A unified approach to shape model fitting and non-rigid registration Marcel LΓΌthi, Christoph Jud and Thomas Vetter University of Basel
Shape modeling pipeline Acquisition Registration Modeling Fitting Correspondence Correspondence π(π, Ξ£)
Shape modeling pipeline Acquisition Registration Modeling Fitting Correspondence Correspondence π(π, Ξ£) β’ Strong prior β’ Weak prior assumptions β’ Parametric β’ Non-parametric β’ Standard optimization β’ Variational approach β’ Explicit probabilistic β’ Implicit model model (regularization)
Shape modeling pipeline Acquisition Registration Modeling Fitting Correspondence Correspondence π(π, Ξ£) β’ Strong prior β’ Weak prior assumptions β’ Parametric β’ Parametric β’ Standard optimization β’ Standard optimization β’ Explicit probabilistic β’ Explicit probabilistic model model
Outline Goal: Replace registration with model fitting β’ Why model fitting β’ Conceptual formulation β Statistical shape models and Gaussian processes β’ How to make it practical β Low rank approximation β’ Application to image registration
Advantage 1: Sampling
Advantage 2: Posterior models
Advantage 3: Simple(r) optimization
Statistical Shape Models β’ Example data: Surfaces in correspondence with Reference Ξ π β¦ Ξ Ξ π 1
Statistical Shape Models β’ Example data: Surfaces in correspondence with Reference Ξ π β¦ Ξ 1 = Ξ π + π£ 1 Ξ π = Ξ π + π£ π
Statistical Shape Models β’ Estimate mean and sample covariance: Ξ π (π¦ π ) Reference + mean deformation π π¦ π = 1 π π¦ π + π£ π π¦ π = π¦ π + π£ π (π¦ π ) π Ξ π (π¦ π ) Ξ π (π¦ π ) Ξ£ π¦ π , π¦ π = 1 π π π¦ π + π£ π π¦ π β π π¦ π π¦ π + π£ π π¦ π β π π¦ π π Covariance of deformations = 1 π π π£ π π¦ π β π£ π¦ π π£ π π¦ π β π£ π¦ π π
Gaussian process view β’ βDeformation modelβ on Ξ π u βΌ π»π π£, Ξ£ π£: Ξ π β β 3 β’ Shape model: Ξ βΌ Ξ π + π£ Model deformations instead of learning them β’ Ξ£(π¦, π§) can be arbitrary p.d. kernel β’ 2 π¦ βπ§ π π¦, π§ = exp (β ) enforces smoothness β’ π 2
Registration using Gaussian processes β’ Previous work: β U. Grenander, and M. I. Miller. Computational anatomy: An emerging discipline. Quarterly of applied mathematics, 1998 β 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
Back to statistical models: PCA Statistical model π[π½ π , β¦ , π½ π ] : π π π (π¦) π π£(π¦) = π£ π¦ + π½ π βπ , π½ π βΌ π(0,1) π β’ Mercerβs Theorem: π π π¦, π§ = π π π π π¦ π π (π§) π=1 β’ Use NystrΓΆm approximation to compute , π π π=1..π , (m βͺ n) π π β’ Low rank approximation of k(x,y)
Eigenspectrum and smoothness 0 100
Advantage 1: Sampling
Advantage 2: Posterior models
Advantage 3: Simple(r) optimization
3D Image registration Experimental Setup: β’ 48 femur CT images β’ Perform atlas matching β’ Evaluation: dice coefficient with groundtruth segmentation
Conclusion β’ Replaced non-rigid registration with model fitting β’ One concept / one algorithm β Parametric, generative model β Works for images an surfaces β’ Extreme flexibility in choice of prior β Any kernel can be used β Future work: Design application specific kernels
Thank you Source code available at: www.statismo.org
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