Nonrigid Registration Using Regularization that Accommodates Local Tissue Rigidity

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Nonrigid Registration Using Regularization that
Accommodates Local Tissue Rigidity
D. Ruan, J. A. Fessler, M. Roberson, J. M.
Balter, M. Kessler 48th AAPM The John S.
Laughlin Science Council Research Symposium 08.
01. 2006
This work is partly sponsored by NIH Grant P01
CA59827
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Registration 101

• Registration studies how to deform one image to
  look like the other(s).
• Registration is useful.
• diagnosis, utilize radiotherapy, simulation.
• Classification
• rigid v.s. nonrigid feature-based v.s.
  intensity-based monomodality v.s. multimodality.

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4D Thorax CT
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So whats the problem?

• Observations
• Good match in general
• Smooth deformation
• Twisted/bending ribs (nonphysical)

Bone Warping
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Why? What has been done?

• Heterogeneous tissue type specific property was
  not accounted for.
• Bone structures should transform more
  rigidly than soft tissues
• Existing work
• Treats different regions independently Little
  97,Huesman 03
• relies on precise segmentation
  boundary issues
• Empirical spatial varying filter Staring 05
• requires knowledge of stiffness map (hard
  thresholding)
• departs from the optimization framework
  hard to control/analyze

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We propose

• Stick with optimization!
• Use spatial varying regularization to
  incorporate tissue specific priors
• where T denotes the deformation field, and
  the objective function E(T) is composed of a
  data fidelity term which measures the intensity
  match and a spatial variant regularization that
  accommodates tissue type dependent elasticity
  information.

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Regularized Registration

• Regularization
• Incorporates prior information on the solution
  into the optimization setup
• Adds a regularization energy to the dissimilarity
  metric
• Two (or more) antagonist goals Bayesian
  interpretation
• Often used to enforce geometric properties

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Conventional Regularization Design

• Geometric regularization depends on the desirable
  property of solution a backtracking process.
• Typical regularization choices
• Smoothness
• Topology preserving Karacali 03
• 
  for
• Volume Preserving Rohlfing 03

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Objective Function Design
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Objective Function Design
Squared difference in pixel intensity
the squared norm for the difference
between reference image and deformed homologous
image when and are looked on as 3D
scaler fields.
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Objective Function Design
Homogenous smoothness Spatial varying rigidity
penalty
spatial varying
weight that specifies the tradeoff between
intensity match and rigidity property. We also
call it local stiffness factor.
regularization term that penalizes the
deviation of the local transformation from being
rigid.
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Design Detail 1local nonrigidity penalty

• Q1 How to quantify rigidity?
• Potential As affineness, volume/angle
  preserving, condition number
• We say The group composed of pure
  translation/rotation and their compositions.
• Q2 How to measure the deviation, what can be
  compute efficiently?
• Potential As residual error for approximation,
  Det(Jacobiant), eigen values
• We say rigid
  is an orthogonal matrix

• sufficient and necessary condition.
• penalty expressed in terms of Jacobian
  components, avoided SVD
• can analytically differentiate, easy to evaluate.

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Design detail 2 stiffness factor
Fact Voxel Intensity (CT) number is highly
correlated with tissue type, hence a good
inference source!
Thorax CT histogram Theoretical CT Values Hsieh
03' Air -1000 Hu Fat Muscle
Bones 2501000 Hu
-10060 Hu

• We use a monotone increasing function to infer
  the degree of desirable rigidity from CT
  numbers.

h tanh simple sharp saturation
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Parameterize Deformation
B-spline parameterization (controlled by knot
distribution and coefficients)
where and
indicates the deformation direction.
A 1D transformation example
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Parameterize Deformation
B-spline parameterization (controlled by knot
distribution and coefficients)
where and
indicates the deformation direction.
A 1D transformation example
Knot
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Optimization method

• Optimization procedure utilizes
• Multi-resolution scheme Kybic 94
• Conjugate gradient descent algorithm

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Experiment Setup

• We register X-ray CT images acquired during
  different breathing phases
• Reference deep inhale breath-hold (80 tidal
  breath) thorax CT
• Homologous exhale
• 512512148 with voxel size 0.20.20.5 cm2
• During preprocessing, crop the reference image to
  size 259175107

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Slice Views of Data Derived Stiffness Map
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Results Intensity Comparison

• Reference v.s. deformed homologous image
• Color Scheme
• Reference blue
• Deformed homologous image green

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Identity (no) Transform
Coronal View
Sagittal View
Axial View
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Affine Transform
Coronal View
Sagittal View
Axial View
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B-Spline Nonrigid Transform no penalty
Coronal View
Sagittal View
Axial View
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B-Spline Nonrigid Transform - Regularized
Coronal View
Sagittal View
Axial View
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Geometry Validation

• CT number is a good reference source we extract
  geometry by thresholding CT number to review
  structure
• Tool AVS isosurface rendering
• Color Scheme
• Reference geometry blue
• Deformed homologous geometry white

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Geometry Comparison B-Spline nonrigid
w/o penalty
w/ proposed penalty
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A Closer Look
Conventional B-Spline
Regularized B-Spline

• Conventional B-Spline offers good local
  intensity match
• Twisted Ribs gt stuck in non-physical local
  minimum, Ouch!
• Regularization achieves desirable compromise
  between intensity match and tissue-type-dependent
  geometry preservation a soft driving force!

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Landmark Validation

• CT data, 11 patient, normal exhale/inhale
• 6 landmarks manually located for each dataset
• Mutual information (MI) used as data fidelity
  metric Multi-modality generalization
• Registration accuracy validated by RMS error of
  deformed landmark location v.s. ground truth
  position.

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Illustration of Landmark Data
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Validation Comparison w/ RMSE

• TPS (expert picked control pts) AAPM 03
• Multi-resolution B-Spline AAPM 04
• Regularized B-Spline

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Conclusion

• A new method accounting for tissue-type dependent
  rigidity with regularization design.
• As an additive penalty, the regularization acts
  as a soft correcting force in bone structure and
  relaxes in elastic regions.
• Inference from intensity value avoids explicit
  segmentation, robust to partial volume effect.
• Design based on Frobenius norm is computationally
  friendly.
• Physical promising results.

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Discussion Future Work

• Visually determined landmarks lie in
  high-gradient regions, could be biased! gt
  Desires more generic validation tools
• Analytical derivation
• motor controlled phantom
• Extension to incorporate direction dependent
  (anisotropic) properties
• What is no CT is available? Alternative inference
  source (open)

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Thanks Please Smile!
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Hard Classification by thesholding
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Handle Sliding Effect? Much Harder
Bone Warping
Avoided w/ Compositing
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• Notations
• pixel location in vector form,
  region of interest
• reference image (in Hu)
• homologous image (in Hu)
• data (in)fidelity
  measure, SSD/MI
• penalty, penalizes the deviation
  of local deformation from being rigid.
• spatial varying stiffness factor,
  3D scaler field.
• deformation field, 3D vector
  field.