fitting and non-rigid registration Marcel Lthi, Christoph Jud and - - PowerPoint PPT Presentation

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fitting and non-rigid registration Marcel Lthi, Christoph Jud and - - PowerPoint PPT Presentation

Nov 05, 2023 131 likes β€’672 views

A unified approach to shape model fitting and non-rigid registration Marcel Lthi, Christoph Jud and Thomas Vetter University of Basel Shape modeling pipeline Acquisition Registration Modeling Fitting Correspondence Correspondence

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A unified approach to shape model fitting and non-rigid registration

Marcel LΓΌthi, Christoph Jud and Thomas Vetter University of Basel

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Shape modeling pipeline

Modeling

𝑂(𝜈, Ξ£)

Registration Fitting Correspondence Correspondence Acquisition

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Shape modeling pipeline

  • Weak prior assumptions
  • Non-parametric
  • Variational approach
  • Implicit model

(regularization)

  • Strong prior
  • Parametric
  • Standard optimization
  • Explicit probabilistic

model Modeling

𝑂(𝜈, Ξ£)

Registration Fitting Correspondence Correspondence Acquisition

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Shape modeling pipeline

  • Weak prior assumptions
  • Parametric
  • Standard optimization
  • Explicit probabilistic

model

  • Strong prior
  • Parametric
  • Standard optimization
  • Explicit probabilistic

model Modeling

𝑂(𝜈, Ξ£)

Registration Fitting Correspondence Correspondence Acquisition

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Outline

  • Why model fitting
  • Conceptual formulation

– Statistical shape models and Gaussian processes

  • How to make it practical

– Low rank approximation

  • Application to image registration

Goal: Replace registration with model fitting

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Advantage 1: Sampling

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Advantage 2: Posterior models

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Advantage 3: Simple(r) optimization

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  • Example data:

Surfaces in correspondence with Reference Γ𝑆

Statistical Shape Models

Ξ“

1

Ξ“π‘œ …

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  • Example data:

Surfaces in correspondence with Reference Γ𝑆

Statistical Shape Models

Ξ“

1 = Γ𝑆 + 𝑣1

Ξ“π‘œ = Γ𝑆 + π‘£π‘œ …

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  • Estimate mean and sample covariance:

Statistical Shape Models

𝜈 𝑦𝑗 = 1 π‘œ 𝑦𝑗 + 𝑣𝑙 𝑦𝑗 = 𝑦𝑗 + 𝑣𝑙(𝑦𝑗)

𝑙

Ξ£ 𝑦𝑗, π‘¦π‘˜ = 1 π‘œ 𝑦𝑗 + 𝑣𝑙 𝑦𝑗 βˆ’ 𝜈 𝑦𝑗 π‘¦π‘˜ + 𝑣𝑙 π‘¦π‘˜ βˆ’ 𝜈 π‘¦π‘˜

π‘ˆ 𝑙

= 1 π‘œ 𝑣𝑙 𝑦𝑗 βˆ’ 𝑣 𝑦𝑗 𝑣𝑙 π‘¦π‘˜ βˆ’ 𝑣 π‘¦π‘˜

π‘ˆ 𝑙 Γ𝑙(𝑦𝑗) Γ𝑙(𝑦𝑗) Γ𝑙(𝑦𝑗) Reference + mean deformation Covariance of deformations

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Gaussian process view

  • β€œDeformation model” on Γ𝑆

u ∼ 𝐻𝑄 𝑣, Ξ£ 𝑣: Γ𝑆 β†’ ℝ3

  • Shape model:

Ξ“ ∼ Γ𝑆 + 𝑣

  • Model deformations instead of learning them
  • Ξ£(𝑦, 𝑧) can be arbitrary p.d. kernel
  • 𝑙 𝑦, 𝑧 = exp

(βˆ’

𝑦 βˆ’π‘§ 𝜏2 2

) enforces smoothness

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Registration using Gaussian processes

  • Previous work:

– U. Grenander, and M. I. Miller. Computational anatomy: An emerging discipline. Quarterly of applied mathematics, 1998 – B. SchΓΆlkopf, F. Steinke, and V. Blanz. Object correspondence as a machine learning

  • problem. Proceedings of the ICML 2005.

Challenge: Space of deformations is very high dimensional

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Back to statistical models: PCA

  • Mercer’s Theorem:

𝑙 𝑦, 𝑧 = πœ‡π‘—πœšπ‘— 𝑦 πœšπ‘—(𝑧)

π‘œ 𝑗=1

  • Use NystrΓΆm approximation to compute

πœ‡π‘— , 𝜚 𝑗 𝑗=1..𝑛 , (m β‰ͺ n)

  • Low rank approximation of k(x,y)

Statistical model 𝑁[𝛽𝑗, … , 𝛽𝑛]: 𝑣(𝑦) = 𝑣 𝑦 + π›½π‘—βˆšπœ‡ 𝑗 𝜚 𝑗(𝑦)

𝑛 𝑗

, 𝛽𝑗 ∼ 𝑂(0,1)

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Eigenspectrum and smoothness

100

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Advantage 1: Sampling

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Advantage 2: Posterior models

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Advantage 3: Simple(r) optimization

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3D Image registration

Experimental Setup:

  • 48 femur CT images
  • Perform atlas matching
  • Evaluation: dice coefficient with

groundtruth segmentation

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Conclusion

  • Replaced non-rigid registration with model

fitting

  • One concept / one algorithm

– Parametric, generative model – Works for images an surfaces

  • Extreme flexibility in choice of prior

– Any kernel can be used – Future work: Design application specific kernels

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Thank you

Source code available at: www.statismo.org