mercer registration

Medical Image Registration: Concepts and Implementation

Feb 28, 2006Jen Mercer

Registration

Spatial transform that maps points from one image to corresponding points in another image

Registration Criteria

Landmark-based– Features selected by the user

Segmentation-based– Rigidly or deformably align binary

structures Intensity-based

– Minimize intensity difference over entire image

Spatial Transformation

Rigid– Rotations and translations

Affine– Also, skew and scaling

Deformable– Free-form mapping

Registration Framework

Transforms

x’=T(x|p)=T(x,y|tx,ty,θ) Goal: Find parameter values that

optimize image similarity metric

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Often require derivative of image similarity metric (S)

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Jacobian and Image Gradient

Identity Transform

Maps every point to itself Only used for testing Fixed set (C): set of points that remain

unchanged by transform

Translation Transform

Fixed set is an empty set

Scaling Transform

Isotropic vs. anisotropic Fixed set is the origin of the coordinates

Scaling and Translation

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Rotation Transform Fixed set is the origin

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Rotations in Polar Coordinates

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Optimization

Search for value of θ that minimizes cost function S

Gradient descent algorithm– Update of parameter

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Combined Scaling and Rotation

D=scaling factor M=cost function Apply transform to a point as:

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Add Translation

Find fixed point of transformation Translation (d) is result of scaling and

rotation

Scaling, Rotation,Translation

P=arbitrary point C=fixed point of transformation D=scaling factor Θ=rotation angle P and C are complex numbers (x+iy) or reiθ

Store derivates of P in Jacobian matrix for optimizer

Rigid if D=1, otherwise similarity transform

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Affine Transformation

Collinearity is preserved x’=A x + T A is a complex matrix of coefficients With fixed point

– x’=A (x–C) + C A is optimized similar to the scaling

factor

Quaternions

Quotient of two vectors– Q= A / B

Operator that produces second vector– A= Q B

Represents orientation of one vector with respect to another, as well as ratio of their magnitudes– Versor-rotates vector– Tensor-changes vector magnitude

Scalars and Versors

Quaternion represented by 4 numbers– Versor

• Direction – parallel to axis of rotation• Rotation angle• Norm – function of rotation angle

– Tensor• Magnitude

Unit Sphere Versor Representation

Versor Composition

Versor definition (vector quotient)– VCB = B / C

– VBA = A / B

– VCA = A / C Versor composition

– VCA = VBA ◊ VCB – Not communative

Versor Addition

Optimization of Versors

Versor exponentiation– V2 = V ◊ V

– V = V1/2 ◊ V1/2

– Θ(V) = θ– Θ(Vn) = nθ

Versor Increment

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Rigid Transform in 3D

Use quaternions instead of phasors P’=V*(P-C)+C P’=V*P+T, T=C-V*C P=point, V=Versor, T=Translation, C=fixed

point Transform represented by 6 parameters

– Three numbers representing versor– Three components of fixed coordinate system

Numerical Representation of a Versor Right versor

Numerical Representation of a Versor

-i = k ◊ j -j = i ◊ k -k = j ◊ i Set of elementary quaternions = [i,j,k]= [eiπ/2 , ejπ/2, ekπ/2]

Numerical Representation of a Versor Any right versor can be represented as

– v=xi+yj+zk– x2+y2+z2=1

Any generic versor can be represented in terms of the right versor parallel to its axis and the rotation angle as– V=evθ

Similarity Transform in 3-D

Replace versor of rigid transform with quaternion to represent rotation and scale changes

x’=Q*(x-C)+C x’=Q*x+T, T=C-Q*C

Image Interpolators

2 functions– Compute interpolated intensity at

requested position– Detect whether or not requested position

lies within moving-image domain

Nearest Neighbor

Uses intensity of nearest grid position Computationally cheap Doesn’t require floating point

calculations

Linear Interpolation

Computed as the weighted sum of 2n-1 neighbors

n=dimensionality of image Weighting is based on distance

between requested position and neighbors

B-spline Interpolation

Intensity calculated by multiplying B-spline coefficients with shifted B-spline kernels

Higher spline orders require more pixels to computer interpolated value

Third-order B-spline kernels typically used because good tradeoff between smoothness and computational burden

Metrics

Scalar function of the set of transform parameters for a given fixed image, moving image, and transformation type

Typically samples points within fixed image to compute the measure

Mean Squares

Mean squared difference over all the pixels in an image

Intensities are interpolated for the moving image

For gradient-based optimization, derivative of metric is also required

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Mean Squares

Optimal value of zero Interpolator will affect computation time

and smoothness of metric plot Assumes intensity representing the

same homologous point is in both images

Images must be from same modality

Mean Squares

Smoothness affected by interpolator

Normalized Correlation

Computes pixel-wise cross-correlation between the intensity of the two images, normalized by the square root of the autocorrelation of each image

