivation
Image registration is called a single direction model because the
reference image is fixed and only the template image is moving.
This causes asymmetry in the results in such a way that if we fix the
template image and try to find the transformation needed for the
reference image to match the template image (backward registration),
the results of forward and backward registration are not the exact
opposite of each other.
Usually in image registration based slice interpolation, first the two
images are registered and then the in-between slice is reconstructed
using interpolation techniques along the displacement field.
Here we combined these two steps together!

Proposed Method
Curvature-based slice interpolation
Assuming 2 input images, R1 and R2,
and assuming linear displacement for
corresponding points in the two images:
E[u] = D[R1 ◦ −u, R2 ◦ u] + αS[u]
where
R ◦ u = R(x + u)
,x being the image grid.
Fig.2 Known slices (top and
bottom) with the unknown
slice (middle)

Proposed Method
SSD is deﬁned as:
D = [R1 ◦ −u, R2 ◦ u] :=
1
2
||R1(x − u) − R2(x + u)||2
L2
=
1
2 Ω
(R1(x − u) − R2(x + u))2
dx
Smoothness term is deﬁned as:
S[u] =
1
2
2
l=1 Ω
(∆ul )2
dx
where ∆ is the curvature operator, and the summation is computed over
two dimensions of the images and the integral is computed inside the
domain of the image.

Proposed Method
In order to minimize the above mentioned joint functional, computing the
Gateaux derivative of E[u] and equaling that to zero to ﬁnd the minimum
point, an Euler-Lagrange PDE equation is resulted like follows:
f (x, u(x)) + αAcurv
[u](x) = 0
where
Acurv
[u] = ∆2
u
and
f (x, u(x)) = (R2(x + u) − R1(x − u)).( R1(x − u) + R2(x + u))

Proposed Method
To solve this PDE, a time-stepping implicit iteration method can be
considered:
∂tuk+1
= f (x, uk
(x, t)) + αAcurv
[uk+1
](x, t)
Using a ﬁnite diﬀerence approximation of the derivative with time step τ
and also collecting the grid points in a lexicographical ordering, the
discretized version is as follows:
(In + ατAcurv
)Uk+1
l = Uk
l + τFk
l , l = 1, 2.

Proposed Method
Assuming m × n pixel images as input, Acurv will be of the size mn × mn!
Therefore a fast Discrete Cosine Transform (DCT) solver is used.
The set of coeﬃcients for the DCT solver:
di,j = −4 + 2cos
(i − 1)π
m
+ 2cos
(j − 1)π
n
Therefore
Uk+1
l = IDCT[V ]
where
Vi,j = Gi,j [1 + ταd2
i,j ]−1
and
G = DCT[Uk
l + τFk
l ], l = 1, 2

Proposed Method
Algorithm
Initialization: τ, α, X, U0 = 0, di,j ;
for k = 0 to convergence do
Computing Forces: Fk
l
% Solving The Linear System %
for l = 1 to 2 do: G = DCT[Uk
l + τFk
l ], l = 1, 2
for i = 1 to m do:
for j = 1 to n do: Vi,j = Gi,j [1 + ταd2
i,j ]−1,
end for
end for
Uk+1
l = IDCT[V ],
end for
end for
Interpolation: Result = R1(X−UFinal )+R2(X+UFinal )
2

Results and Discussion
Examples
Some tests were conducted to see the performance of the method in
comparison to an intensity based method: Linear Interpolation;
Even though there are more sophisticated intensity based methods,
like cubic spline, still, since these methods only work based on the
intensities and not the objects and their changes between frames,
they act the same as linear interpolation.
Tests:
1 Synthetic images: non-rigid circles;
2 Medical images: temporal MRI of heart;
3 Medical images: CT images of brain;
4 Medical images: CT database of brain.

Example 1: non-rigid circles
Fig 3. Reconstruction of 3 in-between slices using linear interpolation
(bottom) and proposed method (top)

Example 2: Medical images: temporal MRI of heart
Fig 4. Temporal MRI of the heart muscle

Fig 5. Results of linear interpolation (left, 84.20) and proposed method
(right, 52.52)

Fig 6. Close-up of the heart muscle for linear (left) and proposed method
(right)

Example 3: Medical images: CT images of brain
Fig 7. Sequence of CT brain images

Fig 8. Results of linear (left, 71.67), non-modiﬁed (middle, 45.36) and
proposed (right 42.72) method

Fig 9. Close-up of the results of linear (left), non-modiﬁed (middle) and
proposed method

Example 4: Medical images: CT database of brain
79 slices of size 217 × 181 pixel;
interpolation of the evenly numbered slices;
more than 2.5 % improvement in mean MSD.
Method Linear Non-modiﬁed Proposed
MSD 118.7852 56.0765(52.78%) 54.6450(53.99%, 2.56%)

Discussion
Moving both frames:
1 Reduces the computational time for convergence;
2 Can avoid local minima caused by large displacements.
Integrating the linear interpolation, reduces the need to interpolate
along the displacement ﬁelds to just a simple averaging.
Step time, τ is ﬁxed here, but can be more optimized using a line
search method for faster and more robust convergence.
Regarding α:
1 Small α ⇒ Non-smooth displacement;
2 Big α ⇒ More rigid displacement.

Conclusion
Conclusion
A new registration-based slice interpolation method is introduced;
Linear displacement between corresponding points is integrated in the
optimization process;
Higher accuracy and speed were achieved in comparison to linear
interpolation and non-modiﬁed methods, respectively.
C/C++ or GPU implementation of the methods are next steps for
achieving higher speeds.

Conclusion
Any Questions?
Thank you very much!

Curvature-Based Registration for Slice Interpolation of Medical Images

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