Nonrigid Registration Using Free-Form Deformations

Introduction

This paper presents an algorithm for non-rigid registration of contrast-enhanced breast MR image sequences. The authors propose a model incorporating both global transformations (represented by affine transformation) as well as local transformation (free-form deformation represented using B-splines). Normalized mutual information was used as the similarity measure across images. The authors demonstrate the algorithm’s superior performance compared to the rigid and affine registration techniques.

Method

Given a contrast enhanced breast MRI, the task of image registration is to find the optimal transformation $\boldsymbol{T}$ which maps any point in the dynamic image sequence $I(x,y,z,t)$ at time $t$ to its corresponding point in the reference image $I(x’, y’, z’, t_0)$ at time $t_0$ Since the motion of the breast is non-rigid, there is a need for a combination of a global term and a local term for transformation.

$$ \boldsymbol{T}(x, y, z) = \boldsymbol{T}_{\text{global}}(x, y, z) + \boldsymbol{T}_{\text{local}}(x, y, z) $$

The global motion model can be generally represented as affine transformations with six additional degrees of freedom for scaling and shearing. For a 3D representation, it can be written as

$$ \boldsymbol{T}_{\text{global}}(x, y, z) = \begin{pmatrix} \theta_{11} & \theta_{12} & \theta_{13}\\ \theta_{21} & \theta_{22} & \theta_{23}\\ \theta_{31} & \theta_{32} & \theta_{33} \end{pmatrix} \begin{pmatrix} x\\ y\\ z \end{pmatrix} + \begin{pmatrix} \theta_{14}\\ \theta_{24}\\ \theta_{34} \end{pmatrix} $$

where the parameters represent the 12 degrees of freedom of the global transformation.

For modeling the local motion, a Free-Form Deformation (FFD) based method using B-splines has been chosen. The underlying idea is to control the deformation of an object by manipulating the associated mesh of control points. Cubic B-splines are used since changing a control point affects only the local neighborhood of the control point. $$ \boldsymbol{T}_{\text{local}}(x, y, z) = \sum_{l=0}^{3} \sum_{m=0}^{3} \sum_{n=0}^{3} B_l(u) B_m(v) B_n(w) \phi_{i+l, j+m, k+n} $$

$$ B_0(u) = (1-u)^3/6$$$$ B_1(u) = (3u^3 - 6u^2 + 4)/6$$$$ B_2(u) = (-3u^3 + 3u^2 + 3u + 1)/6$$$$ B_3(u) = u^3/6 $$

The control points act as parameters of the B-spline and the degree of the non-rigid deformation depends on the resolution of the mesh of control points. Therefore, in order to achieve the best results from the trade-off between the degree of the deformation and the computational cost, a hierarchical multi-resolution approach is adopted.

In order to penalize for deviations from smoothness in deformation, a penalty is used to regularize the deformation.

$$ C_{smooth} = \frac{1}{V} \int_0^X \int_0^Y \int_0^Z \bigg[ \Big(\frac{\partial^2\boldsymbol{T}}{\partial x^2}\Big)^2 + \Big(\frac{\partial^2\boldsymbol{T}}{\partial y^2}\Big)^2 + \Big(\frac{\partial^2\boldsymbol{T}}{\partial z^2}\Big)^2 + $$$$ 2\Big(\frac{\partial^2\boldsymbol{T}}{\partial xy}\Big)^2 + 2\Big(\frac{\partial^2\boldsymbol{T}}{\partial yz}\Big)^2 + 2\Big(\frac{\partial^2\boldsymbol{T}}{\partial zx}\Big)^2\bigg] dx \ dy \ dz $$

where $V$ denotes the volume of the image domain.

In order to compare a pre-contrast enhanced image to a post-contrast enhanced image, the similarity criterion used is mutual information, denoting the amount of information that one image contains about another image. In order to nullify any dependency on the amount of image overlap, a normalized variant of mutual information is used.

$$ C_{similarity} (A, B) = \frac{H(A) + H(B)}{H(A, B)} $$

The final cost function consisting of both the local and the global transformation parameters can be written as

$$ C(\Theta, \Phi) = - C_{similarity}\big(I(t_0), \boldsymbol{T}(I(t))\big) + \lambda C_{smooth}(\boldsymbol{T}) $$

where $\lambda$ is the weighting parameter. Although the authors have empirically chosen $\lambda = 0.01$, this choice is immaterial for low resolutions of the control point mesh owing to the intrinsic smooth nature of the B-splines. At higher resolutions, however, the value of $\lambda$ becomes crucial because of the need for regularization.

In order to evaluate the proposed algorithm, two metrics - mean and variance of the squared sum of intensity differences (SSD), and correlation coefficient (CC) of the the average intensities of the images before and after motion were used.