For two identical images, metric =1 Misalignment, metric <1

Normalized Correlation

-1 added for minimum-seeking optimizers

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Normalized Correlation

Difference Density

Each pixel’s contribution is calculated using bell-shaped function

f(d) has a maximum of 1 at d=0 and minimum of zero at d=+/-infinity

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Difference Density

λ controls the rate of drop off– Corresponds to the difference in intensity

where f(d) has dropped by 50%

Difference Density

Optimal value is N Poor matches = small measure values Approximates the probability density

function of the difference image and maximizes its value at zero

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Width of peak controlled by λ

Multi-modal Volume Registration by Maximization of Mutual Information

Wells W, Viola P, Atsumi H, Nakajima S, Kikinis R

Registering Images from Same Modality Typical measure of error is sum of

squared differences between voxels values

This measure is directly proportional to the likelihood that the images are correctly registered

Same measure is NOT effective for images of different modalities

Relationship Between Images of Different Modalities Example: We should be able to construct a

function F() that predicts CT voxel value from corresponding MRI value

Registration could be evaluated by computing F(MR) and comparing it to the CT image– Via sum of squared differences (or correlation)

In practice, this is a difficult and under-determined problem

Mutual Information

Theory: Since MR and CT both describe the underlying anatomy, there will be mutual information between the two images

Find the best registration by maximizing the information that one image provides about the other

Requires no a priori model of the relationship Assumes that max. info. is provided when the

images are correctly registered

Notation

Reference (fixed) volume: u(x) Test (moving) volume: v(x) x: coordinates of the volume T: transformation from coordinate frame

of reference volume to test volume v(T(x)): test volume voxel associated

with reference volume voxel u(x)

Mutual Information

Defined in terms of entropies

If there are any dependencies, H(A,B)<H(A)+H(B)

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Maximizing Mutual Information

h(v(T(x))) encourages transformations that project u into complex parts of v

Last term of MI eqn contributes when u and v are functionally related

Together, last two terms of MI eqn identify transforms that find complexity and explain it well

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Parzen Windowing Used to estimate probability density

P*(z) Entropy estimated based on P*(z)

Finding Maximum of I(T)

To find maximum of mutual information, calculate its derivative:

Derivative of reference volume is 0, b/c not a function of T

Entropies depend on covariance of Parzen window functions

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Stochastic Maximization of Mutual Information

Similar to gradient descent Steps are taken that are proportional to dI/dT Repeat:

– A {sample of size NA drawn from x}

– B {sample of size NB drawn from x}– T T+λ(dI/dT)

λ is the learning rate Repeated a fixed # of times, or until

convergence

Stochastic Approximation

Uses noisy derivative estimates instead of the true derivative for optimizing a function

Authors have found that technique always converges to a transformation estimate that is close to locally optimal– NA=NB=50 has been successful

The noise introduced by the sampling can effectively penetrate small local minima

MRI-CT Example

Coarse to fine registration Images were smoothed by convolving with

binomial kernel Rigid transform represented by displacement

vectors and quaternions Images were sampled and tri-linear

interpolation was used 5 levels of resolution

– 10000, 5000(*4) iterations

Initial Condition of MR-CT Registration

Final Configuration for MR-CT Registration

Initial Condition of MR-PET Registration

Final Configuration for MR-PET Registration

Application Register 2 MRIs of brain (SPGR and

T2-weighted) to visualize anatomy and tumor– Create at 3-D model for surgical planning

and visualization

3-D Model

Tumor(green), Vessels(red), Ventricles(blue), Edema (orange)

Correlation Conventional correlation aligns two

signals by minimizing a summed quadratic difference between their intensities

If intensity of one signal is negated, then intensities no longer agree, and alignment by correlation will fail

Mutual information is not affected by negation of either signal

Occlusion

Correlation is significantly affected by occlusion because intensity is substantially different

Occlusion will reduce mutual information at alignment– But “mutual information measure degrades

gracefully when subject to partially occluded imagery”

Comparison to Other Methods

Many researchers use surface-based methods to register MRI and PET imagery– Need for reliable segmentation is a drawback

Others use joint entropy to characterize registration– “not robust”: difficulty describing partial overlap– Mutual Information is better because it has a larger

capture range• Additional influence from term that rewards for complexity

in portion of test volume into which reference volume is transformed

Comparison to Other Methods

Woods et al. register MR and PET by minimizing range of PET values associated with a particular MR intensity value– Effective when test volume distribution is

Gaussian– Mutual Information can handle data that is multi-

modal– Woods’ measure is sensitive to noise and outliers

Conclusions

Intensity based techniques work directly with volumetric data (vs. segmentation)

Mutual information does not rely on assumptions about nature of imaging modalities

Have also used this technique to register 3D volumetric images to video images of patients