Curvature-Based Registration for Slice Interpolation of Medical Images
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Transcript of Curvature-Based Registration for Slice Interpolation of Medical Images
Curvature-Based Registration For Slice Interpolation OfMedical 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
Ahmadreza, Zeyun (CEAS-UWM) Slice Interpolation Sept 3, 2014 2 / 27
Introduction
What is slice interpolation?
Using modern image modalities, like CT and MRI, a sequence ofimages are provided from body organs that can be used in building3D models.
The resolution of the images is not identical in all three directions.
This asymmetry in the resolution causes problems such asstep-shaped iso-surfaces and discontinuity in structures in 3Dreconstructed models.
Fig.1 A series of slices of human brain
Ahmadreza, Zeyun (CEAS-UWM) Slice Interpolation Sept 3, 2014 3 / 27
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 isdirectly computed from the intensity values of input images.Linear and cubic spline interpolation methods are two examples of thisgroup.The major advantages of these methods are their simplicity and lowcomputation complexity.They suffer from blurring effects in object boundaries, which makestheir results non-realistic and visually unpleasing.
Object-based interpolation:
In object based methods, extracted information from objects containedin input images are used in order to guide the interpolation into a moreaccurate result.
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Literature Review
Registration based slice interpolation methods are based on two importantassumptions:
The consecutive slices contain similar anatomical features.
The registration method is capable of finding the appropriatetransformation map to match these similar features.
What did we do?
In the present work, we developed a novel image registration basedmethod for slice interpolation using curvature registration method.
Ahmadreza, Zeyun (CEAS-UWM) Slice Interpolation Sept 3, 2014 5 / 27
Image Registration
Image Registration
Given two images, reference (Re) and template (Te), imageregistration is the process of finding a valid and optimal spatialtransformation such that the transformed template image matchesthe reference image.
Two different classes:
Parametric: rigid, affine, landmark-based, FFT-based, spline, etc.Non-parametric: diffusion, fluid, curvature, elastic, etc.
Ahmadreza, Zeyun (CEAS-UWM) Slice Interpolation Sept 3, 2014 6 / 27
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
Ahmadreza, Zeyun (CEAS-UWM) Slice Interpolation Sept 3, 2014 7 / 27
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....
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Motivation
Motivation
Image registration is called a single direction model because thereference image is fixed and only the template image is moving.
This causes asymmetry in the results in such a way that if we fix thetemplate image and try to find the transformation needed for thereference image to match the template image (backward registration),the results of forward and backward registration are not the exactopposite of each other.
Usually in image registration based slice interpolation, first the twoimages are registered and then the in-between slice is reconstructedusing interpolation techniques along the displacement field.
Here we combined these two steps together!
Ahmadreza, Zeyun (CEAS-UWM) Slice Interpolation Sept 3, 2014 9 / 27
Proposed Method
Curvature-based slice interpolation
Assuming 2 input images, R1 and R2,and assuming linear displacement forcorresponding points in the two images:
E [u] = D[R1 −u,R2 u] + αS [u]
whereR u = R(x + u)
,x being the image grid.Fig.2 Known slices (top andbottom) with the unknown
slice (middle)
Ahmadreza, Zeyun (CEAS-UWM) Slice Interpolation Sept 3, 2014 10 / 27
Proposed Method
SSD is defined as:
D = [R1 −u,R2 u] :=1
2||R1(x − u)− R2(x + u)||2L2
=1
2
∫Ω
(R1(x − u)− R2(x + u))2dx
Smoothness term is defined as:
S [u] =1
2
2∑l=1
∫Ω
(∆ul)2dx
where ∆ is the curvature operator, and the summation is computed overtwo dimensions of the images and the integral is computed inside thedomain of the image.
Ahmadreza, Zeyun (CEAS-UWM) Slice Interpolation Sept 3, 2014 11 / 27
Proposed Method
In order to minimize the above mentioned joint functional, computing theGateaux derivative of E [u] and equaling that to zero to find the minimumpoint, an Euler-Lagrange PDE equation is resulted like follows:
f (x , u(x)) + αAcurv [u](x) = 0
whereAcurv [u] = ∆2u
and
f (x , u(x)) = (R2(x + u)− R1(x − u)).(5R1(x − u) +5R2(x + u))
Ahmadreza, Zeyun (CEAS-UWM) Slice Interpolation Sept 3, 2014 12 / 27
Proposed Method
To solve this PDE, a time-stepping implicit iteration method can beconsidered:
∂tuk+1 = f (x , uk(x , t)) + αAcurv [uk+1](x , t)
Using a finite difference approximation of the derivative with time step τand also collecting the grid points in a lexicographical ordering, thediscretized version is as follows:
(In + ατAcurv )Uk+1l = Uk
l + τF kl , l = 1, 2.
Ahmadreza, Zeyun (CEAS-UWM) Slice Interpolation Sept 3, 2014 13 / 27
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
ThereforeUk+1l = IDCT [V ]
whereVi ,j = Gi ,j [1 + ταd2
i ,j ]−1
andG = DCT [Uk
l + τF kl ], l = 1, 2
Ahmadreza, Zeyun (CEAS-UWM) Slice Interpolation Sept 3, 2014 14 / 27
Proposed Method
Algorithm
Initialization: τ, α,X ,U0 = 0, di ,j ;for k = 0 to convergence do
Computing Forces: F kl
% Solving The Linear System %for l = 1 to 2 do: G = DCT [Uk
l + τF kl ], 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 forend for
Uk+1l = IDCT [V ],
end forend forInterpolation: Result = R1(X−UFinal )+R2(X+UFinal )
2
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Results and Discussion
Examples
Some tests were conducted to see the performance of the method incomparison 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 theintensities 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.
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Results and Discussion
Example 1: non-rigid circles
Fig 3. Reconstruction of 3 in-between slices using linear interpolation(bottom) and proposed method (top)
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Results and Discussion
Example 2: Medical images: temporal MRI of heart
Fig 4. Temporal MRI of the heart muscle
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Results and Discussion
Example 2: Medical images: temporal MRI of heart
Fig 5. Results of linear interpolation (left, 84.20) and proposed method(right, 52.52)
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Results and Discussion
Example 2: Medical images: temporal MRI of heart
Fig 6. Close-up of the heart muscle for linear (left) and proposed method(right)
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Results and Discussion
Example 3: Medical images: CT images of brain
Fig 7. Sequence of CT brain images
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Results and Discussion
Example 3: Medical images: CT images of brain
Fig 8. Results of linear (left, 71.67), non-modified (middle, 45.36) andproposed (right 42.72) method
Ahmadreza, Zeyun (CEAS-UWM) Slice Interpolation Sept 3, 2014 22 / 27
Results and Discussion
Example 3: Medical images: CT images of brain
Fig 9. Close-up of the results of linear (left), non-modified (middle) andproposed method
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Results and Discussion
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%)
Ahmadreza, Zeyun (CEAS-UWM) Slice Interpolation Sept 3, 2014 24 / 27
Results and Discussion
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 interpolatealong the displacement fields to just a simple averaging.
Step time, τ is fixed here, but can be more optimized using a linesearch method for faster and more robust convergence.
Regarding α:1 Small α⇒ Non-smooth displacement;2 Big α⇒ More rigid displacement.
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Conclusion
Conclusion
A new registration-based slice interpolation method is introduced;
Linear displacement between corresponding points is integrated in theoptimization process;
Higher accuracy and speed were achieved in comparison to linearinterpolation and non-modified methods, respectively.
C/C++ or GPU implementation of the methods are next steps forachieving higher speeds.
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