Li Yusheng, Defrise Michel, Matej Samuel, Metzler Scott D
Department of Radiology, University of Pennsylvania, Philadelphia, PA 19104 USA.
Department of Nuclear Medicine, Vrije Universiteit Brussel, B-1090, Brussels, Belgium.
Inverse Probl. 2016;32(9). doi: 10.1088/0266-5611/32/9/095004. Epub 2016 Jul 6.
Due to the unique geometry, dual-panel PET scanners have many advantages in dedicated breast imaging and on-board imaging applications since the compact scanners can be combined with other imaging and treatment modalities. The major challenges of dual-panel PET imaging are the limited-angle problem and data truncation, which can cause artifacts due to incomplete data sampling. The time-of-flight (TOF) information can be a promising solution to reduce these artifacts. The TOF planogram is the native data format for dual-panel TOF PET scanners, and the non-TOF planogram is the 3D extension of linogram. The TOF planograms is five-dimensional while the objects are three-dimensional, and there are two degrees of redundancy. In this paper, we derive consistency equations and Fourier-based rebinning algorithms to provide a complete understanding of the rich structure of the fully 3D TOF planograms. We first derive two consistency equations and John's equation for 3D TOF planograms. By taking the Fourier transforms, we obtain two Fourier consistency equations and the Fourier-John equation, which are the duals of the consistency equations and John's equation, respectively. We then solve the Fourier consistency equations and Fourier-John equation using the method of characteristics. The two degrees of entangled redundancy of the 3D TOF data can be explicitly elicited and exploited by the solutions along the characteristic curves. As the special cases of the general solutions, we obtain Fourier rebinning and consistency equations (FORCEs), and thus we obtain a complete scheme to convert among different types of PET planograms: 3D TOF, 3D non-TOF, 2D TOF and 2D non-TOF planograms. The FORCEs can be used as Fourier-based rebinning algorithms for TOF-PET data reduction, inverse rebinnings for designing fast projectors, or consistency conditions for estimating missing data. As a byproduct, we show the two consistency equations are and for 3D TOF planograms. Finally, we give numerical examples of implementation of a fast 2D TOF planogram projector and Fourier-based rebinning for a 2D TOF planograms using the FORCEs to show the efficacy of the Fourier-based solutions.
由于独特的几何结构，双面板正电子发射断层扫描（PET）扫描仪在专用乳腺成像和机载成像应用中具有许多优势，因为这种紧凑型扫描仪可以与其他成像和治疗方式相结合。双面板PET成像的主要挑战是有限角度问题和数据截断，这可能由于数据采样不完整而导致伪影。飞行时间（TOF）信息可能是减少这些伪影的一个有前景的解决方案。TOF平面图是双面板TOF PET扫描仪的原始数据格式，而非TOF平面图是线性图的三维扩展。TOF平面图是五维的，而物体是三维的，存在两个冗余度。在本文中，我们推导了一致性方程和基于傅里叶的重排算法，以全面理解全三维TOF平面图的丰富结构。我们首先推导了三维TOF平面图 的两个一致性方程和约翰方程。通过进行傅里叶变换，我们得到了两个傅里叶一致性方程和傅里叶 - 约翰方程，它们分别是一致性方程和约翰方程的对偶。然后我们使用特征线法求解傅里叶一致性方程和傅里叶 - 约翰方程。沿着特征曲线的解可以明确引出并利用三维TOF数据的两个纠缠冗余度。作为一般解的特殊情况，我们得到了傅里叶重排和一致性方程（FORCEs），从而得到了一个在不同类型的PET平面图之间进行转换的完整方案：三维TOF、三维非TOF、二维TOF和二维非TOF平面图。FORCEs可以用作基于傅里叶的重排算法用于TOF - PET数据缩减、用于设计快速投影仪的逆重排，或用于估计缺失数据的一致性条件。作为一个副产品，我们展示了三维TOF平面图 的两个一致性方程是 和 。最后，我们给出了使用FORCEs实现快速二维TOF平面图投影仪和二维TOF平面图基于傅里叶的重排的数值示例，以展示基于傅里叶的解的有效性。
