Among numerous medical imaging modalities, diffusion weighted imaging (DWI) is extremely sensitive to acute ischemic stroke lesions, especially small infarcts. However, magnetic resonance imaging is time-consuming and expensive, and it is also prone to interference from metal implants. Therefore, the aim of this study is to design a medical image synthesis method based on generative adversarial network, Stroke-p2pHD, for synthesizing DWI images from computed tomography (CT). Stroke-p2pHD consisted of a generator that effectively fused local image features and global context information (Global_to_Local) and a multi-scale discriminator (M2Dis). Specifically, in the Global_to_Local generator, a fully convolutional Transformer (FCT) and a local attention module (LAM) were integrated to achieve the synthesis of detailed information such as textures and lesions in DWI images. In the M2Dis discriminator, a multi-scale convolutional network was adopted to perform the discrimination function of the input images. Meanwhile, an optimization balance with the Global_to_Local generator was ensured and the consistency of features in each layer of the M2Dis discriminator was constrained. In this study, the public Acute Ischemic Stroke Dataset (AISD) and the acute cerebral infarction dataset from Yantai Yantaishan Hospital were used to verify the performance of the Stroke-p2pHD model in synthesizing DWI based on CT. Compared with other methods, the Stroke-p2pHD model showed excellent quantitative results (mean-square error = 0.008, peak signal-to-noise ratio = 23.766, structural similarity = 0.743). At the same time, relevant experimental analyses such as computational efficiency verify that the Stroke-p2pHD model has great potential for clinical applications.