Estimation of the Rigid-Body Motion from Three-Dimensional Images Using a Generalized Center-of-Mass Points Approach

Abstract

We present an analytical method for the estimation of rigid-body motion in sets of three-dimensional SPECT and PET slices. This method utilizes mathematically defined generalized center-of-mass points in images, requiring no segmentation. It can be applied to compensation of the rigid-body motion in both SPECT and PET, once a series of 3D tomographic images are available. We generalized the formula for the center-of-mass to obtain a family of points co-moving with the object's rigid-body motion. From the family of possible points we chose the best three points which resulted in the minimum root-mean-square difference between images as the generalized center-of-mass points for use in estimating motion. The estimated motion was used to sum the sets of tomographic images, or incorporated in the iterative reconstruction to correct for motion during reconstruction of the combined projection data. For comparison, the principle-axes method was also applied to estimate the rigid-body motion from the same tomographic images. To evaluate our method for different noise levels, we performed simulations with the MCAT phantom. We observed that though noise degraded the motion-detection accuracy, our method helped in reducing the motion artifact both visually and quantitatively. We also acquired four sets of the emission and transmission data of the Data Spectrum Anthropomorphic Phantom positioned at four different locations and/or orientations. From these we generated a composite acquisition simulating periodic phantom movements during acquisition. The simulated motion was calculated from the generalized center-of-mass points calculated from the tomographic images reconstructed from individual acquisitions. We determined that motion-compensation greatly reduced the motion artifact. Finally, in a simulation with the gated MCAT phantom, an exaggerated rigid-body motion was applied to the end-systolic frame. The motion was estimated from the end-diastolic and end-systolic images, and used to sum them into a summed image without obvious artifact. Compared to the principle-axes method, in two of the three comparisons with anthropomorphic phantom data our method estimated the motion in closer agreement to than of the Polaris system than the principal-axes method, while the principle-axes method gave a more accurate estimation of motion in most cases for the MCAT simulations. As an image-driven approach, our method assumes angularly complete data sets for each state of motion. We expect this method to be applied in correction of respiratory motion in respiratory gated SPECT, and respiratory or other rigid-body motion in PET.

Fig. 1

Summed images were generated for the MCAT phantom simulations. (Upper row) No motion compensation in summing images at t and t+dt. (Lower row) Motion compensated summing. From left to right, results for 4 noise-levels are plotted. No post-smoothing was applied to the summed images.

Fig. 2

Reconstructions of the motion-present simulations for several noise-levels (Bottom row, from left to right), using the motion estimated from the optimal three generalized center-of-mass points (listed in Table 1). As a comparison, the same data were reconstructed with the ideal motion compensation (Upper row) and without motion compensation (Middle row). No post-smoothing was applied to these images.

Fig.3

(Left) Reconstruction of the motion-free baseline acquisition. (Middle) The same transaxial slice for the image summed for the 4 reconstructions of the data acquired in 4 sequential acquisitions, without motion compensation in summing. (Right) The same slice with compensation for motion. No post-smoothing was applied to the images. Note that the summed images had 4 times the counts of the no motion image.

Fig.4

(Left) Reconstruction of the motion-present data with compensation for the motion which was detected by Polaris. (Middle) No motion Compensation. (Right) Motion compensation with the motion estimated from the generalized center-of-mass points. No post-smoothing was applied to the images.

Fig. 5

Polar maps generated for the images of anthropomorphic phantom shown in Fig. 4. (Left) Motion compensation with Polaris. (Middle) No motion Compensation. (Right) Motion compensation with the generalized center-of-mass points.

Fig. 6

In simulations with the gated MCAT phantom, the end-diastolic (ED) (Upper left) and the end-systolic (ES) (Upper right) images were used. The ES image was translated and rotated (Upper right), and summed into the ED image without compensation for the motion (Lower left), and with compensation for the motion estimated from the generalized center-of-mass points (Lower right). Note the full 3D heart was employed in this study. Translation and rotation were also 3D. Thus the summed ED + ES slice at lower right is not the sum of the ED and ES slices shown at the top of the figure, but the sum of the ED slice at top and its appropriate counter part after 3D translation and rotation.