# Real-Time 3D PET Image with Pseudoinverse Reconstruction  
[PERSON]  
1 Nuclear Physics Group and IPARCOS, Faculty of Physical Sciences, University Complutense of Madrid, CEI Moncloa, 28040 Madrid, Spain; [EMAIL_ADDRESS] (P.G.); [EMAIL_ADDRESS] (J.M.U.)  
[PERSON]  
1 Nuclear Physics Group and IPARCOS, Faculty of Physical Sciences, University Complutense of Madrid, CEI Moncloa, 28040 Madrid, Spain; [EMAIL_ADDRESS] (P.G.); [EMAIL_ADDRESS] (J.M.U.)  
[PERSON]  
1 Nuclear Physics Group and IPARCOS, Faculty of Physical Sciences, University Complutense of Madrid, CEI Moncloa, 28040 Madrid, Spain; [EMAIL_ADDRESS] (P.G.); [EMAIL_ADDRESS] (J.M.U.)  
[PERSON]  
3 QIMP Team, Center for Medical Physics and Biomedical Engineering, Medical University of Vienna, 1090 Vienna, Austria; [EMAIL_ADDRESS]  
[PERSON]  
4 Departamento de Bioingenieria e Ingenieria Aeroespacial, Universidad Carlos III de Madrid, 28911 Leganes (Madrid), Spain; [EMAIL_ADDRESS] (J.V.); [EMAIL_ADDRESS] (M.D.)  
[PERSON]  
4 Instituto de Investigacion Sanitaria Gregorio Maranon, 28007 Madrid, Spain  
4 Centro de Investigacion Biomedicas Carolovasculares Carlos III (CNIC), 28029 Madrid, Spain  
4 Centro de Investigacion Biomedicas en Red de Salud Mental (CIBERSAM), 28029 Madrid, Spain
Tomographic PET image reconstruction methods are usually classified into analytical [7; 8], and iterative methods [9; 10; 11]. In order to be applicable, analytical methods such as filtered back projection (FBP) require many assumptions such as ideal detection, uniform sampling, data completeness, and noise properties to build the system response matrix (SRM) [12]. Although these assumptions are quite reasonable for computed tomography (CT) imaging [13], they are questionable in PET, making these reconstructions of suboptimal quality [10; 14; 15; 16]. Another disadvantage of analytical methods is that the resolution of the images is not uniform across the field of view (FOV), which affects quantification [17]. On other hand, iterative reconstructions provide better results than analytical methods [9; 18], thanks to the incorporation in the reconstruction process of the physics effects involved in the emission, transport, and detection of PET radiation [14] such as positron range [19], non-collinearity, depth-of-interaction (DOI) in the crystals [13], geometrical considerations of the scanner [18; 20], etc. However, iterative methods with realistic SRM are slow, and therefore, their application to real-time fully 3D PET imaging is still challenging [3; 7].  
One of the most common approaches to speeding-up the reconstruction process involves introducing axial rebinning of the data, followed by 2D reconstructions [7; 21; 22]. The simplest method for axial rebinning of PET data is single-slice rebinning (SSRB) [23], where the activity is assigned to the central axial coordinate z of both crystals z1, z2. This method is very fast, however, it is only accurate for activity close to the axis of the scanner. A more accurate alternative to SSRB is Fourier rebinning (FORE) [24], but unfortunately, it is much slower.  
Here, we propose the pseudoinverse (PINV) of the SRM [25] to achieve real-time reconstructions with uniform resolution [26]. The goal was to achieve some of the features of iterative methods with the computational cost of analytical reconstructions, thus providing fast reconstructions while incorporating the physics involved [18].