An Adaptive Gaussian Mixture Model for Non-rigid Image Registration

Abstract

This paper presents a new model based on statistical and variational methods for non-rigid image registration. It can be viewed as an improvement of the intensity-based model whose dissimilarity term is based on minimization of the so-called sum of squared difference(SSD). In the proposed model, it is assumed that the residue of two images can be described as a mixture of Gaussian distributions. Then we incorporate the features of variational regularization methods and expectation-maximization(EM) algorithm, and propose the new model. The novelty is the introduction of two weighting functions and some control parameters in dissimilarity term. The weighting functions could identify low and high contrast objects of the residue automatically and effectively, and the control parameters help to improve the robustness of the model to the choice of regularization parameters. By the introduced parameters and weighting functions, the algorithm could locally adjust the behavior of deformation in different contrast regions. Numerical experimental results of 2D synthetic and 3D MR brain images demonstrate the efficiency and accuracy of the proposed approach compared with other methods.

Appendices

Appendix A

Algorithm 2

(The EM Algorithm)

To maximize the log likelihood (MLE) function $\ln(\mathcal{L}(\varTheta |\mathcal{C}))$, given a realization c of a random vector $\mathcal{C}$ and an initial guess Θ ⁰ for the parameters Θ,

for g=0,1,…, repeat,

1. (E-Step) Compute $\mathcal{Q}(\varTheta ;\varTheta ^{g})$, the conditional expectation of the log likelihood function for the complete data, given the observed c and the MLE approximation Θ ^g. In the discrete case, this takes the form

$$\mathcal{Q}(\varTheta ;\varTheta ^g)=\sum_{\mathbf{y}}\ln(p(\mathbf{c},\mathbf{y}|\varTheta ))p(\mathbf{y}|\mathbf{c},\varTheta ^g).$$

2. (M-Step) Compute a maximizer Θ ^g+1 of $\mathcal{Q}(\varTheta ;\varTheta ^{g})$.

Appendix B

CC is defined by: Given two images I ₁ and I ₂ defined on Ω⊂ℝ^d, we have

where $\langle\cdot,\cdot\rangle_{L_{2}(\varOmega )}$ is the L ₂-inner product, μ(I _l ) and σ(I _l ) are the expectation and standard deviation of image I _l (l=1,2) respectively,