Cardiac motion estimation from medical images: a regularisation framework applied on pairwise image registration displacement fields

Figure 1. Schematic diagram of the proposed regularisation. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_{n-1}\rightarrow t_n$$\end{document}tn-1→tn is image registration from time n-1 to time n. Forward marching accumulates image registration displacements to obtain coordinate change over time. Initialisation then sample these coordinates to fit into the b-spline Fourier model (BSF).

Figure 2. (a) Gradient of volume over time curve for the fetal1 RV at various values of N at grid size of 11 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document}× 11 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document}× 11; (b) Gradient of volume over time curve for the fetal1 RV at various grid sizes when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=4$$\end{document}N=4.

Figure 3. CMAC volunteer 4 tracked geometry from the various steps: (a) forward marching (b), initialisation (c), consistency corrected.

Figure 4. Box plot of the averaged Euclidean distance error (Eu) of all patients, for the various methods applied to the CMAC data. The various steps from our proposed framework, using Elastix as the base pairwise registration: FM Forward Marching with fully Eulerian (Eul), Lagrangian (Lag) and Quadratic weights, Init initialisation, CC consistency correction, and the motion estimation algorithms of CMAC participants: INRIA Inria Asclepios, MEVIS Fraunhofer MEVIS, and UPF Universitat Pompeu Fabra. The whiskers are the lowest and highest datum at 1.5 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document}× of interquartile range assuming normally distributed data. Data beyond the whiskers are considered outliers, labelled with ‘+’. Their average (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt}…

Figure 5. (a) Volume of the fetal1 RV obtained from manual segmentation compared to the proposed method, with and without additional inputs of the additional displacement field from the end diastolic to end systolic time frames; (b) temporal gradient of volume over time curve for the same set of data; (c) surface contour depicting temporally averaged Euclidean distance error between the motion estimated mesh and the manually segmented mesh.

Figure 6. Fetal1 right ventricle geometry overlaid on top of ultrasound at different time points for (a) consistency corrected data without additional input, (b) consistency corrected data with additional vector inputs, (c) manually segmented geometry.

Figure 7. (a) Motion estimated zebrafish1 ventricle on top of its confocal microscope image at different time points. (b)Temporal averaged Euclidean distance error between motion estimated surface and manually segmented surface of zebrafish1 ventricle. The surface color contour represents the magnitude of Euclidean distance error; (c) volume changes of zebrafish1 ventricle.

Figure 8. (a) Chick1 embryo ventricle geometry, obtained from Additional Input, overlaid on top of its ultrasound image at different time points; (b) temporal averaged Euclidean distance difference between the mesh obtained from the proposed framework and manually segmented mesh for chick1 ventricle. The surface color contour relates to the magnitude of Euclidean distance error between motion estimated surface and manual segmented surface; (c) volume over time curve for the chick1 ventricle.