2011 IEEE NUCLEAR SCIENCE SYMPOSIUM CONFERENCE RECORD 1 Mass-Preserving Motion Correction of PET: Displacement Field vs. Spline Transformation Fabian Gigengack, Lars Ruthotto, Martin Burger, Carsten H. Wolters, Xiaoyi Jiang, Senior Member, IEEE, and Klaus P. Sch¨ afers Abstract—In positron emission tomography (PET), motion due to the cardiac and respiratory cycle causes blurred images. Different approaches for motion correction in PET vary in the general concept (optical flow or image registration) or, e.g., in the discretization of motion. Given our mass-preserving transformation model, we evaluate different motion models in this work: dense displacement field (compute for each voxel an individual displacement) vs. spline transformation (i.e. free-form deformation). Thereby a focus is put on the parametrization of the spline transformations where we optimize the number of spline coefficients and the regularization parameter. We make a quantitative comparison of the motion estimates of the different motion models based on data of the established XCAT software phantom. For both motion models (displacement field (DF) and spline transformation (ST)) the registration results are evaluated by 1) the total processing time and 2) the Euclidean distance to the ground-truth vectors provided by the XCAT phantom. We found that the spline transformation model is superior to the displacement field strategy in terms of processing time and accuracy. Index Terms—motion correction, mass-preservation, image registration, spline transformation, hyperelastic regularization, PET I. I NTRODUCTION In positron emission tomography (PET), motion due to the cardiac and respiratory cycle causes blurred images. Various algorithms for motion correction in PET were recently devel- oped [1], [2]. In this context, we proposed the incorporation of prior knowledge (preservation of mass) into the registration process with the Variational Algorithm for Mass-Preserving Image REgistration (VAMPIRE) [3], [4], [5]. The different approaches for motion correction in PET vary in the general concept (optical flow [2] or image registration [1], [3]) or, e.g., in the motion model. Given our mass- preserving transformation model [3], we evaluate in this work This work was partly funded by the Deutsche Forschungsgemeinschaft, SFB 656 MoBil (projects B2 and B3) and projects BU2327/2-1, JU445/5-1 and WO1425/1-1. F. Gigengack is with the European Institute for Molecular Imaging (EIMI) and the Department of Mathematics and Computer Science, University of M¨ unster, Germany. L. Ruthotto is with the Institute of Mathematics and Image Computing (MIC), University of L¨ ubeck, Germany. M. Burger is with the Institute for Computational and Applied Mathematics, University of M¨ unster, Germany. C. H. Wolters is with the Institute for Biomagnetism and Biosignalanalysis, University of M¨ unster, Germany. X. Jiang is with the Department of Mathematics and Computer Science, University of M¨ unster, Germany. K. Sch¨ afers is with the European Institute for Molecular Imaging (EIMI), University of M¨ unster, Germany. Corresponding author: fabian.gigengack@uni-muenster.de. (a) Template image T (b) Reference image R Fig. 1. The template image T is registered to the reference image R. different motion models in the sense of [6]: dense displace- ment field (compute for each voxel an individual displacement) vs. spline transformation (i.e. free-form deformation). Thereby a focus is put on the parametrization of the spline transforma- tions where we optimize the number of spline coefficients and the regularization parameter. Based on the ground-truth motion vectors, provided by the established XCAT software phantom [7], we make a quan- titative comparison of the motion estimates of the different motion models. II. MATERIALS AND METHODS A. XCAT Phantom Data Two gates with varying cardiac and respiratory phase are generated with the XCAT software phantom [7]: one template image T , showing the systolic heart phase at maximum inspiration, and a reference image R in the diastolic heart phase at mid-expiration, see Fig. 2(a). The simulated tracer uptake values were derived from a real 18 F-FDG patient scan [3]. The ideal images of the XCAT tool were blurred with a Gaussian kernel to simulate the partial volume effect (PVE) (FWHM ≈ 3.85 mm). The blurred images were forward projected into data space, where Poisson noise was simulated. In a final step, the sinograms were reconstructed with an EM algorithm [8], [9] which can be downloaded at [10]. The original images were cropped to a size of 80 × 80 × 44 with a voxel size of 3.375 mm. B. VAMPIRE - Variational Algorithm for Mass-Preserving Image REgistration A template image T : Ω → R is registered onto an assigned reference image R :Ω → R, where Ω ⊂ R 3 is the image domain. This yields a transformation y : R 3 → R 3