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


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


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.


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.


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.


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 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!


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)


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.


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))


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.


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


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


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.



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) andproposed (right 42.72) method




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


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.


Conclusion

Any Questions?Thank you very much!


Curvature-Based Registration for Slice Interpolation of Medical Images

Engineering

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