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Medical Image Registration Kumar Rajamani

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

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x y z Time axis : aging, development … … … … ……… Anatomy Label Function Pathology, Histology … Cross-subject, same modality Same subject, cross modality … Common coordinate system Image Registration Data Fusion Atlases for population studies Motion Correction for image reconstruction Surgery Planning

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Affine Scale / Skew Example Registration Transformations Rigid Deformable Similarity Rigid + Scale Original

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5 Image comparison Difference before registration Difference after registration Registered moving image

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Before Registration B-Splines based Registration After Registration *GE Owned

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B-Splines based Registration ( Hill et al.)

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Dense NR Registration ( Hill et al.)

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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

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Spatial Transformation Rigid –Rotations and translations Affine –Also, skew and scaling Deformable –Free-form mapping

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Registration framework pieces 2 input images, fixed and moving Metric - determines the “fitness” of the current registration iteration Optimizer - adjusts the transform in an attempt to improve the metric Interpolator - applies transform to image and computes sub-pixel values 11

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Registration — General Problem Problem: Find transform T such that image f[p] matches m[T(p)] Domain of an individual subject Common coordinate system fm transformmetric optimizer Interpolate

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Registration Framework

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Transforms x’=T(x|p)=T(x,y|t x,t y,θ) Goal: Find parameter values that optimize image similarity metric

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Optimizer Often require derivative of image similarity metric (S)

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Understanding the Transform Jacobian J shows how changing p shifts a transformed point in the moving image space. This allows efficient use of a pre-computed moving-image gradient to infer changes in corresponding-pixel intensities for changes in p Now we can update dS/dp by just updating J 16

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Jacobian and Image Gradient

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Transforms Before we discuss specific transforms, let’s discuss the… Fixed Set = the set of points (i.e. physical coordinates) that are unchanged by the transform The fixed set is a very important property of a transform 18

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Identity Transform Maps every point to itself Only used for testing Fixed set = everything (i.e., the entire space)

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Translation Transform Fixed set = empty set Translation can be closely approximated by: –Small rotation about distant origin, and/or… –Small scale about distant origin –Both of these do have a fixed point Optimizers will frequently (accidently) do translation by using either rotation or scale –This makes the optimization space harder to use –The final transform may be harder to understand 20

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Translation Transform Fixed set is an empty set

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Scaling Transform Isotropic vs. anisotropic Fixed set is the origin of the coordinates C C

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Scaling and Translation

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2D Rotation Transform Rotation transforms are typically specific to either 2D or 3D Fixed set = origin = “center” = C 24 C C

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Rotation Transform Fixed set is the origin

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Rotations in Polar Coordinates

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Transforms Affine Transform S=Id reduces to Rigid case, ( e.g. Head motion correction ) Affine Transforms also encode scale/skew factors Scaling Rotation Translation Original Rigid

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Optimization Search for value of θ that minimizes cost function S Gradient descent algorithm –Update of parameter –G is the variation from the gradient of the cost function – is step length of algorithm

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Sequence of translations and metric values at each iteration of the optimizer

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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

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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 re iθ 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

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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

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Scalars and Versors Quaternion represented by 4 numbers –Versor Direction – parallel to axis of rotation Rotation angle Norm – function of rotation angle –Tensor Magnitude

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Unit Sphere Versor Representation

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Versor Composition Versor definition (vector quotient) –V CB = B / C –V BA = A / B –V CA = A / C Versor composition –V CA = V BA ◊ V CB –Not communative

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Versor Addition

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Optimization of Versors Versor exponentiation –V 2 = V ◊ V –V = V 1/2 ◊ V 1/2 –Θ(V) = θ –Θ(V n ) = 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

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Numerical Representation of a Versor Right versor

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Numerical Representation of a Versor -i = k ◊ j -j = i ◊ k -k = j ◊ i Set of elementary quaternions = [i,j,k]= [e iπ/2, e jπ/2, e kπ/2 ]

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Numerical Representation of a Versor Any right versor can be represented as –v=xi+yj+zk –x 2 +y 2 +z 2 =1 Any generic versor can be represented in terms of the right versor parallel to its axis and the rotation angle as –V=e vθ

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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

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Image Interpolators 2 functions –Compute interpolated intensity at requested position –Detect whether or not requested position lies within moving-image domain

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

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Linear Interpolation Computed as the weighted sum of 2 n-1 neighbors n=dimensionality of image Weighting is based on distance between requested position and neighbors

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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

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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

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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

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Mean Squares Smoothness affected by interpolator

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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

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Normalized Correlation -1 added for minimum-seeking optimizers

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Normalized Correlation

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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 d is difference in intensity b/w F and M

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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%

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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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Difference Density Width of peak controlled by λ

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Multi-modal Volume Registration by Maximization of Mutual Information Wells W, Viola P, Atsumi H, Nakajima S, Kikinis R

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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

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Figure 8.9 from the ITK Software Guide v 2.4, by Luis Ibáñez, et al. 62 MI Inputs T1 MRIProton density MRI

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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

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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

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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)

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Mutual Information Defined in terms of entropies If there are any dependencies, H(A,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)

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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 N A drawn from x} –B {sample of size N B drawn from x} –T T+λ(dI/dT) λ is the learning rate Repeated a fixed # of times, or until convergence

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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 –N A =N B =50 has been successful The noise introduced by the sampling can effectively penetrate small local minima

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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

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Initial Condition of MR-CT Registration

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Final Configuration for MR-CT Registration

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Initial Condition of MR-PET Registration

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Final Configuration for MR-PET Registration

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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

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3-D Model Tumor(green), Vessels(red), Ventricles(blue), Edema (orange)

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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

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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”

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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

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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

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Registration Tools/ Softwares Elastix (ITK) ANTS (ITK) Plastimatch Slicer3D (ITK) MevisLab Fiji

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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

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April 2011 87 Kumar Rajamani GE Global Research kumartr@gmail.com

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