For most image registration problems a smooth one-to-one mapping is desirable, a diffeomorphism. This can be obtained using
priors such as volume preservation, certain kinds of elasticity or both. The key principle is to regularize the strain of
the deformation which can be done through penalization of the eigen values of the stress tensor. We present a computational
framework for regularization of image registration for isotropic hyper elasticity. We formulate an efficient and parallel
scheme for computing the principal stain based for a given parameterization by decomposing the left Cauchy-Green strain tensor
and deriving analytical derivatives of the principal stretches as a function of the deformation, guaranteeing a diffeomorphism
in every evaluation point. Hyper elasticity allows us to handle large deformation without re-meshing. The method is general
and allows for the well-known hyper elastic priors such at the Saint Vernant Kirchoff model, the Ogden material model or Riemanian
elasticity. We exemplify the approach through synthetic registration and special tests as well as registration of different
modalities; 2D cardiac MRI and 3D surfaces of the human ear. The artificial examples illustrate the degree of deformation
the formulation can handle numerically. Numerically the computational complexity is no more than 1.45 times the computational
complexity of Sum of Squared Differences.
[Show abstract][Hide abstract] ABSTRACT: This paper presents a unifying approach for calculating a wide range of popular, but seemingly very different, similarity measures. Our domain is the registration of n-dimensional images sampled on a regular grid, and our approach is well suited for gradient-based optimization algorithms. Our approach is based on local intensity histograms and built upon the technique of Locally Orderless Images. Histograms by Locally Orderless Images are well posed and offer explicit control over the three inherent and unavoidable scales: the spatial resolution, intensity levels, and spatial extent of local histograms. Through Locally Orderless Images, we offer new insight into the relations between these scales. We demonstrate our unification by developing a Locally Orderless Registration algorithm for two quite different similarity measures, namely, Normalized Mutual Information and Sum of Squared Differences, and we compare these variations both theoretically and empirically. Finally, using our algorithm, we explain the empirically observed differences between two popular joint density estimation techniques used in registration: Parzen Windows and Generalized Partial Volume.
IEEE Transactions on Pattern Analysis and Machine Intelligence 06/2013; 35(6):1437-1450. DOI:10.1109/TPAMI.2012.238 · 5.78 Impact Factor
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