3.2 Dense motion tracking
3.2.2 Registration-based motion tracking
For cardiac motion analysis, image registration based methods are widely used to extract cardiac motion [Wang and Amini, 2012]. Non-rigid registration using free-form deformations was proposed by [Rueckert et al., 1999]. In this approach, a multilevel B-splines based FFD is used to describe the location motion, while the global motion is modelled by an affine transformation. A hierarchical registration is achieved by optimising a cost function measuring the similarity between two images as well as the smoothness of the deformation required to align the images. The similarity measure used is based on NMI.
Chandrashekara et al. [Chandrashekara et al., 2004b] used a 4D B-spline motion model and nonrigid image registration proposed in [Rueckert et al., 1999] for the analysis of myocardial deformations. In this approach, the cost function represents the normalised sum of the NMI between the registered SA and LA images. In their work, tag localisation and deformation field reconstruction are performed simultaneously within the optimisation. Another advantage is that no assumptions about the nature of the tag pattern are made so that the information of other image modalities can be easily incorporated.
Perperidis et al. [Perperidis et al., 2005] extended Rueckert’s work [Rueckert et al., 1999] to 4D B-spline based FFD for the inter-subject spatiotemporal alignment of cardiac MR image sequences. The 4D registration is resolved into decoupled temporal registration and spatial registration. The temporal transformation consists of an affine transformation for correcting scaling and translation differences between two sequences and a 1D B-Spline transformation accounting for the temporal misalignment caused by length differences of the cardiac phases. The spatial transformation also consists of an affine transformation and a 3D B-Spline transformation addressing differences in the size, orientation and translation of the hearts and differences in the shape of the hearts respectively. Two different optimisation methods were proposed: one performed both registrations simultaneously; the other performed the temporal registration followed by the spatial registration.
3.2. Dense motion tracking 79
a 4D template image by repeating the ED image and estimates the cardiac motion from this 4D template to all other time points simultaneously. Compared to other registration-based algorithms which estimate the motion by sequentially registering one frame to another, the proposed 4D framework makes the motion recovery temporally smooth and consistent. An attribute vector, consisting of intensity, boundary, and geometric moment invariants (GMIs), is defined for each voxel to reflect the underlying structure at different scales. The similarity used for registration and the distinctiveness for each voxel are calculated based on the attribute vector. The registration is performed in a multi-resolution fashion by first selecting the most distinctive points and gradually adding less-distinctive points.
Peyrat et al. [Peyrat et al., 2010] used a 4D spatiotemporal registration approach based on the Diffeomorphic Demons algorithm for intersubject registration. The 4D registration is decoupled into two components: temporal registration and spatial registration. The temporal registration is performed using global cardiac physiological state parameters: First, it temporally aligns two sequences with a linear transformation based on the ECG and the transformation is then refined by temporally aligning the volume curves. This temporal alignment ensures that the following intersubject spatial registration is performed between frames at corresponding states during the cardiac cycle and thus with similar geometrics. For the spatial registration illustrated in Figure3.4, Peyrat et al. first applied Diffeomorphic Demons to find a dense trajectory of points in each of the two sequences; and then these trajectories are used as constraints to compute the intersubject 3D spatial transformation. So they formulated the 4D spatial registration as a multichannel registration of 3D images and solved it with Multichannel Diffeomorphic Demons.
Metz et al. [Metz et al., 2011] proposed a method based on a 3D (2D+time) or 4D (3D+time) free-form B-spline deformation model to extract cardiac motion. In order to avoid a bias towards a specifically chosen reference time point, the registration is performed directly on the dynamic image. They used a stochastic gradient descent method with adaptive step size estimation for constrained optimization. The similarity metric is computed to minimize the intensity variances over time. Optionally, cyclic motion constraints can be imposed for model building.
Figure 3.4 – (a) Discretization of the 4D spatial registration with the spatial transformations
Sj between the sequences at time tj, and the trajectories Mj,k and M
0
j,k between frames at times
tj and tk. The intersequence transformation Sj maps the reference Ij to the target I
0
j at time tj
knowing the trajectories of points given by the intrasequence motion transformations Mj,j+1 and
Mj,j+10 between the times tj and tj+1. (b) Under trajectory constraints, the 4D registration can
be parametrized by a single spatial transformation Sj4D and thus formulated as a multichannel
3D registration problem. Once Sj4D is estimated, the other transformations Sk4D can be computed
from Sj4D with the trajectory constraints: Sk4D =Mj,k0 o S4Dj o Mj,k−1 [Peyrat et al., 2010].
Yigitsoy et al. [Yigitsoy et al., 2011] also applied FFDs based B-splines [Rueckert et al., 1999] to model the motion, with an additional dimension along the temporal direction. This groupwise registration approach ensures the smoothness and consistency in time. The authors used accumulated pairwise estimates (APE) as a similarity measure, which was introduced as a suitable metric for groupwise framework. The APE computes the sum of the mean squared intensity differences between each of all combination pairs of warped images. To deal with the high-dimensional simultaneous registration, they combined a gradient-based optimization procedure with a stochastic sampling in the spatial domain, while only 10% of the total pixels are randomly selected in each iteration for calculating the image similarity.
The Temporal Diffeomorphic Free Form Deformation (TDFFD) [De Craene et al., 2012] extended the FFD registration technique [Rueckert et al., 1999] by summing spatiotemporal B-spline kernels to model the velocity field and hence enforced time consistency. The image similarity metric is calculated as the sum of squared differences between the intensities of each of all time frames and a reference frame. In addition, the incompressibility of myocardial tissue