Purpose: To assess the performance of full dose (FD) positron emission tomography (PET) image synthesis in both image and projection space from low-dose (LD) PET images/sinograms without sacrificing diagnostic quality using deep learning techniques. Methods: Clinical brain PET/CT studies of 140 patients were retrospectively employed for LD to FD PET conversion. 5% of the events were randomly selected from the FD list-mode PET data to simulate a realistic LD acquisition. A modified 3D U-Net model was implemented to predict FD sinograms in the projection-space (PSS) and FD images in image-space (PIS) from their corresponding LD sinograms/images, respectively. The quality of the predicted PET images was assessed by two nuclear medicine specialists using a five-point grading scheme. Quantitative analysis using established metrics including the peak signal-to-noise ratio (PSNR), structural similarity index metric (SSIM), region-wise standardized uptake value (SUV) bias, as well as first-, second- and high-order texture radiomic features in 83 brain regions for the test and evaluation dataset was also performed. Results: All PSS images were scored 4 or higher (good to excellent) by the nuclear medicine specialists. PSNR and SSIM values of 0.96 ± 0.03, 0.97 ± 0.02 and 31.70 ± 0.75, 37.30 ± 0.71 were obtained for PIS and PSS, respectively. The average SUV bias calculated over all brain regions was 0.24 ± 0.96% and 1.05 ± 1.44% for PSS and PIS, respectively. The Bland-Altman plots reported the lowest SUV bias (0.02) and variance (95% CI: -0.92, +0.84) for PSS compared with the reference FD images. The relative error of the homogeneity radiomic feature belonging to the Grey Level Co-occurrence Matrix category was -1.07 ± 1.77 and 0.28 ± 1.4 for PIS and PSS, respectively Conclusion: The qualitative assessment and quantitative analysis demonstrated that the FD PET prediction in projection space led to superior performance, resulting in higher image quality and lower SUV bias and variance compared to FD PET prediction in the image domain.