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Medical Image Registration: Concepts and Implementation

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Affine. Also, skew and scaling. Deformable. Free-form mapping. Registration Framework ... Affine Transformation. Collinearity is preserved. x'=A x T ... – PowerPoint PPT presentation

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Title: Medical Image Registration: Concepts and Implementation


1
Medical Image Registration Concepts and
Implementation
  • Feb 28, 2006
  • Jen Mercer

2
Registration
  • Spatial transform that maps points from one image
    to corresponding points in another image

3
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

4
Spatial Transformation
  • Rigid
  • Rotations and translations
  • Affine
  • Also, skew and scaling
  • Deformable
  • Free-form mapping

5
Registration Framework
6
Transforms
  • xT(xp)T(x,ytx,ty,?)
  • Goal Find parameter values that optimize image
    similarity metric

7
Optimizer
  • Often require derivative of image similarity
    metric (S)

8
Jacobian and Image Gradient
9
Identity Transform
  • Maps every point to itself
  • Only used for testing
  • Fixed set (C) set of points that remain
    unchanged by transform

10
Translation Transform
  • Fixed set is an empty set

11
Scaling Transform
  • Isotropic vs. anisotropic
  • Fixed set is the origin of the coordinates

12
Scaling and Translation
13
Rotation Transform
  • Fixed set is the origin

14
Rotations in Polar Coordinates
15
Optimization
  • Search for value of ? 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

16
Combined Scaling and Rotation
  • Dscaling factor
  • Mcost function
  • Apply transform to a point as

17
Add Translation
  • Find fixed point of transformation
  • Translation (d) is result of scaling and rotation

18
Scaling, Rotation,Translation
  • Parbitrary point
  • Cfixed point of transformation
  • Dscaling factor
  • Trotation angle
  • P and C are complex numbers (xiy) or rei?
  • Store derivates of P in Jacobian matrix for
    optimizer
  • Rigid if D1, otherwise similarity transform

19
Affine Transformation
  • Collinearity is preserved
  • xA x T
  • A is a complex matrix of coefficients
  • With fixed point
  • xA (xC) C
  • A is optimized similar to the scaling factor

20
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

21
Scalars and Versors
  • Quaternion represented by 4 numbers
  • Versor
  • Direction parallel to axis of rotation
  • Rotation angle
  • Norm function of rotation angle
  • Tensor
  • Magnitude

22
Unit Sphere Versor Representation
23
Versor Composition
  • Versor definition (vector quotient)
  • VCB B / C
  • VBA A / B
  • VCA A / C
  • Versor composition
  • VCA VBA ? VCB
  • Not communative

24
Versor Addition
25
Optimization of Versors
  • Versor exponentiation
  • V2 V ? V
  • V V1/2 ? V1/2
  • T(V) ?
  • T(Vn) n?
  • Versor Increment

26
Rigid Transform in 3D
  • Use quaternions instead of phasors
  • PV(P-C)C
  • PVPT, TC-VC
  • Ppoint, VVersor, TTranslation, Cfixed point
  • Transform represented by 6 parameters
  • Three numbers representing versor
  • Three components of fixed coordinate system

27
Numerical Representation of a Versor
  • Right versor

28
Numerical Representation of a Versor
  • -i k ? j
  • -j i ? k
  • -k j ? i
  • Set of elementary quaternions
  • i,j,k eip/2 , ejp/2, ekp/2

29
Numerical Representation of a Versor
  • Any right versor can be represented as
  • vxiyjzk
  • x2y2z21
  • Any generic versor can be represented in terms of
    the right versor parallel to its axis and the
    rotation angle as
  • Vev?

30
Similarity Transform in 3-D
  • Replace versor of rigid transform with quaternion
    to represent rotation and scale changes
  • xQ(x-C)C
  • xQxT, TC-QC

31
Image Interpolators
  • 2 functions
  • Compute interpolated intensity at requested
    position
  • Detect whether or not requested position lies
    within moving-image domain

32
Nearest Neighbor
  • Uses intensity of nearest grid position
  • Computationally cheap
  • Doesnt require floating point calculations

33
Linear Interpolation
  • Computed as the weighted sum of 2n-1 neighbors
  • ndimensionality of image
  • Weighting is based on distance between requested
    position and neighbors

34
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

35
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

36
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

37
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

38
Mean Squares
Smoothness affected by interpolator
39
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 lt1

40
Normalized Correlation
  • -1 added for minimum-seeking optimizers

41
Normalized Correlation
42
Difference Density
  • Each pixels contribution is calculated using
    bell-shaped function
  • f(d) has a maximum of 1 at d0 and minimum of
    zero at d/-infinity
  • d is difference in intensity b/w F and M

43
Difference Density
  • ? controls the rate of drop off
  • Corresponds to the difference in intensity where
    f(d) has dropped by 50

44
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

45
Difference Density
  • Width of peak controlled by ?

46
Multi-modal Volume Registration by Maximization
of Mutual InformationWells W, Viola P, Atsumi
H, Nakajima S, Kikinis R
47
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

48
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

49
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

50
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)

51
Mutual Information
  • Defined in terms of entropies
  • If there are any dependencies, H(A,B)ltH(A)H(B)

52
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

53
Parzen Windowing
  • Used to estimate probability density P(z)
  • Entropy estimated based on P(z)

54
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

55
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?(dI/dT)
  • ? is the learning rate
  • Repeated a fixed of times, or until convergence

56
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
  • NANB50 has been successful
  • The noise introduced by the sampling can
    effectively penetrate small local minima

57
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

58
Initial Condition of MR-CT Registration
59
Final Configuration for MR-CT Registration
60
Initial Condition of MR-PET Registration
61
Final Configuration for MR-PET Registration
62
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

63
3-D Model
Tumor(green), Vessels(red), Ventricles(blue),
Edema (orange)
64
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

65
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

66
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

67
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

68
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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